The plethystic inverse of a formal power series

DISCRETE MATHEMATICS ELSEVIER

Discrete Mathematics 139 (1995) 413-442

The plethystic inverse of a formal power series Julia S. Y a n g Department qf Mathematics, West Virginia University, Morgantown, W V 26506, USA Received 8 September 1992: revised 29 June 1993

Abstract

A bijective definition of Littlewood's plethysm of symmetric functions in terms of unlabelled colored combinatorial structures is introduced. The plethystic inverse is understood by examining associated partially ordered sets and using M6bius inversion techniques. Resum6 Nous introduisons une d6finition bijective du pl6thysme de Littlewood des fonctions symetriques en faisant appel aux structures color6es non-6tiquet6es. Nous donnons une interpr6tation de l'inverse pl6thystique en utilisant les techniques de l'inversion de M6bius sur des ensembles partiellement ordonn6es bien choisis.

La th6orie combinatoire de la distribution et de l'occupation de 'balles' dans des 'boites' fournit une interpr6tation classique pour les fonctions sym6triques. De ce point de vue, on 6num6re les faqons de placer des boules distinctes dans des boites distinctes de sorte que certaines conditions soient satisfaites (par exemple: pas plus d'une boule dans chaque boite). Les mots boule et bofte sont pris ici dans un sens trds g6ndral. Par exemple, les boules peuvent 6tre interpr6t6es comme les 'sommets 6tiquetds' d'un graphe et les boites comme des 'couleurs' assign6es fi ces sommets. Les fonctions sym6triques 6num6rent alors, fi la Pdlya [16, 17], les structures colorees 6tiquet6es selon les sch6mas de distribution des couleurs. Un traitement syst6matique de ce point de vue a 6t6 fait par Doubilet [4]. Plus r6cemment, Bonetti et al. [2] ont fourni des interprdtations ensemblistes pour les sommes infinies, les produits infinis, et les exponentiations de fonctions sym6triques. Ceci a permis de produire des preuves bijectives de plusieurs identit6s classiques satisfaites par les fonctions sym6triques.

'~Partially supported by a National Science Foundation Graduate Fellowship. This material formed part of the author's doctoral dissertation. 0012-365X/95/$09.50 © 1995--Elsevier Science B.V. All rights reserved SSD1 0 0 1 2 - 3 6 5 X ( 9 4 ) 0 0 1 4 5 - 9

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Nous examinons, dans le pr6sent texte, les classes d'6quivalence de structures 6tiquet6es color+es soumises/t des actions de groupes. Ceci nous am6ne fi formuler une d~finition bijective de la notion de pl&hysme des fonctions sym6triques au sens de Littlewood. Cette th6orie des fonctions sym&riques est ensuite 6tendue de faqon ~i inclure les ensembles partiellement ordonn6s ainsi qu'une interpr6tation de la notion d'inverse pl6thystique diff~rente de celle donn6e par Gessel et Labelle [5] (voir aussi Labelle [12,9, 11]). Les orbites des actions de groupes consid6r6es peuvent ~tre interpr6t6es intuitivement comme des graphes (structures) non-&iquet6s ~i sommets color6s soumis ~ des schemas sp6cifiques de coloriages (par exemple, on peut demander que les sommets soient de couleurs distinctes). Ceci constitue une extension des foncteurs analytiques, ou 6tiquetages de Joyal [8]. Les op6rations d'addition, de multiplication, et d'exponentiation sont imm6diatement 6tendues aux structures non-&iquet6s ~ sommets color6s, que nous appelons types color,s (colored types). Nous introduisons une composition appropri6e remarquablement simple des types color6s qui correspond au pl&hysme de Littlewood des fonctions g6neratrices, l~tant donn6es deux families M e t N de types color6s, le pl6thysme de N dans M donne lieu a des M-structures dont les sommets color,s ont 6t6 remplac~s par des N-structures appropri~es; deux 'sommets' ont la m6me 'couleur' lorsque les N-structures associ6es sont les 'm~mes'. Ce pl6thysme constitue une g~n6ralisation de l'interpr&ation combinatoire usuelle de celui de Littlewood. L'un de nos r6sultats principaux consiste en une nouvelle interpr6tation combinatoire pour l'inverse pl&hystique. Cette interpr6tation fait appel aux techniques de l'inversion de M6bius dans les ensembles partiellement ordonn6s introduites par Rota [19]. Les treillis de structures combinatoires color6es apparaissant ici sont construits en utilisant les m6thodes introduites par Mendez et Yang [14]. La Section 1 contient des rappels de notions de base sur les fonctions sym6triques et la th6orie des esp6ces polynomiales. Le pl6thysme est introduit dans la Section 2 et est illustr6 par plusieurs exemples dont celui du pl6thysme d'une fonction sym&rique dans une fonction sym6trique homog6ne qui correspond aux assembl6es (ou ensembles) de types color6s. La Section 3 traite de l'inverse pl&hystique par le biais de treillis appropri+s.

1. Background material

1.1. Symmetric functions and Littlewood's plethysm Let X = {xi}i~ be a finite or infinite set of variables with index set, I. A symmetric function is a formal power series in X with integer coefficients which is invariant under permutation of the variables and of bounded degree. The symmetric functions form a ring denoted by A(X).

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Let ~DtN denote the set of sequences of non-negative integers indexed by I with only finitely many non-zero entries. For :~=(cq)s(~xN, x ~ denotes the monomial Iqix~'. The most important symmetric functions are as follows: 1. Monomial symmetric function ma. A (number) partition ),=(21,)`2 .... ) is a finite or infinite sequence of non-negative integers in non-increasing order with only finitely many non-zero elements. A partition may also be expressed as 2=(1"12 . . . . . t where the partition has nl elements equal to i. The monomial symmetric.Junction mz(X) or ma is defined as mx=5~,x ~, where the sum ranges over all :~eOtN with exactly ni elements equal to i for all i. 2. Elementary symmetric function a,. For a non-negative integer n, the nth elementary symmetric function a,(X) or a, is the sum of all products of n distinct variables. Let I~[=~] ~i for ~ e O I N . Then a 0 = 1, and a,--~l~l= n x ~, where the sum ranges over all ~ e 0 1 N with only 0 or 1 for entries. The generating function for the elementary symmetric functions is E a. = I-[ (I +.,q).

3. Complete homogeneous symmetric function h,. For each non-negative integer n, the nth complete homogeneous symmetric function h.(X) or h. is the sum of all monomials of total degree n; h0 = 1 and h. = 5~1,1=, xL The generating function for the homogeneous symmetric functions is

4. Power sum symmetric function p.. For each positive integer n, the nth power sum symmetric function p.(X) or p, is defined as P.= Zl x~. Littlewood's plethysm. Let F(X)=~k>ofk(X) and G ( X ) = Z k > I gk(X) be two sums of symmetric functions in the variable set X, where fk(X) and gk(X) are symmetric functions of degree k (these sums are called generating functions of symmetric functions). Suppose that the index set is countably infinite. A special kind of multiplication for these generating functions, called the plethysm or composition, can be defined. The basic idea is that if G(X) is expressed as a sum of terms of the form xL then the plethysm F* G(X) is obtained by substituting the terms of G(X) into the variables xi in F(X). Formally, write G(X) as the sum of monomials: G(X)=Z~ w~xL where the sum ranges over all c~sO~ N. Define a set of d u m m y variables y~, such that

I-](l+yiz)=I](l+x=z) w,. To simplify notation, suppose that the index set I is linearly ordered. Littlewood's plethysm is defined as

F* G(x~,x2 .... )=F(y~, y2 .... ).

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When F(X) is homogeneous of degree n and G(X) is homogeneous of degree k, then F • G(X) is homogeneous of degree nk. See [13] for more about symmetric functions and plethysm.

1.2. Polynomial species The theory of polynomial species provides set-theoretic interpretations of operations on symmetric functions. In this subsection, the definition for polynomial species and the combinatorial operations of addition, multiplication, and exponentiation will be reviewed [2]. Let Ens be the category of sets and functions, and let B be the category of finite sets and bijections. Define a covariant functor Hom: B-~Ens as H o m [ E ] = { f : E--*X} where E is a finite set, and f is a function. Let M : B - ~ E n s be a functor (frequently, M is a (Joyal) species M : B ~ B , and the image of a finite set E under M is called the set of all M-structures with label set E). Let R be a subfunctor of M × H o m . Given a R-structure (re,f), the function f may be visualized as 'coloring' the labels (or nodes) of the M-structure with elements from the variable set X. A bijection g:E--.E' of finite sets induces the morphism

R[g]E:R[E]-+R[E'] (m, f : E ~X)~--~(M[g]m, f o g-1). The subfunctor R is called a polynomial species when for any finite subset F _ X, the set {(m,f)~R [E]: i m f _ F} is finite; i m f d e n o t e s the set of variables in the image o f f A polynomial species is said to be symmetric when for any bijection u : X ~ X and for any (m,f)~R[E], (m, uof)~R[E]. Given any polynomial species R and any non-negative integer n, let Rt. j be the polynomial species where Rt.1[E]=R[E ] if IEl=n, and R t . j [ E ] = 0 otherwise. Let Ro denote the polynomial species where Ro[E]=R[E] if E is nonempty, and Ro [ E l = 0 otherwise. The generating functions associated with a polynomial species will now be covered. For a finite set E and an arbitrary function f:E--.X, let g e n ( f ) = [ I e ~ j ( e ) = x ~ where ~=(cq)eO~ N, and cq is the cardinality of the set {f-l(xi)}. Given any R-structure (re,f ) call f the coloring, and call ~ or x" the color scheme. Set gen(m, f ) = gen(f). Define gen(R [ E ] ) =

~

gen (m, f).

(m,f)eR[E]

Note that every monomial in the sum has a finite coefficient by the definition of a polynomial species. Also, if R is symmetric, then gen(R In]) is a symmetric function of degree n where R [n] = R [{ 1,2 .... , n} ].

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The 9enerating.(unction of a polynomial species R is defined as the formal power series Gen(R)= ~ gen(R[n])/n!. n>~O

Polynomial species form a category whose morphisms are natural maps r:R--,S such that for any finite set E, rE: R[E]--*S[E] is a bijection where TE(m,f)=(m',f) for (m,fleR[E]. Write R=S when ~ is a natural isomorphism. R and S are said to be equipotent (R - S) if gen(R [n]) = gen(S [n]) for all non-negative integers n. This is a weaker condition. The combinatorial operations of addition, multiplication, and exponentiation will now be covered. Let R~M1 x Hom and S~_M2 x Horn be two polynomial species. Let E be a finite set. The sum and product of R and S are given by

(R + S)[ E] =R[E]wS[E], (R'S)[E]=

~

{((m,m'),f):(m, flE1)~R[E1],(m',flE2)~S[E2] ].

E1 + E 2 = E

The generalizations to finite sum and finite product are straightforward. Let 1 be the polynomial species: if E=O, otherwise.

l[E]={~(0,f0)}

A family of polynomial species (Ri)i~1 is said to be summable if for any finite subset F of the variable set X, the set I(E, F ) = {ieI: there exists (m,f)eRi[E] with i m f c _ F l is finite. In words, there are only finitely many i6I where an Ri-structure uses any subset of the colors in F. The intinite sum and infinite product H(1 + R i ) of a summable family (R~)i~l are defined as

"

i61(E,F)

where the unions range over all finite subsets of X. Suppose that a polynomial species, S, is without constant term (S[O] =0). An assembly of S-structures with associated partition x is an ordered pair (a,f) where (mB,fB)~S[B] for every BE~, a={mB}, and f]B=f~ for all B ~ . The k-th divided power 7k(S) is the polynomial species of assemblies of S-structures where the associated partition contains k blocks. The exponential exp(S) is the polynomial species of all assemblies of S-structures. Note that when IX] = 1, and M1 and m 2 a r e Joyal species, these operations reduce to the corresponding operations on species. The generating functions for these operations behave as desired.

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Theorem 1.1. Let R and S be polynomial species. Then Gen (R + S) = Gen (R) + Gen (S), Gen(R • S)= Gen (R). Gen(S).

When

(Ri)iE I

is a summable family of polynomial species,

Gen (~, R i ) = ~ Gen(R i), Gen ( 1-I (1 + Ri))--1-I (1 + Gen(Ri)).

When S is a polynomial species without constant term, Gen(~k(S)) = Gen(S)k/k!, Gen (exp (S) ) = exp (Gen (S)).

2. Colored types and plethysm In the standard combinatorial approach to Littlewood's plethysm (which will be phrased in the language of species), given two Joyal species M and N, where N [0] = 0, the plethysm corresponds to unlabelled M o N-structures whose nodes are colored by the variable set X in all ways. a In this section, a more refined plethysm is introduced where we may restrict which color schemes appear. More specifically, let R be a symmetric polynomial species. The group E! of permutations on E acts on R[E] by the morphism of structures. The orbits of R [E] under this group action may be identified with unlabelled R-structures. This is a generalization of types which correspond to unlabelled structures, and of analytic functors or Joyal labellings which correspond to unlabelled structures colored in all ways. Call the unlabelled R-structures colored types. The plethysm of colored types has a remarkably simple combinatorial interpretation. Let R and S be two symmetric polynomial species. Associate a color in X to each unlabelled S-structure. Then the operation of plethysm takes an unlabelled R-structure and replaces each colored node with an unlabelled S-structure of the same associated color. This operation is related to the plethysm of symmetric functions. While polynomial species are associated with exponential generating functions, orbits of polynomial species (and hence, colored types) correspond to ordinary generating functions. Let orb(S [E]) denote the set of orbits of S [E] under the group action. Set gen(__q)=gen(q) where q is a representative of the orbit __q.The generating function for the orbits of S-structures is defined as Gen(orb(S)) = ~

~

gen(_q).

n>_O q_eorb(S[n])

x Recall t h a t M o N c o n s i s t s of all p a i r s (a, m) w h e r e a is a n a s s e m b l y of N - s t r u c t u r e s a n d m is a n M - s t r u c t u r e o n the a s s o c i a t e d p a r t i t i o n of a.

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Note that this generating function equals the (ordinary) generating function for the colored types of S where each unlabelled S-structure contributes its color scheme to the sum. It will be shown that the plethysm of the generating functions for the colored types of R and S equals the generating function for the plethysm of the colored types as defined above. These notions will now be covered rigorously.

2.1. Colored O'pes In practice, it is inconvenient to work directly with orbits in these kinds of arguments. For example, let M : B ~ B be a Joyal species. In the standard approach to Littlewood's plethysm, one begins with the cycle index series ZMeQ[xl,x2 .... ] for M, and the usual plethysm or wreath product for formal power series in infinitely many variables (not Littlewood's plethysm) commutes with the combinatorial operation of substitution [7]. Then the results for symmetric functions are obtained by replacing x, with the power sum symmetric function p,(X). Combinatorially, this corresponds to taking a labelled structure with automorphism a and then coloring its nodes by independently coloring the cycles of a; this is equivalent to examining unlabelled structures colored in all ways. In [1], Bergeron extends this method by examining structures based on permutations. In this section, a different approach is taken where we work with the symmetric functions directly rather than going through the intermediate step of power sum substitution. Combinatorially, this corresponds to beginning with a colored labelled structure, and then examining an automorphism of the structure which preserves the coloring. The advantage of this method is that the colorings do not depend upon an underlying permutation of the labelled structure; this results in more control over the coloring schemes. For example, the elementary symmetric functions can be interpreted in this context. From this point forward, assume that all polynomial species are symmetric unless otherwise specified, and let X = {xi}i~l be a countably infinite variable set. Definition 2.1. Let S be a polynomial species. The (polynomial) species of colored types S is defined as follows: S [ E ] = { ( t r , m,f): (m,f)ES[E], ~ E ! , and a(m,f)=(m,f)} for E a finite set. In other words, a S-structure is a pair (o,s) where a is an automorphism of the S-structure, s. The exponential generating function of the polynomial species S is equal to the ordinary generating function for the orbits of S. This may be proved by Burnside's lemma: if a finite group G acts on a finite set B, then the cardinality of the set of orbits of B is equal to the cardinality of the set {(a, b): a6 G, b~B, tr. b = b } divided by the cardinality of G.

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For polynomial species R and S, and for a summable family of polynomial species (Ri)~t, it is simple to show the isomorphisms

R+S=R+S, R.S=R.S,

Z R, = Z / ~ i ,

I~(I+R,)=I-I(I+R,).

Intuitively, these operations correspond to disjoint sum and to ordered sequences of unlabelled, colored structures. The polynomial species exp(S) corresponds to unlabelled assemblies of S-structures and will be covered shortly. Before defining the combinatorial operation of plethysm, colored types will be assigned colors in the variable set X. For now, assume that S is nontrivial. Theorem 2.1. Any symmetric polynomial species S may be decomposed into its orbits S = Z Si, iel

where the Si are polynomial species (not necessarily symmetric), and the sum ranges over the index set I of the variable set X. More formally,for (m, f ) ~ S i [ E] and (m',f')~S~[ E], (re,f ) and (m',f') are in the same orbit iff i=j. This may be proved by simple cardinality arguments. Definition 2.2. Given a decomposition S = ~ Si, let xi be the associated color of an S~-structure. Let fll equal the size of the subjacent set of an Srstructure. Let ?i be equal to the cardinality of the set Si[fl~]. Set 6~ equal to the number of automorphisms of any representative (s,f)eS~[fl~], and let Yl be the color scheme used by this representative (Yi = gen(f)). The decomposition of S into its orbits induces a decomposition of colored types: i

As before, let xi be the associated color of a ffi-structure. The monomial Yl is simply the color scheme of any Si-structure; the variables y i will be called the dummy variables for ft. By Burnside's lemma, Gen(ffi)

=

t~i))iYi/fli! = Y i .

Example 2.1. Let A be the elementary symmetric species of distinctly colored sets. A [0] = { (0,f0) } and for a nonempty finite set E, A [El = {(E, f } :fis injective}. ,4 may be identified with distinctly colored, unlabelled sets. Since two A-structures with the same subjacent set and same coloring scheme are in the same orbit, A - - ~ A~, where the sum ranges over all ~ ) t t ~ with only 0. and 1 for entries, and y~=xL Then / L = I~1, ~'~ = I~1!, and 6~= 1. In addition, Gen(.4) equals the generating function for the elementary symmetric functions.

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Example2.2. Let U be the uniform symmetric speciesof colored sets. For a finite set E, U[E]={(E,f): f:E-,X}. U corresponds to unlabelled, colored sets. Since two U-structures with the same subjacent set and the same coloring scheme are in the same orbit, U = ~ U,, where the sum ranges over all ~ ® ~ I ~ , and y,=x ~. Then ~;~=l~l!/(1-I~i!) and 6~=I]oti!. Furthermore, Gen(U) is equal to the generating function for the homogeneous symmetric functions.

Example 2.3. Let M be any Joyal species. M x Horn corresponds to unlabelled, multicolored M-structures, where multicolored means that the objects have been colored in all possible ways. Its generating function is ZM(p~,p2,P3.... ), where Z~4 is the classical cycle indicator series for M and p, denotes the nth power sum.

Example2.4.

Let P be the linear symmetric species of monochromatic linear orders. P [ 0 ] = 0 , and for a nonempty finite set E, P[E]={(l,f): l linear order on E, lira f [ = 1 }. Gen(/~) is equal to the generating function for the power sum symmetric functions.

Example 2.5. Let D be the symmetric polynomial species of permutation-enriched functions. D [ 0 ] = {(0,J0)} and for E:~0, D I E ] = {(s,f)}, where f: E--,X is any function; each fiber f = l(x) has a permutation associated with it; and s is the set of these permutations. When n is a positive integer, let p(n) equal the number of partitions ofn where all the entries are positive; by convention, p(0)= I. An unlabelled monochromatic permutation is completely determined by its associated number partition 2=(it'), where the permutation has ri/-cycles (i~N). /) corresponds to unlabelled assemblies of monochromatic permutations where no two permutations use the same color. Therefore, Gen(/5)=H

(E p(n)x~) = H I _xk ' i, iel kE~

since

~p(n)x"= 1-[ ( l + Xk-k-x2k + x3k-+-''')" k~

2.2. Plethysm q/polynomial species Let R and S be nontrivial polynomial species where S is without constant term. Recall that the species of colored type S can be decomposed (S=y~ Si). The monomials Yl (where yi is the color scheme of an S~-structure) are the dummy variables for Gen(S). To see this, consider assemblies of S-structures with at most one structure of each 'associated color'.

r=Fl(1 +g,).

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Then G e n ( T ) = I1 (1 + Yi) = l-[ (1 +xat) w'. i

at

A rigorous definition for the plethysm of polynomial species of colored type may now be given. Definition 2.3. Let _ff and S be two polynomial species of colored type. The plethysm /~* S [ E ] consists of structures of the form (a, a, f, #, m, f), where 1. (a, f ) is an assembly of S-structures with automorphism a, i.e.,

(a,a,f)~exp(S)[E]. Let n be the associated partition. ^ 2. The decomposition of S induces a coloring f : n ~ X ; namely, if (s, h)~Si[B] is a member of the assembly built on the block Ben, thenf(B)=xi. Furthermore, since the automorphism a transports an S-structure in the assembly to itself or to a neighboring S-structure of the same type, a induces a permutation # : n ~ n of the associated partition such that f . t~=J~ Specifically, t~(B)=B' when there exists representatives b~B, b'~B' such that a(b)= b'. 3. The /~-structure (~,m,f) is an element of R [ n ] where the coloring f and the automorphism # are as specified above. Theorem 2.2. Let R and S be polynomial species. Then

Gen(/~ * S)= Gen (/~) • Gen(S). Proof. The generating function for/~ may be expressed as Gen(/~) =

2

u~xat/[~l!=

at

2 at

uat

1-I (xi)at' 1~1! 1 i

Consider a nontrivial eE0)/N with I~l=k. Then uat is the number of structures

(#, m , f ) e R [ k ] with coloring scheme ct. Choose a particular coloringf':[k]---,X with this coloring scheme. Let u'at be the number of structures in /~[k] which use the coloring f ' . Then k!

u~=u'- - -

(1)

lq ~i!" This follows from partitioning the elements o f / ~ [ k ] according^ to which coloring is used. Each block will be of the same size. Given a coloring f " , there is a bijection u: [ k ] ~ [ k ] such that f " = f ' o u-1. The function g from the block o f f ' structures to the block off" structures

g(O,m,f)=(#o u , M [ u ] m , f o u -1) is a bijection. Since there are k ! / [ I cq! distinct colorings, Eq. (1) follows.

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Let m be the size of the subjacent set for an S-assembly with associated color scheme :~, i.e., m=~,~ifll. The number of structures (cr,a, J i & m , [ ) e R * S [ m ] where the S-assembly (a,f) has associated color scheme c~ is given by m!

u; H

H

I-l y?.

The first term counts the number of allowed functionsJ~: g--,X. An allowed function must satisfy the condition that each block of the partition which is mapped to xi will be of size/3i. Only in this way is it possible to have an S-assembly with associated coloring f A composition of a set is a set partition where the blocks are linearly ordered. There is a bijection between compositions of [m] with ~1 blocks of size/31, and then ~2 blocks of size/32, etc., and allowed functions with fibers that are linearly ordered. Specifically, given a composition, assign the first 21 blocks to x~, the next 0(2 blocks to x2, and so on. There are m!/(I]/3i!") compositions. To compute the number of allowed functions, divide out by the number of linear orderings of the fibers. Given a coloring f: ~--,X, the second term counts the number of elements (& m, f ! in /~[rc] which use the specified coloring. Given an ,q-structure (& re,f: g ~ X ), the third term counts the number of ways that S-structures can be placed on the blocks of the partition re, such that the associated coloring of the partition is f. Let (a, f ) be an assembly of S-structures with associated coloring.~ and let (& m, f ) be an R-structure on the associated partition with the specified coloring. Recall than an automorphism a of the assembly (a,f) transports an S-structure in the assembly to itself or to a neighboring member of the same type. This automorphism will have 6- for its induced permutation iffss(a, f ) with subjacent set B is transported to s'da, j') with subjacent set B' exactly when 6 ( B ) = B'. There are 6i ways to transport s to s' when s (and hence s') have associated color xi. Therefore, the fourth term ( I] 6~ ~) equals the number of automorphisms a of the assembly (a, J'l such that the induced map on the associated partition is 6". The fifth term is the associated monomial for an /~, SZstructure (a,a,l~d,m,f) where the S-assembly has associated color ~. Use Eq. (1) to simplify the main expression. The generating function for these structures is given by

u~ H (6~7~Y~' 1

1

Sum over all ~ e O r N to get the generating function for R * S and the desired result. []

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2.3. Examples

Theorem 2.3. A[1 ]

acts

as a two-sided identity for any polynomial species S (without

constant term): S * AD]=A[I]* S = S .

Theorem 2.4. Let R, S, and T be polynomial species where T is without constant term, and let n and k be positive integers. Then 1. /~[n] * /D[k] =/~[k] * /~[.] ~---

?[n'k],

2. T * P t , I - Pt,] * T, 3. ( R + S ) * T = R * T + S * T, 4. ( R . S ) , T = ( R , T ) . ( S , T).

Proof. (1) Given a/6[.~./~tk]-structure (a, a,f, (~,m,J~), concatenate the linear orders in the assembly a in the order specified by m, to obtain the corresponding /~[,-kV structure. Conversely, given a P[, k]-structure, 'cut' the linear order into continuous segments of size k and give the segments the induced ordering to obtain a/~t.] * Ptk~structure. (2) A crown is an assembly of T-structures provided with an automorphism permuting its members circularly. By the same argument used by Joyal for types [7], a T* Pt,rstructure is isomorphic to a crown with n members marked by an initial member. In turn, the species of these marked crowns is equivalent to fir.I* ~. When T is asymmetric, the equivalence is an isomorphism. (3) Clear from the definition of plethysm. (4) Follows from splitting an ( R ' S ) * T-structure into an ordered pair where the first element is the part affected by /~ and the second element is the part affected by S. [] Let S be any nontrivial polynomial species such that S [0] = 0. Then

g=Iq (l + g,), iel

Gen(/~* S)= l-I (1 +Yi). iel

Intuitively, ,4. S corresponds to unlabelled assemblies of S-structures where no two members are the 'same'. Example 2.6. For any positive integer k, A • Ark j may be associated with unlabelled assemblies of distinctly colored k-sets, where no two sets use the same color scheme. The symmetric function of degree nk in the corresponding generating function (i.e., gen(,,t* ,'ttkj Ink])) equals a, * ak; this counts the number of unlabelled assemblies (as specified above) with n members.

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425

U[k]

Example 2.7. A* corresponds to unlabelled assemblies of colored k-sets, where no two sets use the same color scheme. Therefore, gen(,4, UEkl[-nk])= a, * h k counts the number of unlabelled assemblies of colored k-sets where there are n sets in the assembly, and no two sets use the same color scheme. Let S be any polynomial species without constant term. The polynomial species ~

~

exp(S) is isomorphic to U * S. Intuitively, exp(S) corresponds to unlabelled assemblies of S-structures; therefore, the plethysm of a symmetric function into a homogeneous symmetric function counts unlabelled assemblies exp(S) = U* S = I - I (1 + 7~(S~)+ 72(Si)+ ..-). i %

The polynomial species ~,(Si) is equivalent to S~ since I ~ gen(S~'[m])=~in gen(Sin [ m ] ) = 6 i n n.gen(7,(Si)[m])=gen(7,(Si)[m]),

where m = nfli. Therefore, Gen(exp(S))=Gen

(

H(I+Si+S{+'")

)n' =

1-yi"

Example 2.8. The polynomial species U • Arkj corresponds to unlabelled assemblies of distinctly colored k-sets. Therefore, h. * ak counts the number of unlabelled, n member assemblies of distinctly colored k-sets.

Example 2.9. U • U[k] corresponds to unlabelled assemblies of colored k-sets. Therefore, h, * hk counts the number of unlabelled, n member assemblies of colored k-sets.

3. Plethystic inverse Let F ( X ) = Yfk(X) be a generating function of symmetric functions in the variable set X, where eachfk(X) contains only nonnegative integer coefficients. The symmetric function a l ( X ) acts as a two-sided identity for F(X):

F(X) * a , ( X ) = a ~ ( X ) . F ( X ) = F(X). The generating function G(X) is said to be the plethystic inverse of F(X) when ~(x)

• F(X) = a l (X).

It is simple to show that if F(X) has a plethystic inverse, then fo(X)=O and f a ( X ) = a l ( X ) . Furthermore, by using a recursive argument on the degree of the symmetric functions, the following theorem can be shown.

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Theorem 3.1. Every F(X) with fo(X)= 0 and f l (X)= a i(X) has a plethystic inverse. Chen [3] and Read [18] have examined inverses of formal power series under the wreath product. Joyal [8] and Labelle [9, 10] have studied inversion of species in the context of virtual species, and inversion of indicator series for species under the wreath product. Hanlon [6] examined the inverse of cycle indicator series for certain (multicolored), 'tree-like,' rooted, connected, graphs. His method interpreted the inverse in terms of related partial orders. Here, a generalized technique is employed to study a wider range of structures. In particular, more coloring schemes are permitted, and structures are not necessarily tree-like. (Also, the straight M6bius inversion arguments used here allow us to avoid certain complex counting arguments.) The general method for interpreting inverses in this section was introduced in [14]. First of all, the notion of a polynomial species is extended by taking as the base category not families of colored structures, but families of colored finite partially ordered sets with unique minima and maxima. The generating function for such a MSbius polynomial species is computed by replacing the notion of cardinality with the evaluation of the M6bius function. This permits interpretation of symmetric functions with negative terms. Then given a (symmetric) polynomial species/~, a M6bius polynomial species/~t- aj is associated to it which is, in a natural way, its inverse. A partial order is defined on assemblies of R-structures provided with automorphisms by first defining something like a monoid on these structures. This partial order leads to a M6bius polynomial species whose generating function is the plethystic inverse of the generating function for/~. This section begins with a careful description of M6bius polynomial species. Then the general technique for building a partial order from a monoid-like structure is reviewed before applying it to the case of polynomial species and plethysm. Several examples such as colored linear orders, enriched trees, and graphs are then given.

3.1. Mbbius polynomial species In this subsection, polynomial species are extended to colored families of partially ordered sets by combining polynomial species with M6bius species [2, 14]. A M6bius species is a functor M : B ~ I n t from the category of finite sets and bijections to the category of finite families of finite posets with 13 and 1, where the morphisms are bijective functions f : A ~ B such that i f f ( l ) = l ' , then I and I' are isomorphic as posets. A MSbius polynomial species is defined as a subfunctor R ~ M × H o m such that for any finite subset F of the variable set X, the set {(m,f)~R[E]: imf_~F} is finite. R is said to be symmetric when for any bijection u:X--,X, and for any structure (m,f)~R[E], (m, u of)~R[E]. From now on, assume that a M6bius polynomial species is symmetric.

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427

Set g e n ( m , f ) = p (m)gen(f) where (m,f) is an R-structure, and/~(m) is the M6bius function of the interval m evaluated at the two extremes. Define gen(R[E])=

~"

gen(m, f).

(m,f)sR[E]

Since R is symmetric, gen(R[n]) is a symmetric function of degree n with (possibly negative) integer coefficients. The MSbius generating function for R is defined as G e n ( R ) = ~ gen(R[n])/n!.

n~>0

Addition and multiplication from the theory of polynomial species can be extended to M6bius polynomial species; the corresponding generating function identities behave as desired since the M6bius valuation acts with respect to the sum and product of partially ordered sets in the same way that set cardinatity does for ordinary sets. Furthermore, the additive and multiplicative inverses of polynomial species can be interpreted [20]. 3.2. C-monoid~

The general technique for defining a partial order from a monoid-like structure will be briefly reviewed. Let d be a category of categories such that for any element A in d the morphisms of A are isomorphisms. Furthermore, suppose that [] : z¢ x ,~/--,~' is a bifunctor such that for any A 1 , A z E d , A l DA 2 is a subcategory of the product category A 1 x A 2. (Subsequent notation will reflect this.) In addition, suppose that for any A 1, A 2, A 3 E d , there is a natural isomorphism zt :(A1 [] A2) [] A3---~A ~ [] (A2 [] A3), ((ax,a2),a3)~--~(al,(az,a'3)),

where ai~obj Ai, a3eobj A3, and a3 is isomorphic to a3 in A 3. (Assume that iN is coherent. This does not present problems since only concrete natural examples are considered here.)

Definition 3.1. A c-monoid in d is a triple (A, c, I) consisting of a nonvacuous element A 6 d , a functor c:A []A--,A, and an I~_A such that 1. (a) Associativity. The following diagram commutes: (A~A)DA

~ ,A[](A[]A)

]c[]id AZJA Write a l "a2 for c(al,a2).

id:C,ADA

]c c

) A

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(b) Left cancellation. For any ( a l , a 2 ) , ( a l , a ' 2 ) ~ A D A , if a l . a 2 = a l . a ' 2 , then 0 2 = a~. 2. (a) Right identity. The product c induces a natural isomorphism ?:A [] I---,A. In particular, for any a~A, 7(a, iR(a)) = a iff a. iR(a) = a. (b) Left identity. The product c induces a natural isomorphism f l : I D A ~ A . In particular, for any a ~ d , fl(iL(a),a')=a iff a' is isomorphic to a in A and iL(a)" a' = a. (c) No proper divisors of unity. If there is an i e l and (al,a2)eA [] A such that a~ .a2=i, then al =i. A c-monoid (A,c,I) in d induces a partial order ~<~ on the objects of A, where al <~ca3 iff there exists an a2~A such that c ( a l , a 2 ) = a 3. Example 3.1. When M : B ~ B is a Joyal species, el(M) denotes the category whose objects are M-structures and whose morphisms are isomorphisms of M-structures. Recall that the substitution M ( N ) of two species M and N (N [-0] = 0) is defined as M(N)EE]=~

1-] N [ B ]

x

MEn],

n B¢~

where the sum ranges over all partitions of the finite set E. For this example, let U denote the uniform species ( U [ E ] = {E}). Let ~¢0 be the category whose objects are of the form el(Uo(M)) where M is a species such that IM [0-11 = 0 and IM [1]t = 1. The morphisms of the category are the functors h:el(Uo(M))~el(Uo(N)) induced by the morphisms Uo(g): Uo(M)---,Uo(N) which are themselves induced by the natural transformations g : M ~ N . For el(Uo(M)) and el(Uo(N)) in d o , the product [] is defined as el(Uo(M)) [] ei(Uo(N))=el(Uo(N(M))). A c-monoid for el(Uo(Uo)) may be constructed as follows. Let n be a partition of a set E, and let t be a partition of ~z. The coinduced partition ind,(t) is a partition of E whose blocks are the subsets C of E of the form C = I) { B : B 6 D } as D ranges over the blocks of t. For example, if r r = { B I , B 2 , B 3 } and t = { { B 1 , B 2 } , {B3} }, where B1={1,2}, B2={3,4}, and B3={5}, then ind~(t)= { {1, 2, 3, 4}, {5} }. Given (rr, t ) e Uo(Uo(Uo))[E], the product is defined as c(lr, t)=ind~(t). This product induce the usual refinement order on partitions. Lemma 3.1. For ao~objA, let P,o denote the poset {al~objA: ( a o , a l ) ~ A D A } with the order induced by <~c; let I,o = {aEobj A: ao <-%~a} be the upper order ideal of ao. Then the map q~:P,o~l~o defined as q~(al)=a when ao .al =a is an order isomorphism. It is precisely this upper interval condition which allows us to use M6bius inversion arguments when interpreting inverses.

J.S. Yang ,' Discrete Mathematics 139 (1995) 413- 442

429

3.3. ~ l . and the partial order First of all, a little terminology. Recall that any nontrivial (symmetric) polynomial species S may be decomposed into its orbits: S=y~ Si. In subsequent discussions, any polynomial species will come with an implicit decomposition. When S is without constant term; an assembly of S-structures (a,f) with associated partition n has an associated coloring f: n ~ X wheref(B)= xi for B ~ when the corresponding member of the assembly built on B is an S~-structure (refer to the definition of plethysm for the relevance o f f ) . Let a be an automorphism of the assembly (a,f). The associated partition of(a, a, f ) is defined as the associated partition of(a, f). The structure (a, a,.f) has an induced permutation c ~ : ~ n of the associated partition where d-(B)= B' when there exists b~B and b'~B' with a(b)=b'. Let R and S be polynomial species where S is without constant term. R* S is isomorphic to R * S where R * S is defined as

R * S[E] = {(a, f, m , f ) : (a,f)~exp(S)[E], ( m , f ) ~ R [ ~ ] I, where n is the associated partition of [a,f) and f is the associated coloring. In particular, the polynomial species exp(S)o is isomorphic to Uo * S = 3o * S; these two species will be used interchangeably. For any polynomial species R, let el(R) denote the category whose objects are R-structures, and whose morphisms are isomorphisms of R-structures. Let , ~ . be the category whose objects are of the form el(Uo*/~), where R is a (symmetric) polynomial species such that g e n ( R [ 0 ] ) = 0 and gen(R[1])=al(X). The morphisms of ~eJ. are the functors h:el(3o */~)~el(3o * S) derived as follows: consider the natural map Uo(g)" Uo * R ~ U o * S induced by a morphism g : R ~ S in the category of polynomial species. Uo(g) induces the natural map ¢-~o(g): 3o */~'--* Uo * S, where

3o(g)~ (;0 • fi{ E]-~3o • g [ E ] , (a, (a, f ) ) w-~(a, Uo(g)E(a, f)). This is well defined by functoriality. Uo(g) induces the functor h" el(Uo */~)--* el(Uo * S) which is a morphism of the category ~ ' . . For el(Uo */~) and el(Uo * S) in ~4., define the product L~ as el(Uo */~) G el(Uo * 5D= el(Uo * (S*/~)). When ( a , a , f ) is an assembly of R-structures provided with an automorphism, let ](a,a, f)] or ]a] equal the number of blocks in the associated partition. Always assume that a c-monoid (el(Uo */~), c,l) in d . is graded: for any

I(~, a, f), (~, a',f))~ 3o * tfi* fi)[E],

cti~,a,f), (~,a ', f ) ) = ( a , a ", f ) e U o- * R [ E ]

and

la'l=]a"l.

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J.S. Yang / Discrete Mathematics 139 (1995) 413-442

Example 3.2. Recall that P is the linear symmetric species of monochromatic linear orders. (el(rio */;), c, I)is a c-monoid where given (a, a,f, & l,f)6/~*/~[E], the product concatenates the linear orders in the assembly a in the order specified by I. Note that all members of (a, f ) are linear orders of the same length and color since f is a constant function. Also, the automorphisms a and # must be identity permutations. It is simple enough to generalize the product to /]o */~* P--structures. The identity elements for c-monoids are easily described. Lemma 3.2. Let (el(t]o*/~),c,l) be a c-monoid in d . . Then for any (a, a,f)e ~Yo*/~[E], 1. iL(a, a,f) is the unique element of ~IEI(R)[E] with coloring f and automorphism a. %

2. iR(a,a,f) is the unique element of ?l,,l(R)[n] with coloring f and automorphism where n is the associated partition of(a, a,f ),f : n ~ x is the induced coloring of(a, a,f ), and ~: n ~ ~ is the induced permutation of a.

Proof. In both cases, the colorings and automorphisms are specified by the graded condition for c-monoids. Consider lL(a,a,f).(a,a " A ,,f)=(a,a,f). ^ Since a and a' are isomorphic, the size of the subjacent set of a' equals ]E]. Since (~,a',f) is built on blocks of the associated partition of iL(a, a, f), iL(a, a, f ) must be an assembly with ]El members. By the definition of D, ig(a,a,f)=(&a',f)efJo*R[g]. Since the c-monoid is graded, the assembly (a, fr ) ^ contains I~l members. [] A c-monoid (el(t]0*/~),c,l) in d . induces a partial order on assemblies of R-structures provided with automorphisms by setting (a, a 1,f) ~
PR [a, E,f] = ({ (a, a,f)e (]o */~[E] }, ~
J.S. Yang/ Discrete Mathematics 139 (1995) 413-442

431

1. For any bijection g: E--*F between.finite sets, the map pR[g]:p[a,E,f]__.p[goaog-l,F,f,~g

a],

r~-+ Uo*/~rg-]r is an order isomorphism. 2. P[a, E , f ] has a () equal to the unique element of TIEI(R)[E] with coloring l a n d automorphism tr. 3. For any p~P[tr, E , f ] , the upper-order ideal l p = { p ' e P [ a , E , f ] : P<~¢P'I is isomorphic to P[&Tt,f], where 7z is the associated partition of p, f is the associated coloring, and ~ is the induced permutation. Proof. (1) By the functoriality of c.

(2) By the left identity property of a c-monoid and Lemma 3.2. (3) It suffices to consider the restriction of ~b in Lemma 3.1 to P [ a , E , f ] . 3.4. Plethystic inverse of a polynomial species Given an R-structure (re, f ) , let {(m,f)} denote the assembly containing only the R-structure (re, f). For an /~-structure (tr, m, f), let {(tr, m , f ) denote the assembly {(m, f)} provided with the automorphism tr. Definition 3.2. Let (el(Uo */~), c, I) be a c-monoid in d , . The reverse M6bius polynomial species/~[-u is defined as /~[ '][E] = {([0, {(~, m , f ) } ] , f ) : ( a , m , f ) e R [ E ] , [0, {(m m,f)} ] _c P[m E , f ] ], /~[-1 ] [g] (1-[~, {r } ] , f ) = ( [0, {R[g]r}],J'o g - ' ) , where g: E + F is a bijection of finite sets. Theorem 3.3. Let (el(Uo * R), c, I) be a c-monoid in d , .

Then

Gen(/~[- 11), Gen(/~) = a a(X). Proof. First of all, some notation will be given. Let f: E ~ X be any function provided

with a permutation ~r:E--*E such that f o a = f The color scheme o f f is equal to (ctl)6®l[~ when ctl elements of E are mapped to xi b y f A cycle of tr is said to have color xl when every element of that cycle is mapped to xi b y f The cycle index of (or,f ) is equal to (qo)60~ ×r ~ where qlj is the number of j-cycles in a of color xi. Note that Recall that a Uo */~-structure (tr, a,f) has an associated c o l o r i n g f a n d an induced permutation & The associated color scheme 2(tr, a,f) is defined as the color scheme ofJ: In words, 2(¢r,a,f) keeps track of the associated colors of the R-structures in the assembly (a, f). The induced cycle index ~o(a, a,f) is defined as the cycle index of(&f).

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J.S. Yang / Discrete M a t h e m a t i c s 139 (1995) 4 1 3 - 4 4 2

Even though the associated color scheme is completely determined by the induced cycle index, both notions will be useful in subsequent arguments. By Theorem 3.2, PR[a,E,f] only depends upon the cycle index of (or,f). Let c~(~t N be any color scheme and q~Ot×s N be any cycle index such that cq=Y.jqu. Set P.,,=PR[~r,E,f] where (cr,f ) has color scheme c~ and cycle index q. Then by Theorem 3.2, P.,~ will satisfy the following upper interval condition: for any p~P~,,, lp is isomorphic to Patp~,~,~p)where Ip is the upper order ideal of p in P~,~. A few facts will be needed before proceeding to the main part of the proof. (1) For any triple (~,E,f) where E is a finite set, f : E ~ X , and ¢r is a permutation of E such that fc, a =f, let

II P~ I-~, E,f] II = ~

ix(O, r),

I,l=l

where/~ is the M6bius function for PR[¢r, E,f]. The generating function for/~t- ll may be expressed as Gen(/~t- 11= ~ d.,, ItP.,, tlx~/[a[ !, where the sum ranges over all color schemes c~O~ IN and cycle indexes r/e(~t ×x IN, and the coefficient d.,, equals the number of triples (a, [[c~[],f) with color scheme and cycle index q. Given a function f: Ileal]--. X with color scheme c~,set c., ~equal to the number of permutations a of [[c~[] such that (or,f ) has cycle index r/. Then

d . , , = ~I~l! i!c.,,,

1-[ ~,!

where c . , , = [ - I Jtjljrlijrhj

!.

(2) Let n be a positive integer. The number of structures (or,s,f #, n,f)e w h e r e f h a s color scheme c~ and (#,f) has cycle index q is given by ,zi

~Jo*/~[n],

•

This is proved by using an argument similar to the one for Theorem 2.2. The first term counts the number of allowable functions f: ~---,X. Given ~ the second term counts the number of permutations d : r c ~ such that (~,f) has cycle index tl. Given

(d,f), the third term equals the number of exp(R)-structures (~,s,f) with induced permutation # and associated coloringf. Use (1) to simplify the expression. (3) Suppose that (~,s,f) is a Uo*/~-structure with associated color scheme c~=(cq,e 2 .... ). The color scheme of this structure may be computed from e and the dummy variables for R since a member of the assembly (s,f) of type i will contribute Yi to the color scheme. Therefore, the color scheme of (~r,s,f) equals y~'y~ .... . Now for the main part of the proof, pick any nontrivial c ~ N, and set n---I~l. Consider the behavior of (el(Uo */~), ~<<) restricted to structures with subjacent set [n] and color scheme e: P~= ({ (o', s,f)eUo

*/~[n] : gen(f) =x~},~<~).

J.S. Yang /'Discrete Mathematics 139 (1995) 4 1 3 - 4 4 2

433

This poser is the disjoint sum of posets of the form PR[a, [ n ] , f ] where g e n ( f ) = x ~. Write ~ and # for the zeta and M6bius functions of P~, and write P[a,n,f] for PR[a,[n],f]. In the following calculation, the sum will be over the subposets PIg, n,f] of P~. Therefore, ~ and/a on P[a, n , f ] will be equal to the zeta and M6bius functions for that subposet:

C~l~l,xx~/n!= 2

E

2

~(O'a)#(a'r)"<~/n!

P [ a , n , f ] r E P [ a , n , f ] O<~a<~r

1,1=1

:

Z

Z

P[a,n,f] a~P[~,n,f]

....

a<.r

Irl=l =

yl 52 "" /tiT P [ a , n , f ] ~',q' a e P ( a , n , f ]

a<~r

a(a) = ~'

Irl 1 =

co(a)=~'

:EZ

2

IIP~,,,,II "yx )'2 " " / n .

P [ a , n , f ] ~',~1' a ~ P ( a , n , f ] A(a).=~' o~(a)=tl'

=

I{a~P~'2(a)=x',~o(a)=q']]" ]IP,,,,, "Y, Y2 ""/n v.

=

a..., II ~'.'

'

..11 H(y,) '

i

[zl!

The second equality follows from (3), the fourth from the upper interval condition, and the sixth from (2). Sum over all z (i.e., over all P,) to get the desired result. E-]

3.5. Examples In this section, assume that any species (Joyal or polynomial) is without constant term. The following notation will be used: M is a Joyal species; R is a symmetric polynomial species; r is a (colored, labelled) R-structure; a and ~ are automorphisms; U is either the Joyal or the polynomial uniform species of sets; a is an assembly of structures which m a y or m a y not be colored; in both cases, use depends u p o n context. Two members r and r' of an assembly are said to be of the same type if there exists an a u t o m o r p h i s m of the assembly which transports r to r'; intuitively, this means that the unlabelled versions of r and r' are the 'same'. A singleton is a structure with only one node.

3.5.1. Linear orders Let L be the Joyal species of linear orders. In this subsection, linear orders endowed with different coloring schemes will be examined. The c-monoid product for each example will be concatenation; if (a, a,f 6, l,f)~R*/~[E] for one of the linear order

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symmetric species given below, then the product is (a,l',f) where l' is obtained by concatenating the linear orders in the assembly a in the order induced by I. This product can be easily extended to t~o */~*/~-structures. Note that the automorphism of any linear order is trivial and can be safely ignored for now. Example 3.3 (Power sum). Let (9(L)=/7, where P is the symmetric species of monochromatic linear orders. Let (a, r)E(_9(L)[n], where ~ must be the trivial permutation and r is a monochromatic linear order. A linear partition is a partition together with a linear order on the elements in each block. Any element of the interval [0, {(a, r)}] is a (colored) linear partition 'obtained' by 'cutting' the monochromatic linear order r into continuous segments of the same length. The interval is isomorphic to the lattice of divisors of n; its MObius valuation is #(n), where # is the classical MObius function of number theory, and Gen((9(L))= ~ p.,

Gen((9(L)t-lJ)= ~ I~(n)p,.

and

n~>l

n~>l

Example 3.4 (Multicolored linear orders). Let JC'(L)=L × Hom. Since this corresponds to (unlabelled), multicolored, linear orders, the color of each node can be picked independently. Therefore,

'

n>~l

Let (a, r)~(L). Any element of the interval [0, {(~, r)}] is a linear partition obtained by 'cutting' r into continuous segments. Given two such linear partitions (cr,a) and (cr, a'), (~,a)~c(a,a') if there is a way to 'snip' the segments in a' to obtain a. Since specifying where 1 is cut completely determines (a, a), and since more cuts result in a structure lower in the partial order, [-0, {(a, r)} ] is isomorphic to the Boolean poset B,_ 1- Therefore, Gen(J-/(L)t- 11)=

( - ) " - l p ~ = 1+ p l . n~>l

Example 3.5 (Not distinctly colored). Let ~(L)=/~, where R is the symmetric species of singletons and linear orders which are not distinctly colored. Then Gen(~(L))= ~ p~- ~ n!a,. n>~l

n~>2

The c-monoid is well defined. To prove this, it suffices to show that the product c maps ~(L)*~(L)[n] into .~(L)[n]. Let (e,a,~,r)e~(L)*2~(L)[n] for n~>2. If the assembly a of colored linear orders consists only of singletons, then r is a linear order with two members of the same color. These correspond to two nodes in the underlying set [n] of the same color; this coloring does not change in the resulting concatenated linear order. If the assembly contains a nonsingleton, that linear order has two nodes of the same color, and again the resulting linear order has the desired property.

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435

Let (~r,r)~(L)[n]. Any element of the interval [0,{(or, r)}] is a linear partition obtained by 'cutting' r into continuous segments such that (a) each segment is a singleton or contains two nodes of the same color, and (b) if there is more than one segment, then at least two segments are of the same type. There arguments can be generalized to linear orders that are singletons or have at least k nodes of the same color for any positive fixed integer k/> 2. Example 3.6 (Not monochromatic). Let @(L) =/~, where R is the symmetric species of singletons and linear orders which are not monochromatic. Then Gen(~(L))= ~ p~- ~ n~>l

p,.

n>~2

The concatenation product is well defined. Let (a, a, cL r)~(L)* @(L)[n] for n >/2. If the colored linear partition a consists only of singletons, then the R-structure r forces two nodes of In] to use different colors. If the assembly contains a nonsingleton, that linear order has two nodes of different colors. In both cases, the resulting linear order is not monochromatic. Let (a,r)E@(L)[n]. Any element of the interval [0,{(~r,r)}] is a linear partition obtained by 'cutting' r into continuous segments such that (a) each segment is a singleton or contains two nodes of different color, and (b) if there is more than one segment, then at least two segments are of different type. These arguments can be generalized to linear orders that are singletons or use at least k different colors for any positive fixed integer k ~>2. Example 3.7. Let J - ( L ) = / ~ , where R is the symmetric species of singletons and colored linear orders where the first two nodes use the same color. Count such structures by picking the color for the first two nodes, and then independently picking the colors for the remaining nodes. Then. P2 p2p~-Z=pa+l_pl

Gen(,Y-(L))=pl + n>~2

Let (or,r)c~--(L)[n]. Any element of the interval [0, {(a, r)} ] is a linear partition obtained by 'cutting' r into continuous segments such that (a) each segment is a singleton or has the first two nodes colored the same, and (b) if there is more than one segment, then the first two segments (in the ordering induced by r) are of the same type. The partial order is determined by c; note that this interval is not necessarily a subposet of the corresponding interval for J/t'(L). These arguments can be generalized to linear orders that are singletons or use the same color for the first k nodes for any positive fixed integer k ~>2.

Example 3.8. Let ff(L)=/~, where R is the symmetric species of singletons and colored linear orders where the first and last nodes use the same color, f#(L) has the same generating function as J-(L).

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The concatenation product is well defined. Let (a, a, ~, r)e~g(L) * ff(L) [n] for n ~>2. The linear order r induces a linear order on the members of the assembly a; it also forces the first and last members (called 11 and 12, respectively) to be of the same type. By the definition of the plethysm, the first and last nodes of Ii use the same color; likewise for Iz. Since the two linear orders are of the same type, the first nodes of each must use the same color. Therefore, the first node of la and the last node of I 2 u s e the same color; these are simply the first and last nodes of the resulting linear order under the concatenation product. Example 3.9 (Palindromes). A palindrome is a word that reads the same backwards or forwards. Let R be the symmetric species of colored linear orders where the colors of a linear order - - with the induced ordering - - form a palindrome. Then /~ corresponds to palindromes from alphabet X. Count palindromes of length n by picking the first and nth letters, then independently picking the second and ( n - 1 ) t h letters, etc. Note that after pairing letters, odd length palindromes will have an extra letter in the middle. As a result, G e n ( / ~ ) = p , ~" p~+ Z n~>o

.~>1

pg=P,+Pz 1 --P2

Given (or,r)e/~[n], let w = wl w2 ... w,(wi~X) denote its corresponding palindrome. Then any element of the interval [0,{(~,r)}] can be viewed as an assembly of subwords of w formed by 'snipping' w into k continuous segments such that (a) each segment is a palindrome, and (b) the ith and (k+ 1 - i ) t h segments (in the induced ordering) are the same words for 1 ~
3.5.2. New examples from old examples For the colored, linear orders considered above, the automorphisms were always the identity permutations. For a general symmetric polynomial species R, the automorphisms associated with /~'o */~-structures might seem to add difficulty in constructing an appropriate c-monoid. However, they can be ignored in practice. More specifically, one can define a simpler c-monoid for el(Uo* R) which automatically induces a c-monoid on el(/~o*/~). The formal setup is as follows. Let d , be the category whose objects are of the form el(Uo* R), where R is polynomial species such that gen(R [0]) = 0 and gen (R [ 1 ]) = a 1(X). The morphisms of d , are the functors h:el(Uo*R)~el(Uo*S) induced by the morphisms Uo(9) : Uo * R ~ Uo * S, which are themselves induced by natural transformations

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g" R ~ S in the category of polynomial species. For el(Uo * R) and el(Uo * S) in ~/',, define the product ~ as el(Uo * R) [] el(Uo * S) = el(Uo * (S * R)). When ( a , f ) is an assembly of R-structures, let I(a,j)l or lal equal the number of blocks in the associated partition. Always assume that a c-monoid (el(U0 * R), c', I') in A, is graded: for any ((a, f), ( a ' f ' ) ) i n U o * ( R * R ) [ E ] , c'((a,f),(a',f'))=(a",f)eUo*R[E]

and

la'l=la"l.

Theorem 3.4. A c-monoid ( e l ( U o * R ) , c ' l ' ) in ~J, will induce a c-monoid (el(/]o */~), c, I) in ~ / , . Specifically, given (a,a,[;&a',f)6~Jo*,R[E ] and c' (a,f a',f) = (a", f),

c(a, a,.f ~, a',f) = (~, a", f). The converse also holds. Proof. The product c is well defined; a is an automorphism of (a",f) since c' is a functor. The converse holds by examining the behavior ofc on 0o */~*/~-structures, where the provided automorphisms are trivial permutations. The Joyal species of linear orders has an associated c-monoid in d where the product is concatenation. Recall that this same product was used to define c-monoids in d , for certain colored, linear orders. This can be generalized. Theorem 3.5. A c-monoid (el(Uo(M)),c',I') in ~/: Jbr the Joyal species M : B ~ B will induce a c-monoid (el(Uo * R ), c, I) in ~ , j b r each of the jollowing symmetric polynomial species: 1. Jg(M)=/~, where R = M x H o m is the symmetric species of multicolored M-structures. 2. Cg(M)= R, where R is the symmetric species of monochromatic M-structures. 3. (a) 2(M)=/~, where R is the symmetric species of singletons and M-structures which are not distinctly colored. (b)-~k(M)=/~, where R is the symmetric species of singletons and colored M-structures where at least k nodes use the same color Jbr any positive.fixed integer k>~2. 4. (a) ~ ( M ) = / ~ , where R is the symmetric species of singletons and colored M-structures which are not monochromatic. (b) ~k(M)=/~, where R is the symmetric species of singletons and colored M-structures which use at least k different colors fi~r any positive.fixed integer k >~2.

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In each case, the product for the appropriate c-monoid is given by c(a,a,f,#,a',.f)=(cr, c'(a,a'), f). Proof. The original c-monoid in d o d , where

induces a c-monoid (el(Uo*R),c",I") in

c"(a,f,a',f)=(c'(a,a'),f). Use arguments similar to those employed in the previous subsection to show that c is well defined. Then use Theorem 3.4 to induce the desired c-monoids in d , . Let (a, m,f)eJg(M) [E]. Note that ifa is the identity permutation, then the interval [IJ, {(a,m,f)}] under the ordering induced by c will be isomorphic to the interval [13,{m}] under the ordering induced by c'. [] As a result of this theorem, many examples of c-monoids in d , can be generated since many examples of c-monoids in d o are already known [14].

3.5.3. Sets Let D denote the Joyal species of sets. Recall from Example 3.1 that sets have a c-monoid structure in d o where the product is the coinduced partition. This can be used to define several c-monoids for colored sets. Example 3.10 (Homogeneous). ~/' (D) =/~, where R is the symmetric species of multicolored sets. Gen(~'(D)) is the generating function for homogeneous symmetric functions. Let (a,r)~J#(D)[n]. The interval [0,{(a,r)} ] is the poset of partitions of the colored set r which have automorphism a (under the usual refinement order). Nava studied these posets in [15]. When a is the identity permutation, this interval is isomorphic to the lattice of partitions of In]. While in some sense the product c was defined without paying attention to the automorphisms, the automorphisms will make a critical difference in the behavior of intervals of (el(&7o*.//(D)), ~
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Example 3.12.

,,~(D) corresponds to unlabelled, colored sets which are singletons or are not distinctly colored. ~(D) corresponds to unlabelled, colored sets which are singletons or are not monochromatic. Therefore, Gen(,~(D))= ~

h.-

n~>l

Gen(~(D))= ~ n~>l

~

a.,

n~>2

h,-

Z P," n~>2

Let (a, r)~°)(D)[n]. Any element of the interval [I), {[a, r ) ] ] is a partition of the set r, where (a) cr is an automorphism, (b) each block is a singleton or not distinctly colored, and (c) if the partition has more than one block, then at least two of those blocks are of the same type.

Example 3.13 (Pointed sets). A pointed set

is a set with a specially marked element. Let D* denote the Joyal species of pointed sets. A pointed partition is a partition where each block has a specially marked element; this is simply an assembly of D*structures. (el(Uo(O*)),c,I) is a c-monoid in s ¢ where given (a,s)~O*(D*)[n], the product is the set I-n] where the marked element is the marked element of the marked block of a. This easily extends to assemblies of D** D*-structures. A pointed set is isomorphic to a bush (a rooted tree with no grandchildren). Therefore, the c-monoid for pointed sets generalizes to c-monoids for various kinds of colored bushed, including ~f(D*)=/~, where R is the symmetric species of singletons and bushes where the root and at least one child use the same color.

3.5.4. Trees Let A be the Joyal species of rooted trees. Rooted trees have a c-monoid structure in ,,~ where given (a, tl)~A(A)[n], the product t=c(a,t~) will have all of the edges in a plus a few more defined as follows: for every pair of trees m~ and m 2 in a such that the associated blocks form an edge of tl, insert an edge between the roots of m l and m2. The root of t will be the root of the tree in a whose associated block is the root of t 1.

Example 3.14.

JC(A)=/~, where R is the symmetric species of multicolored trees. Any element (~r,a,f) of ( e l ( U o * J / / ( A ) ) , 4 c ) is a forest of multicolored trees with automorphism a.

Example 3.15. Let J-(A)=/~, where R is the symmetric species of singletons and colored rooted trees where the root and all of the root's children use the same color. Let ~,~(A)=/~, where R is the symmetric species of singletons and colored rooted trees where the root and at least one of its children use the same color. The rooted tree c-monoid induces c-monoids in o~/. for each of these examples since two colored trees of the same type have roots of the same color.

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Example 3.16 (Enriched trees). Thefiber of a tree vertex is the set of its children. For a species M, a tree is said to be M-enriched if each of its fibers is provided with an M-structure. A c-monoid structure d o can be constructed for many kinds of enriched trees including trees enriched with linear orders (i.e., ordered trees), trees where each vertex has a multiple of k children (k a positive integer), partition-enriched trees, permutation-enriched trees, and even trees enriched by ordered trees [14]. These c-monoids in turn induce associated c-monoids in d , . For example, the species AL of ordered trees has a c-monoid structure defined as follows. Consider (a,r)~ALAL[E] where a is a forest of ordered trees and r is an ordered tree on the associated partition. Then c(a, r ) = r' where the rooted tree part of r' is defined as before. Suppose that x is the root of some tree t in the forest a. The 'old' (i.e., original) children of x have a linear order from t. The 'new' children of x inherit a linear order from r. Concatenate these two linear orders (old, then new) to enrich all of the children o f x in the new tree r'. I f x is not the root of a tree in a, then its children and their ordering remain unchanged in r'. Let R be the species of ordered trees where the root and its first child use the same color;/~ has an associated c-monoid in ~ ' , where the product is the natural generalization of the product for AL. Example 3.17 (Vertebrates). A pointed, rooted tree is a rooted tree with a specially marked element; this is also called a vertebrate. Let V be the Joyal species of vertebrates. V has a c-monoid structure in d o . Let (a,r)eV(V)[n], where a is an assembly of vertebrates and r is a vertebrate on the associated partition. The product treats the rooted tree parts as before; the new marked element is the marked element of the marked block. Use this to define a c-monoid for N(A)=/~, where R is the symmetric species of colored vertebrates where the head and tail use the same color. This is well defined by an argument similar to the one used for N(L).

Example 3.18 (Tree-like graphs). Hanlon's treatment of 'tree-like graphs' and their inverses [6] can be expressed in the language of species and e-monoids. Let M be the Joyal species of simple, rooted, connected graphs. Define a c-monoid el(Uo(M), c, I) be replacing the word 'tree' with the word 'M-structure' in the definition of the c-monoid for rooted trees. If a subfunctor N~_M has a well defined c-monoid by restricting the domain of c to Uo(N(N))-structures, then N is called 'tree-like'. J I ( N ) is the symmetric species of multicolored (tree-like) N-structures. Given (a,r)~l(N)[n], any element of the interval [0,{(a,r)}] is an assembly of multicolored N-structures where (a) a is an automorphism of the assembly, and (b) there exists a ,///(N)-structure such that the product of the assembly with it yields (a, r). Jg(A) (multicolored trees) fall into this setting (however, the other examples do not).

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3.5.5. More examples Example 3.19 (Graphs). Let G be the Joyal species of simple, connected graphs. G has a c-monoid structure in ~ o where given (~gB. Berr},,ql )~G(G)[E], the product is the

graphg~G[E]where {x,y} isanedgeofqif ~x,)'~ isanedgeofsomegraphg~,orthe associated blocks of ~z containing x and y form an edge in ,q~. ~',(G) corresponds to unlabelled, simple, connected graphs colored in all ways. ~)~(G) corresponds to those graphs which are singletons or where at least k nodes use the same color: @k(G) correspond to those graphs which are singletons or use at least k different colors. Note that a different c-monoid for simple, rooted, connected graphs can be defined by treating the graphs as specified here and picking the root as was done for bushes in Example 3.13. Example 3.20. A parenthesization of a set is a linear order of the set (e,g., xl x2 ..- x,) viewed as a product of n elements along with a parenthesization such that there is no ambiguity as to the meaning of the product. Let M be the Joyal species of parenthesizations. This has an associated c-monoid structure where the product is the substitution of parentheses, M is isomorphic to the species of ordered binary trees with labels on the external vertices; this is isomorphic to the species of ordered rooted trees. ~/~(M) corresponds to parenthesized words from alphabet X. Since M is isomorphic to certain kinds of trees, J//(M) is isomorphic to the same kinds of multicolored trees. Many more examples such as other kinds of pointed structures and generalizations of parenthesizations fall into this framework.

Acknowledgments Thanks to G. Labelle for his gracious assistance with this paper, and to D. Senato and A. Venezia for their helpful comments. Special thanks to G.-C. Rota for suggesting this problem and for his interest in this material.

References [1] F. Bergeron, A combinatorial outlook on symmetric functions, J. Combin. Theory Set. A 50 (1989~ 226-234. [2] F. Bonetti, G.-C. Rota, D. Senato and A. Venezia, On the foundations of combinatorial theory X: a categorical setting for symmetric functions, Stud. Appl. Math. 86 {1992) 1 29. [3] W. Chen, On the combinatorics of plethysm, Ph.D. Thesis, Masssachusens Institute of Technology. Cambridge, MA, June 1991. [4] P. Doubilet, On the foundations of combinatorial theory VII: symmetric functions through the theory of distribution and occupancy, Stud. Appl. Math. 37 (1981) 193-202. [5] 1. Gessel and G. Labelle, Lagrange inversion for species, unpublished manuscript.

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[6] P. Hanlon, A Cycle index sum inversion theorem, J. Combin. Theory Ser. A 30 (1981) 248 269. [7] A. Joyal, Une theorie combinatoire des series formelles, Adv. Math. 42 (1981) 1-82. [8] A. Joyal, Foncteurs analytiques et especes de structures, in: G. Labelle and P. Leroux, eds., Combinatoire Enum6rative, Proceedings, Montreal, Quebec, 1985, Lecture Notes in Mathematics, Vol. 1234 (Springer, Berlin, 1986) 126-159. [9] G. Labelle, On the generalized iterates of Yeh's combinatorial K-species, J. Combin. Theory Ser. A 50 (1989) 235-258. [10] G. Labelle, Some new computational methods in the theory of species, in: G. Labelle and P. Leroux, eds., Combinatoire Enumerative, Proceedings, Montreal, QuEbec, 1985, Lecture Notes in Mathematics, Vol. 1234 (Springer, Berlin, 1986) 192-209. [11] G. Labelle, Sur la sym&rie et l'asym&rie des structures combinatoires, Theoret. Comput. Sci., to appear. [12] G. Labelle, Counting asymmetric enriched trees, J. Symbolic Comput., to appear. [13] I.G. MacDonald, Symmetric Functions and Hall Polynomials (Oxford Univ. Press (Clarendon), London, 1979). [14] M. Mendez and J. Yang, M6bius species, Adv. Math. 85 (1991) 83-128. [15] O. Nava, On the combinatorics of plethysm, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, August 1986. [16] G. Pblya, Kombinatorische anzahlbestimmungen fur gruppen, graphen und chemische verbindungen, Acta Math. 68 (1937) 145-254. [17] G. P61ya and R.C. Read, Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds (Springer, New York, 1987). [18] R.C. Read, Enumeration with cycle index sums: first steps towards a simplified method, unpublished manuscript. [19] G.-C. Rota, On the foundations of combinatorial theory I: theory of M6bius functions, Z. Wahrscheinlichkeitstheorie 2 (1964) 340-368. [20] D. Senato, A. Venezia and J. Yang, M6bius polynomial species, unpublished manuscript. [21] J. Yang, Symmetric functions, plethysm, and enumeration, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, June 1991.