Localizing combinatorial properties of partitions

DISCRETE MATHEMATICS ELSEVIER

Discrete Mathematics 160 (1996) 1 23

Perspectives

Localizing combinatorial properties of partitions F r a n k K. H w a n g a'*, Uriel G. R o t h b l u m b'c' 1, Yi-Ching Yao d "AT&T Bell Laboratories, 600 Mountain Ave., P.O. 636, Murray Hill, NJ 07974, USA 1'Faculty o f Industrial Engineering and Management, Teehnion Israel Institute o/' Technology. Ha(['a 32000, Israel ~RUTCOR Rutgers Center jor Operations Research, Rucgers UniversiO,, P.O.B. 5062, ~%k,wBrunswick. NJ 08904, USA d Department of Statistics, Colorado State Universio,. Fort ('ollins. COL 80523, USA

Received 25 May 1994; revised 2 March 1995

Abstract The purpose of this paper is to develop a framework for the analysis of combinatorial properties of partitions. Our focus is on the relation between global properties of partitions and their localization to subpartitions. First, we study properties that are characterized by their local behavior. Second, we determine sufficient conditions for classes of partitions to have a member that has a given property. These conditions entail the possibility of being able to move from an arbitrary partition in the class to one that satisfies the given property by sequentially satisfying local variants of the property. We apply our approach to several properties of partitions that include consecutiveness, nestedness, order-consecutiveness, lull nestedness and balancedness, and we demonstrate its usefulness in determining the existence of optimal partitions that satisfy such properties.

1. Introduction

A partition is a collection of disjoint sets and a subset of a partition is called a subpartition of the given partition. We refer to properties of subpartitions as local properties of the given partition. In this paper we study properties of partitions that depend on local behavior, i.e., on properties of (small) subpartitions. This approach is common in the study of combinatorial structures. Specifically, it is of major advantage to be able to 'localize' a property of a combinatorial structure by obtaining characterizations or tests of the property through small, simple substructures. For example,

* Corresponding author. Research of this author was supported by AT&T Bell Laboratories and ONR Grant N00014-92-J1142. 0012-365X/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved SSDI 0 0 1 2 - 3 6 5 X ( 9 5 ) 0 0 1 4 6 - 8

2

F.K. Hwang et al./Discrete Mathematics 160 (1996) 1-23

a graph is bipartite if and only if it contains no odd cycle, and a graph is nonplanar if and only if it contains a topological subgraph which is homeomorphic to a complete graph with five vertices or to a complete bipartite graph with two sets of three vertices. Similarly, the Chinese Remainder Theorem states that a system of (congruent) equations x = ai (mod bl), for i = 1, ..., n, has a solution if and only if each pair of the equations has a solution. A second issue we consider concerns a framework for establishing that a class of partitions, say the class of optimal partitions with respect to some objective function and subject to some constraints, satisfies a given property. In particular, we study situations that facilitate movement from a partition that does not satisfy the given property to one that does by steps that alter only small subpartitions so that a local variant of the underlying property is satisfied. Here, again, the approach is used in other combinatorial problems. For example, the Simplex Method searches for an optimal solution for linear programming problems by moving from one feasible basis to another through 'local' changes, namely, exactly one variable in the basis is replaced at each iteration. Similarly, Lemke's Algorithm for solving linear complementarity problems moves between almost complementary feasible bases through 'local' changes where exactly one variable is replaced at each iteration. Also, a major factor in the success of the theory of Schur convexity and majorization is the availability of a simple test for majorization, namely, a set of numbers A majorizes a set of numbers B if and only if there exist A = Ao, A 1, . . . , A k - 1, Ak = B such that for each i = O , 1. . . . . k - 1, Ai majorizes Ai+l while Ai and Ai+l differ only in two positions. Our paper is of particular use in the analysis of optimization problems over partitions, and the verification of the existence of optimal partitions with a particular property. Optimization problems over partitions arise in many applications including system reliability (e.g., [13]), circuit design (e.g., [10]), inventory grouping (e.g., [1, 5, 6]), location problems (e.g., 1-4]), statistics (e.g., [-2]), scheduling (e.g., [8, 12]), group testing (e.g., [7, 11, 13]) and clustering (e.g., [9, 17]). In general, these optimization problems are difficult due to an enormous number of feasible partitions. A way to deal with the huge partition spaces is to look for small subspaces where it can be shown, nevertheless, that they contain optimal partitions. So, given an optimization problem over partitions, it is useful to search for properties that are satisfied by some optimal partition. The best partition among those satisfying such a property, attained for example by enumeration, will then be an optimal partition. This approach is particularly useful when the number of partitions satisfying the property is substantially smaller than the number of all considered partitions. See the references mentioned earlier in this paragraph for implementations of this approach. For example, by the inclusion-exclusion principle, the number of partitions of N = {1, ... ,n} into p nonempty, unordered parts can be shown to be p-1

# (n,p) = p.'

( - 1)k k=O

(p - k)",

F.K. Hwang et al./ Discrete Mathematics 160 (1996) 1-23

3

which is exponential in n for each fixed p > 1. O n e well-studied property of partitions is consecutiveness which is satisfied when each part of a given partition consists of numbers that are consecutive, for example, see [5, 9, 14]. In the subspace of consecutive partitions, each partition of N into p nonempty, unordered parts corresponds to a way of inserting p - 1 bars into the n - 1 spaces between the n elements that are to be partitioned. The n u m b e r of such partitions is then (p-1), "- 1 which is polynomial in n for each fixed p. W h e n p is not specified, the n u m b e r of all partitions of N is # ( n ) = ~ - 1 # (n,p); these numbers form a series, k n o w n as the Bell Numbers, of which the first 10 elements are 1, 2, 5, 15, 52,203,877, 4140, 21147, 115975. The n u m b e r of all consecutive partitions of N is ~v" = 1 (p_ n 1) = 2" 1, which is substantially smaller; in particular, the first 10 of these numbers are 1,2, 4, 8, 16, 32, 64, 128,256, 512. We introduce our framework in Section 2, and apply our methods to important properties of partitions in Sections 3 7. In addition to consecutiveness, these properties include nestedness, order-consecutiveness, full nestedness and balancedness that will be formally defined later. In Section 8 we apply the results to classes of optimal partitions. Finally, some extensions and conclusions are discussed in Section 9.

2. Consistency and sortability of properties of partitions We start by introducing some terminology a b o u t partitions. T h r o u g h o u t we consider a finite set N of n distinct positive integers. A p-partition of a subset N' of N is a collection of p sets zt = {zq . . . . . 7zp} where rq . . . . ,~zp are disjoint, n o n e m p t y sets whose union is N'. In this case we refer to p as the size o f n and to the sets zrl . . . . . 7r~,as the parts of Tz. If the n u m b e r of elements in the parts of the p-partition 7r = ~drl . . . . . ~zr I are nl . . . . . np, respectively, we refer to the multiset {nl . . . . . nr} as the shape of zr; of course, in this case ~ jp= l nj = IN' I. We sometimes omit the reference to the set N' as the set that is partitioned and refer to a p-partition. Also, we sometimes delete the reference to the size of the partition and refer to a partition of N' or to a partition. We re-emphasize that the set N will be fixed t h r o u g h o u t our paper. O u r results apply to the case where the parts of the partition are ordered. In this case, one refers to a p-partition as an ordered set 7z = (zq, ..., ~p) where the zri's are as above. We do not state the explicit results corresponding to ordered partitions in our paper as the corresponding modifications are straightforward. Of course, when partitions are counted, their n u m b e r grows by a factor of p! when ordered partitions are considered. Let rt = ft7r1 . . . . , ~zp} be a partition. For a subset J of I 1. . . . . p }, let ztj =- ~trtj:j ~ J ~; in particular, if J has k elements, ztj is a k-partition of 0j~J 7rj and we call rrj a ksubpartition of Tz, or briefly a subpartition of zr. So, a subpartition of rt is a subset of zr. In this paper we study properties of partitions. Formally, a property is a univariate relation over partitions, i.e., it corresponds to a set of partitions which are said to satisfy the property. Naturally, each such property defines a corresponding adjective which describes partitions that satisfy the property, and we will use such adjectives in

4

F.K. Hwang et al./Discrete Mathematics 160 (1996) 1-23

the intuitive sense. F o r example, partitions that satisfy nestedness or consecutiveness relation (introduced and studied later in this paper) will be called nested or consecutive, respectively. We say that a p r o p e r t y Q of partitions is hereditary if every subpartition of a partition that satisfies Q must satisfy Q. We say that Q is k-consistent if for each p ~> k, a p-partition ~z satisfies Q whenever all of its k-subpartitions satisfy Q. The next l e m m a considers sensitivity of k-consistency as k changes. L e m m a 2.1. Suppose Q is a property of partitions and k and k' are positive integers with

k' >~k. I f Q is hereditary and k-consistent, then Q is U-consistent. Proof. Suppose rc is a partition of size p >~ k' all of whose k'-subpartitions satisfy Q, and let ~c' be any k-subpartition of re. A u g m e n t i n g ~' with arbitrary k' - k parts of which are not in re' yields a k'-subpartition of n, say re". Then n" is a k'-subpartition of n, hence, it satisfies Q. As 7r' is a k-subpartition of ~r" and re" satisfies Q, the a s s u m p t i o n that Q is hereditary implies that rr' satisfies Q. We see that every k-subpartition of rr satisfies Q; hence, the k-consistency of Q assures that rr satisfies Q. [] A p r o p e r t y Q of partitions has a natural extension to classes of partitions where we say that a class of partitions 17 satisfies Q if it contains a partition which satisfies Q. E x a m p l e s of classes of partitions of a given subset N' of N include the class of all partitions of N ' and the classes of partitions of N' with given size or given shape. Also, objective functions over these classes of partitions define classes of corresponding optimal partitions; see Section 8 for further details a b o u t optimal partitions. O n e of the key goals of the current p a p e r is the development of a f r a m e w o r k for determining that a class of partitions satisfies a given p r o p e r t y Q by considering subpartitions. The idea is to facilitate movements, via recursive changes in subpartitions, from a partition which does not satisfy p r o p e r t y Q to one that does. We next formalize this a p p r o a c h and introduce conditions that assure that the process will produce a partition satisfying the desired property. Let rr be a p-partition of a subset N' of N, let J be a subset of { 1. . . . . p } and let Q be a p r o p e r t y of partitions. A J-resorting of ~, is a p-partition 7r' of N' such that ~ = 7rj for e v e r y j ~ {1 . . . . ,p}\J. T h e idea is to rearrange the parts indexed by J into IJI new parts while leaving the remaining parts of ~z unaltered. A J-Q-resorting of ~r, is a J-resorting of ~ for which ~z}satisfies Q, whereas ~s does not satisfy Q. F o r a positive integer k, call a partition ~' of N' a k-resorting or a k-Q-resorting of n if, respectively, ~' is a J-resorting or a J-Q-resorting of r~ for some subset J of { 1. . . . . p } having k elements. Let Q be a p r o p e r t y of partitions and let k be a positive integer. A n o n e m p t y class of partitions 17 is said to be k-Q-sortable if for every partition ~ e 17, 17 contains a finite set of partitions rc° = re, rc1. . . . . n' such that ~ satisfies Q and for s = 0, ..., t - 1, rcS+ 1 is a k-Q-resorting of rcs.

F.K. Hwang et al./Discrete Mathematics 160 (1996) l 23

5

Lemma 2.2. Let Q be a property of partitions, let k be a positive integer and let II be a nonempty class of partitions which is k-Q-sortable. Then /7 satisfies Q. Let Q be a property of partitions and let k be a positive integer. A nonempty class of partitions/7 is said to be k-Q-sortable locally if for every partition ~ e / 7 of size p ~> k and every subset J of {1. . . . . P l of k elements for which ~ does not satisfy Q, /7 contains a J-Q-resorting of ~. Here, 'k-locally' is to be interpreted as a reference to k-subpartitions. A property Q of partitions is called k-sortable if every nonempty class of partitions which is k-Q-sortable locally is k-Q-sortable. The next result shows that k-sortability implies k-consistency. Lemma 2.3. Let k be a positive integer and let Q be a k-sortable property ~ffpartitions. Then Q, is k-consistent. Proof. Let ~z be a partition of size p >~ k all of whose k-subpartitions satisfy Q and let / / = ~z}. As the only partition in H has no k-subpartitions that do not satisfy Q,/7 is (vacuously) k-Q-sortable locally. As Q is k-sortable, it follows that / / = ~TrI is k-Qsortable; in particular, /7 satisfies Q (see Lemma 2.2). Thus, 7r which is the only partition in /7 satisfies Q. [] Sufficient conditions for k-sortability are next developed. Suppose k is a positive integer, Q is a property of partitions and /7 is a class of partitions which is k-Qsortable locally. One can then iteratively generate a sequence of partitions i n / 7 such that each is a k-Q-resorting of the previous one, initiating with an arbitrary partition in/7 and terminating when a partition is reached all of whose k-subpartitions satisfy Q. So, whenever termination can be guaranteed, the procedure will output a partition all of whose k-subpartitions satisfy Q; in particular, if Q is k-consistent, such a partition must satisfy Q and we will be able to conclude that/7 is k-Q-sortable. The natural tool to assure termination is to show that no partition can recur, hence, the finiteness of the set of all partitions will guarantee termination. Unfortunately, in general this is not the case and the generated sequence of partitions may cycle; see Example 6.1 in Section 6. But, conditions that assure that cycling will not occur are next identified. Recall that a partial order is a transitive, irreflexive relation. Theorem 2.4. Let k be a positive integer, let Q be a property of partitions and let ~ he

a partial order over the set of partitions which do not satis/~v Q. Further, assume that]or every partition ~ of size p >~ k that does not satisfy Q, there is a subset J ~[" ~L1. . . . . p ~ with k elements such that: (a) ~zj does not satisfy Q, and (b) 7r >7 ~' fi~r every J-Q-resorting ~' of ~z which does not satisfy Q. Then Q is k-sortable.

6

F.K. Hwang et al./Discrete Mathematics 160 (1996) 1-23

Proof. L e t / 7 be a nonempty set of partitions which is k-Q-sortable locally and we will show that /7 is k-Q-sortable. Let z~ be an arbitrary partition in /7. We construct a sequence z~° = z~, z~1, ... of partitions i n / 7 in the following way. Stop if a partition satisfying Q is reached. Alternatively, if a p-partition ~s is reached which does not satisfy Q, our hypothesis implies that some subset j s of {1 . . . . ,p} of size k has the property that (~s)js does not satisfy Q and ~s ~> ~, for every JS - Q-resorting ~' of ~s that does not satisfy Q. Now, as (zrs)js does not satisfy Q, ~ ~ / 7 and H is k-Q-sortable locally,/7 contains a partition zcs+ ~ which is a J~-Q-resorting of ~ . In particular, the selection of js implies that if ~s+l does not satisfy Q, ~ ~> ~s+ 1. So we can construct a ~>-decreasing sequence of partitions in/7. As the number of partitions is finite, no partition can recur and the construction must terminate. At termination we have a partition that satisfies Q. So, indeed,/7 is k-Q-sortable. [] Suppose the assumptions of Theorem 2.4 are satisfied a n d / 7 is a nonempty set of partitions which is k-Q-sortable locally. The proof of Theorem 2.4 suggests a construction of a (~>-decreasing) sequence of partitions a construction that starts with an arbitrary partition i n / 7 and applies iterative resorting. The generated sequence will eventually stop with a partition that satisfies Q. Thus, we get a method for identifying a partition i n / 7 which satisfies Q. We emphasize that the identification of an initial partition in the given (nonempty) set of partitions/7 can be a major endeavor, and we are not addressing this issue here. For example, when considering the problem of finding a partition that satisfies a given property in the class of optimal partitions (with respect to some specific objective), we are not addressing the issue of finding an optimal partition. The above method shows how to iteratively move, within the class of optimal partitions, from an arbitrary optimal partition to one that satisfies the desired property. Useful partial orders over partitions that satisfy the assumptions of Theorem 2.4 are linear orders that are defined through particular real-valued or lexicographic objective functions; see the forthcoming sections.

3. Consecutive partitions A subset S of a subset N' of N is called consecutive with respect to N' if for all S and k ~ N' that satisfy kl < k < k2, we have that k ~ S. A partition z~of N' is called consecutive if each part of ~ is consecutive with respect to N'. kl,k 2 E

L e m m a 3.1. Consecutiveness is hereditary. Theorem 3.2. Consecutiveness is 2-consistent. Proof. Let ~ be a partition of size p >~ 2 of a subset N' of N where every 2-subpartition of ~ is consecutive and letj ~ { 1. . . . , p }. To see that ~j is consecutive with respect to N'

F.K. Hwang et al./Discrete Mathematics 160 (1996) t 23

7

assume that k~, k2 6 gj and k ~ N' satisfy k~ < k < k 2 but kCztj. Then k ~ 7rj, for some j' 4: j, implying that {~j, gj, } is a 2-subpartition of ~r which is not consecutive. We have a contradiction which proves that gj is, indeed, consecutive with respect to N'. []

Corollary 3.3. Consecutiveness is k-consistent for ever), k >~ 2. Proof. The conclusion is immediate from Lemmas 2.1, 3.1 and T h e o r e m 3.2.

[]

A partition is called pairwise consecutive if each of its 2-subpartitions is consecutive. L e m m a 3.1 and T h e o r e m 3.2 combine to show that a partition of size p ~> 2 is consecutive if and only if it is pairwise consecutive. We next show that consecutiveness is 2-sortable by verifying the sufficient condition of T h e o r e m 2.4. The underlying argument is essentially the one used in [5, 6] to prove the existence of consecutive optimal partitions for certain optimization problems over partitions. We will need some further definitions. For a finite n o n e m p t y set of integers S, let min(S) and max(S) be the minimal and maximal elements of S, respectively, and define the range of S, denoted range(S), as range(S) - max(S) - rain(S). The range of the empty set is defined to be - 1. Finally, the range of a partition ~, denoted range(g), is the sum of the ranges of its parts.

Theorem 3.4. Consecutiveness is 2-sortable. Proof. Suppose n is a partition of a subset N' of N of size p ~> 2 which is not consecutive. As consecutiveness is 2-consistent (Theorem 3.2), there exist two indices Jl, J2 ~ [1 . . . . . p} such that the 2-subpartition {n jl, n j2 } of n is not consecutive. Let j L]I,J2)f~ and consider any J-C-resorting ~' of 7r, where C stands for consecutiveness. Then the 2-subpartition {n~l, n~2 } of n' is 2-consecutive and z

"

"

g)--Tri

forj~{1 ..... P}'\{jl,J2).

(3.1}

We will show that range(~') < range(g). By (3.1), gjlwg~2 = 7~'jlw~)2 and we denote this (common) set by S. As {g)l, zt~2 } is consecutive, range(g)1) + range(~/2) < range(zt~l wg~2) = range(S).

(3.2)

Also, as {gjl,~rj2} is not 2-consecutive, range(g~l) + range(~j2) ~> range(zrjl w gj2) = range(S),

(3.3)

(where equality can happen when either zti~ or ~2 is a singleton). It follows from (3.1) (3.3) that r a n g e ( z t ' ) - range(g) = [ r a n g e ( ~ l ) + range(Tz)2)] - [range(Tzjl) + range(zr~2)] < 0.

8

F.K. Hwang et al./Discrete Mathematics 160 (1996) 1-23

So, the assumptions of Theorem 2.4 are satisfied with the partial order > defined on the set of all partitions by n > n' when range(n) > range(n'). Hence, Theorem 2.4 implies that consecutiveness is 2-sortable. [] The above proof of Theorem 3.4 establishes a stronger assertion than the necessary condition of Theorem 2.4, namely we have that for every partition ~ which is not consecutive, range(n') < range(n) for every 2-C-resorting n' of n, not just for J-Cresortings for a particular subset J of size 2.

4. Nested partitions In the current section we study 'nestedness' of partitions. The property was introduced by Boros and H a m m e r [-4] whose definition is based on a 'pairwise' condition, i.e., a condition that depends on pairs of parts of an underlying partition. In particular, their definition trivially implies 2-consistency. Here, an alternative 'global' definition is used. It is shown to be equivalent to the one of Boros and Hammer. Let S and S' be two finite sets of integers. We say that S' penetrates S, written S' --* S, if S and S' are disjoint and there exist a, c 6 S and b ~ S' such that a < b < c. We observe that if N' is a subset of N and S is a subset of N', then S is consecutive with respect to N' if and only if N ' k S does not penetrate S. We call a partition g nested if penetration defines a partial order on its parts, i.e., if penetration is transitive and irreflexive on n. Hwang and Mallows [15] showed that for a positive integer p, the number of nested (unordered) p-partitions of N = {1, ... ,n} is (~"1) (~)/n, which is a polynomial function of n. L e m m a 4.1. Nestedness is hereditary. We next provide several characterizations for nestedness of partitions. L e m m a 4.2. Let n = {ha, ..., rip} be a partition o f N'. Then the following are equivalent: (a) n is nested, (b) n has no pair o f parts which penetrate each other. (c) there exist no four elements a, b, c and d in N' such that a < b < c < d and a and c are in one part o f n while b and d are in another, and (d) for every pair of parts n i and nj o f n, ni ~ nj if and only if min(n~) < min(ni) ~< max(hi) < max(n j).

(4.1)

Proof. ( a ) ~ (b): Suppose n is a nested, i.e., the penetration relation a m o n g parts of n is transitive and irreflexive. Now, if n has two parts, say n~ and n j, which penetrate

F.K. Hwang et a l . / D i s c r e t e Mathematics 160 t1996,* 1 23

9

each other, then transitivity would imply that n~--+ 7t~, in contradiction to the irreflexivity. (b) ~ (c): Trivially, if there exist element a, b, c and d in N' such that a < b < c < d and a and e are in one part of n while b and d are in another, then the corresponding parts penetrate each other. (c) ~ (d): Assume that there exist no four elements a, b, c and d in N' such that a < b < c < d and a and e are in one part of n while h and d are in another, and suppose n~ and ni are parts of n. Trivially, (4.1) implies that n~ --, nj. Next assume that ni--+nj. Then for some a eng, rain(n j ) < a < m a x ( h i ) . Now, if either rain(hi) < min(nj) or max(Ttj)< max(Try) we get a direct contradiction to {c). As, trivially, min0zi) ~< max(hi), (4.1) follows. ( d ) ~ ( a ) : The relation defined on the parts of n by (4.1) is obviously transitive. As penetration requires that the corresponding sets be disjoint, it is trivially irreflexive. [] Condition (c) of L e m m a 4.2 was used by Boros and H a m m e r [-4] to define nested partitions. L e m m a 4.2 implies that their definition is equivalent to ours. Also, Condition (b) of L e m m a 4.2 is a constraint on pairs of parts, implying the following fact.

Theorem 4.3. Nestedness is 2-consistent. Corollary 4.4. Nestedness is k-consistent for every k >~ 2. Proof. The conclusion is immediate from L e m m a s 2.1, 4.1 and T h e o r e m 4.3.

[]

A partition is called pairwise nested if each of its 2-subpartitions is nested. L e m m a 4.1 and T h e o r e m 4.3 combine to show that a partition of size p ~> 2 is nested if and only if it is pairwise nested. The following lemma establishes two useful properties of penetration between parts of a nested partition. L e m m a 4.5. Suppose n = {ha . . . . . rip} is a nested partition. Then: (a) I f two distinct parts o f n, say n~l and nj2, are penetrated by a common part Of n, then either n i 1 - ~ n j2 o r n j2 -'~ n j l (b) ] f ~ i l , ; ¢ i 2 and n23 are parts of n such that nil ~ n j 2 and hi3-~n2l k2Tci2, then n i 3 --~ n.i 2 .

Proof. (a) Suppose hi, nil and 7rj2 are parts of n such that ni ~ hal and ni --, n j2. By possibly interchanging Jl and J2 we may and will assume that rain(nil) < rain(hi2), and we will show that nj2 + n j ~ . By L e m m a 4.2, the assertion that hi--, nil and ni--,n~2 implies that r a i n ( n / 2 ) < min(ni) and m a x ( h i ) < max(nil). Hence, if

10

F.K. Hwang et al./ Discrete Mathematics 160 (1996) 1-23

max(n jl) < max(njz), then min(ujl) < m i n ( n j 2 ) < min(ni) ~< max(n/) < max(n jl) < m a x ( n j2), implying that nil and njz penetrate each other, in contradiction to the assertion that n is nested; see Lemma 4.2(b). This contradiction proves that, max(nj2) < max(nj1). Hence, min(njl) < min(njz) ~< max(hiE) < max(nil) and, therefore, Lemma 4.2(d) implies that, indeed, 7rjz -~ nja. (b) Suppose nj~, 7~j2 and 7rj3 are parts of n such that n i l --~ ~Zj2 and n j3 ~ nj~ w 7~j2. As nil ~ nj2, Lemma 4.2 implies that min(nj2) < min(njl) ~< max(njl) < max(nj2); hence, min(rcjlWnj2)=

min(nja)

and

max(njl Unj2)= max(nj2).

(4.2)

Also, a s 7~j3--->njlk_JT~j2 there exist elements a, c e n j l w n j 2 and b E n j 3 such that a < b < c. These inequalities combine with (4.2) to show that min(Trja) = min(njl w nj2) < b < max(njl w nj2) = max(nj2), implying that nj3 --+ 7rj2.

[]

Condition (a) of Lemma 4.5 states a necessary condition on the penetration partial order on the parts of nested partitions. It is readily observed that every partial order satisfying this condition is realizable by the penetration relation over the parts of some nested partition. Let n = {hi . . . . . rip} be a partition of a subset N' of N and let N" be a subset of N'. The restriction o f n to N" is the partition n N'' =-- {hi c~ N " : i = 1, . . . , p and ni ~ N " ~ 0}. Of course, n N'' is a partition of N", but not necessarily a p-partition. We index the parts of n u'' by the indices corresponding to the parts of n, i.e., for i e { 1, ..., p } with n i c ~ N " ¢ 0 we write (nN")i =-- n g ~ N " , so, the indices of the parts of nN" are not necessarily consecutive integers. We next show that nestedness is 2-sortable by verifying the sufficient condition of Theorem 2.4. Theorem 4.6. N e s t e d n e s s is 2-sortable. Proof. Let N' be a given subset of N. We start by identifying two indices i(n) andj(n) of parts of each partition n of N' which is not nested. First let n(n) =- max{a = 1,2 . . . . :n {1...... -l/~n' is a nested partition}.

It is easily seen that n(n) >~ 4 and, as n is not nested, n(n) <<.max(N'). Let U(n)-= {1. . . . . n(~)-- 1 } n N ' , V(n) -- {1 . . . . . n(n)} ~ N '

F.K. Hwang et al./ Discrete Mathematics 160 (1996) 1-23

11

and let i(~) be the index of the part of ~ which contains n(7~). Trivially, rc, l~)c~U(r~) ~ 0, implying that {i = 1. . . . ,p: [rtv<~)]~ ~ 0} = {i = 1, ... ,p: [~v(~t]i ¢ 0} and we denote this c o m m o n set by K(~). It follows that 7rU<~ and 7rv(~) are ]K(~z)lpartitions, 7~vt~l is nested, ~v(~) is not nested, and [rcv~)]i = [Trvl"l]~ for i ~ K(rr)\{i(rc)}. As ~vt~ is nested, L e m m a 4.1 implies that all its 2-subpartitions are nested, i.e., 7rv~l has no pair of parts that penetrate each other (recall L e m m a 4.2). We conclude that no pair of parts o f ~ vt~) where neither is indexed by i(~z) penetrate each other, i.e., all such pairs form nested 2-subpartitions of 7rv~l. Thus, each 2-subpartition of rcvl"~ which is not nested must contain [~v~)]i~). Let I(r~) - {j ~ K(~)\i(rc): { [rcv~)]i<~), [1tYrol]j} is not nested~, and rn(7~) = ]I(lr)]. As nestedness is 2-consistent, rcv{=~ contains a 2-subpartition which is not nested, hence, I(~) ¢ 0 and m(rc) ~> 1. We next show that I(rc) = {j e K(~): [Trv(~]/<, I ~ [~zv~)]j} = {j e K(Tz): [Trv(")]i~, ~ ~ [~ru~")]~}.

(4.3)

First, part (b) of L e m m a 4.2 implies that [rcv(")]~l, ~--* [~zv("l]~ for e a c h j e I(~r). To see the reverse implication, suppose [~v(~)]~, I ~ Irev~"~] ~ f o r j ~ K(~c). Then there exist a, c ~ [~rv<~]y and b ~ [ ~ v ~ ) ] i ~ such that a < b < c. As c e U(~z), c < n(~r). Hence, a]j = [r~v("~]~ and this set is penetrated by [rrv~l]~(~ I if and only if it is also penetrated by [~rvl~)]/<~)\{n(~z)} = [~zu~l]il,). The second equality of (4.3) and part (a) of L e m m a 4.5 imply that ~[~ul~l].1" j ~ I(~r)} is well ordered by penetration. Thus, I(~) contains a (unique) index .](rr) such that [rcv("~]~, 1 = [~v<,~]~(,) is not penetrated by any part [rrvl~)]~ or 7rv~"~ where j ~ l(Tr). We are now ready to establish the sufficient condition of T h e o r e m 2.4. Let rr be a partition of N' which is not nested, and consider any {i(~r), j(Tt)} - N-resorting_~ of 7r which is not nested, where N denotes nestedness. Then the 2-partition {r~,~, ~ , ~ I is not nested, the 2-partition {_g~),_~)} is nested, and ~zz= ~z~ for i e [ 1 , p}~ {i(r~), j(rr)}. We will show that n(_~)/> n(r~), and if n(r~) = n(r~) then m(_~) < m(rc). This conclusion will establish the sufficient condition for 2-sortability obtained in T h e o r e m 2.4 with J = {i(~z), j(~z)} and the partial order ~ defined on the set of partitions which are not nested by having ~z ~> _~ if either n(_~) > n(~), or n ~ ) = n(g) and m(_~) < m(7~). F o r brevity, let U = U(g) and V --- V(rr). We next argue that ~z_~ v is a nested partition. As nestedness is 2-consistent (Theorem 4.3), it suffices to show that all 2-subpartitions of rj__u_ v are nested. This is obvious for subpartitions which do not contain (_rev)~l~) or (Trv)~t~)._ Also, the selection of _re assures that the 2-subpartition {(_~v)~<~I, (~v)~l~} is nested. The remaining cases are separated into two:

12

F.K. Hwang et al./Discrete Mathematics 160 (1996) 1 23

Case I: {(nv)j, (nv)i~)} or {(nv)j, (nv)j~)} for j ~ I(n)\{ j(n)}. We first argue that in this case (nv)j does not penetrate (nv)i~w(nv)j(,), as alternatively, part (b) of L e m m a 4.5 and the facts that n v is nested and that (nv)i~ ~ (nv)j~ would imply that (nv)j ~ (nv)j(~, in contradiction to the selection of j(n). It follows that (7:v)j = (nv)j does not penetrate (nv)i<~) or (nv)j~) which are subsets of (nv)i<~>• (nv)#(~) , implying that both {(_nv)i~), (_nv)j} and {(_nu)j~),(_nv)j} are nested. Case II: {(_nv)j, (__nv)~l~)or {(_nv)j, (_nv)j~)} for j ~ K ( g ) \ [ I ( n ) ~ {i(n), j(n)}]. In this case (4.3) implies that (7~v)i~l does not penetrate (nv)j. Hence, the transitivity of the penetration relation and the fact that (Trv)~(~) penetrates (nv)j~), imply that (gv)~(~) does not penetrate (nv)~. So, (nv)i~)u(rcv)~ = n(~_)i~)u(nv)~) does not penetrate (nv)j = (nv)j, implying that both {(gv)i(~), (nv)j} and {(Ttv)j(~), (nv)j} are nested. We saw that _nU is nested, hence, n(_n) >~ n(g). We next show that if n(_n) -- n(n), then m(_n) < m(n). So, assume that n(_n) = n(n). Then U ~ ) = U, V(Tr) = V and _nv is not nested. We first show that I(n_)~{i(n),j(g)} = 0 . As n(n)en(n_)e(gv)i(~_) and (nv)i(~)w(nv)j(~) = (nv)i(~)u(nv)j(~) ~_ (Trv)i(~) ~_ {n(n)} = {n(_n)}, i(_n)is either i(n) or j(n). Suppose i(_n) = i(n). Then, trivially, i(n)dfI~). As {ui(,),_nj(.)} is 2-nested, so is {(nv)~(~), (nv)j(~)} = {(nv),(,), (nv)j(,)}, implying that j(rc)d~I~). Alternatively, if i(_n)=j(n), then, trivially, j(n)¢I~). As {ni(,),nj(,)} = {_ni(,),_ng(,)} is 2-nested, so is {(nv)~(,),(nv)~(,)} = {(nv),(,),(nv)j(,)}, implying that i(rc)¢l(n_). So, indeed, I(_n) ~ {i(n), j(n)} = 0. In particular, (uv)j = (nv)j

for each j ~ I(_n).

We next show that l(_n) is a subset of I(n). Assume that i e l(_n). Then (4.3) (applied to _n) implies t h a t (~zu)i(_~)~ (TctY)i -- (rcu)i . As (gu)i(n_) is contained in (7cu)i(.)W(rcu)j(.), we conclude that (nv)it,)~(nv)jt,)~(nv)i, i.e., either (nv)i(~) or (nv)j(~) penetrates (nv)~. But, as (nu)i(~) ---,(nv)j(~), the transitivity of penetration between parts of the nested partition n U implies that in either case (nv)~t,) --, (nv)~. By (4.3) (applied to n),

ieI(n). We have established that I ~ ) is a subset of I(n)\{j(n)}. Hence, m(_n)< re(n), completing our proof. []

5. Order-consecutive partitions A partition n = {nl . . . . , Up} of a subset N' of N is called order-consecutive if there is an enumeration of its parts, say na . . . . . n v, such that for t = 1. . . . . P, ~)~=1 nj is consecutive with respect to N'. Of course, a partition n = {nl . . . . . rip} (with a given enumeration of its parts) is order-consecutive if there is a permutation a of { 1. . . . . p } such that for t = 1. . . . . P, U}= 1 n,(j) is consecutive with respect to N'. Chakravarty et al. [6] introduced order-consecutiveness which they called semi-consecutiveness. H w a n g and Mallows [15] showed that for a positive integer p, the n u m b e r of orderconsecutive (unordered) p-partitions of N ..... {1, n} is 2j=i p-1 (2;-/-2) (2.-/-2), which is a polynomial function of n.

F.K. Hwang et al./Discrete Mathematics 160 (1996) 1 23

13

L e m m a 5.1. Order-consecutiveness is hereditary. The next l e m m a c o m p a r e s order-consecutiveness with consecutiveness and with nestedness. Its straightforward p r o o f is omitted. L e m m a 5.2. (a) Each consecutive partition is order-consecutive. (b) Each order-consecutive partition is nested. The following two examples d e m o n s t r a t e that the converse of the conclusions of L e m m a 5.2 does not hold. E x a m p l e 5.1. Let N = {1,2,3,4 I. Then the partition n = {71l,g2,Tr3} with nl = ~tl I, 7re = { 2 , 4 } and n3 = {3} is order-consecutive (in the order n3,n2,n~), but not consecutive. E x a m p l e 5.2. Let N = { 1 , 2 , 3 , 4 , 5 } . Then the partition n = { n l , n 2 , n 3 } ¢ "t and n3 = {4} is nested, but not order-consecutive. re1 : [ 1,3, 5 I, n2 = ~2~

with

Each 2-subpartition of the partition in E x a m p l e 5.2 is order-consecutive. As the partition itself is not order-consecutive, we conclude that order-consecutiveness is not 2-consistent. The next example shows that order-consecutiveness is neither 3-consistent. As T h e o r e m 2.3 shows that k-sortability implies k-consistency, our examples show that order-consecutiveness is neither 2-sortable nor 3-sortable. E x a m p l e 5.3. Let N = {1,2,3,4,5,6}. Then the partition n = {n~,n2,n3,n4} with n~ = {1,3}, n2 = {4, 6}, n3 = {2} and n4 = {5} is not order-consecutive, though all of its 3-subpartitions are. The following l e m m a shows that nestedness can be a u g m e n t e d by additional conditions to obtain a characterization of order-consecutiveness. The result will be used to prove that order-consecutiveness is 4-consistent. L e m m a 5.3. A nested partition n is order-consecutive !f and only if the Jbllowiny two conditions hold: (a) there exists no triplet n j l , nj2 and nj3 o f distinct parts o f n such that n jl --+ n j3, TCj2 ~ Tgj3 and nj3 --+ (Trjl wnj2). (b) there exists no quadruplet nil, h i e , n j3 and n j4 o f distinct parts o f 7r such that nil ~ 7rj3, hi2 ~ n14, nj3 ~ (nj~ u n j 2 ) and nj~ --, (zi 1 uTrs2).

Proof. T o prove the necessity part let N' be a subset of N and let n = {nl . . . . . rCp] be a nested partition of N' which is order-consecutive. It follows that there exists

14

F.K. Hwang et aL / Discrete Mathematics 160 (1996) 1-23

a permutation a of {1, ... ,p} such that Q) n~tj) is consecutive with respect to N'

for t = 1, ... ,p.

(5.1)

j=l

Let z be the inverse permutation of a. It follows from (5.1) that if {j} and I are subsets of {1, ... ,p} and ~j ~ ((Ji~1 hi), then z(j) < max{z(i): i e I}. To verify (a) assume that there exist distinct parts ~rja,~j2 and ~j3 of ~ where 7 r ~ l ~ j 3 , n j 2 ~ n j 3 and ~z~3~(~jlu~z~2). Then z(jl) < z(j3), z(j2) < z(j3) and z(j3) < max{z(jl),z(j2)}, yielding a contradiction which establishes condition (a). Similarly, to verify (b) assume that there exist distinct parts 7rj~,~j2,~j3 and ~j4 of 7~ where nil ~ ~3, ~j2-* ~j4, ~j3 ~ (~jl k") ~j2) and ~j4- "-'4"(~jl k.-)~j2)' Then Z ( j l ) < z(j3), z ( j 2 ) < z ( j 4 ) , z ( j 3 ) < max{z(jl), z(j2)}

and

z(j4) < max {z(jl ), z(j2)}. Hence,

z(jl) < z(j3) < max{z(jl),z(j2)} and z(j2) < z(j4) < max{z(jl),z(j2)}, yielding a contradiction which establishes condition (b). We prove the sufficiency part by induction on the number of parts of the underlying partition. Of course, the case where there are at most two parts is trite. Now, assume that if conditions (a) and (b) are satisfied by a nested partition having less than p ~> 3 parts, then the partition is order-consecutive and consider a nested p-partition 7r of N' which satisfies conditions (a) and (b). We will show that z is order-consecutive. We first prove that n has a part ~rj, which does not penetrate its complement in N', namely N'\nj.. Let 7rj~ be the part of rc which contains min(N'). We consider two cases: Case I: max(N')E Zjl. In this case min(z~jl)= min(N'), max(zcjl)= max(N') and rCjx is penetrated by all parts of re. It follows from condition (a) (with Jl replacing j3) that 7b-1 does not penetrate the union of any pair of other parts of re, immediately implying that ~zjl does not penetrate Uj ~ j~ ~rj = N' \~j~. So, with j* = jr, n j, does not penetrate N' \z~j.. Case II: max(N')¢~rjt. Let rc~2 be the part of 7~ which contains max(N'). As min(rcj~) = min(N') < min(rc~2) and max(z~j2) = max(N') > max(rc~), part (d) of Lemma 4.2 implies that neither of rcj~ and ~zj2 penetrates the other. It follows that max(re, l) < min(Tr~2), for otherwise min(zc~x) < rain(z j2) < max(rcjx) < max(zc~2) implying that rC~l and 7z~2 penetrate each other. We establish, by contradiction, that either rcj~ or rc~2 is consecutive with respect to N'. Assume that this assertion is false. Then there exist parts n j3 and 7rj4, where rcj3 ~ nil and ~zj4---, ~zj2. If j3 ---J4, then part (a) of Lemma 4.5 yields that either z~ ~ ~tj2 or zc~2~ zc~, a contradiction. Hence, J3 # j , . As min(N') = min(zt~) < min(rcj3) < max(rc~) < min(rcj2) < max(~zj,) < max(n j2) = max(N'), n j l ~ n j3 L.3nj4 a n d n j2 ---4-~j3 LJ nj4. , in contradiction to condition (b). This contradiction proves the claim that either nj~ or 7tj2 is consecutive with

F.K. Hwang et al./Discrete Mathematics 160 (1996) 1 23

15

respect to N'. Since min(~jl) = min(N') and max(rrj2 ) = max(N'), it follows that with J* = J l or j* = j 2 , ~j* does not penetrate N \ ~ , . We have seen that there is a part ~j, o f ~ such that ~j, does not penetrate N' \~zj,. Let N" - N j r j,. It follows that consecutiveness of subsets of N" with respect to N" and N' coincide. Now, {~rj: j e {1, ... , p } \ j * } is a partition of N" which (obviously) satisfies conditions (a) and (b); hence, our induction assumption implies that there is a function or{l, . ., p -.1 }.~ [.1 , . ,p}\~;*~,~j ~ such that for t = l , • . . . ~ p - 1 , ~)iJ- 1 7r is consecutive with respect to N' for all t = 1. . . . . p. Corollary 5.4. A partition ~ is order-consecutive i/'and only if it is nested and satisfies conditions (a) and (b) of Lemma 5.3. Proof. The corollary follows immediately from Lemmas 5.2 and 5.3.

U]

Theorem 5,5. Order-consecutiveness is 4-consistent. Proof. Let 7r be a partition of N of size p >~ 4 where every 4-subpartition of 7z is order-consecutive. By Corollary 5.4, every 4-subpartition of ~ is nested and satisfies conditions (a) and (b) of L e m m a 5.3. As nestedness is 4-consistent (Corollary 4.4), 7r is nested. Also, as conditions (a) and (b) of L e m m a 5.3 are satisfied for each 4-subpartition of ~z, it immediately implies that ~ itself satisfies these conditions. By applying the sufficiency of L e m m a 5.3, we conclude that 7r is order-consecutive. [] Corollary 5.6. Order-consecutiveness is k-consistent for every k >~ 4.

Proof. The conclusion is immediate from Lemmas 2.1.5.1 and Theorem 5.5.

[]

A partition is called quadruplewise order-consecutive if each of its 4-subpartitions is order-consecutive. L e m m a 5.1 and Theorem 5.5 combine to show that a partition of size p >~ 4 is order-consecutive if and only if it is quadruplewise order-consecutive.

6. Fully nested partitions We say that a partition ~ is fully nested if it is nested and the partial order induced on its parts by the penetration relation is a linear order, that is, for every pair of parts ~1 and ~2, either ~jx --* 7~j2 o r 7~j2 ---+7gi1. We observe that any fully nested partition is characterized by an insertion of 2p brackets into the n + 1 spacings between the n numbers, where the p left brackets must precede the p right ones, Since one left and one right brackets occupy the outside spacings it follows that there are (2~712) fully nested partitions. Of course, every fully nested partition is order-consecutive.

16

F.K. Hwang et al./Discrete Mathematics 160 (1996) 1-23

Table 1

n1 n2 ~Z3 n4

1

2

3

4

5

1

1 1 1

6

7

2

3

1

1 1 2

3 2 2

1 1 1

Unique 2-subpartition that is not fully nested

2

1

1

{(hi)l, (nl)2}

2 3 3

2 2 1

l

{(TC2)2, (7~2)3 }

1

{(n3)l, (~z3)2}

1

~(n4)2, (n4)3 }

L e m m a 6.1. Full nestedness is hereditary. T h e o r e m 6.2. Full nestedness is 2-consistent. Proof. Full nestedness is the intersection of nestedness and the requirement that for every pair of distinct parts of the underlying partition, one of the two parts must penetrate the other. T h e o r e m 4.3 shows that nestedness is 2-consistent and trivial a r g u m e n t s show that so is the second requirement. Thus, full nestedness is the intersection of two 2-consistent properties, immediately implying that it is 2-consistent. [] C o r o l l a r y 6.3. Full nestedness is k-consistent for every k ~ 2.

Proof. The conclusion is immediate from L e m m a s 2.1, 6.1 and T h e o r e m 6.2.

[]

The next example shows that full nestedness is not 2-sortable. E x a m p l e 6.1. Table 1 defines four 3-partitions n 1, 7~2, g3 7C4 of N' ~ { 1, 2, 3, 4, 5, 6, 7}, by stating the assignment of the elements of N ' to the parts of each partition. Each of these partitions has a unique 2-subpartition which is not fully nested, and the following partition in the a b o v e table is obtained by sorting the elements of the parts of that 2-subpartition to obtain a fully nested 2-subpartition (where 7t1 stands as a follower of n4). Thus, the set II {7~1,/r 2, 7C3, 7C4} is 2-F-sortable locally, where F stands for full nestedness. As H does not contain a fully nested partition, H is not 2-F-sortable (see L e m m a 2.2). [] =

7. B a l a n c e d partitions

A partition is called binary if each of its parts contains exactly 2 elements. Of course, if there is a binary partition of a given set, then the set contains an even n u m b e r of elements. As the class of binary partitions is closed under subpartitioning, i.e., a subpartition of a binary partition is binary, the definitions and results of Section 2

F.K. Hwang et al. ,/Discrete Mathematics 160 (1996) 1-23

17

can be a p p l i e d to the class of b i n a r y p a r t i t i o n s a n d p r o p e r t i e s t h e r e o f (rather t h a n the class of all partitions). In the c u r r e n t section we restrict a t t e n t i o n to b i n a r y partitions. A b i n a r y p - p a r t i t i o n n of a subset N ' of N is called balanced if for k = 1. . . . . p, the kth largest a n d the kth smallest element of N ' are in the same p a r t of n. O f course, a b i n a r y p - p a r t i t i o n n = {nl . . . . . np} is b a l a n c e d if a n d only if there is a p e r m u t a t i o n a of {1 . . . . . p} so that for k = 1. . . . . p, n~(k~ has a r e p r e s e n t a t i o n [ak,bk} such that al
.-.


... < b 2 < b l .

(7.1)

O f course, there is only one b a l a n c e d ( u n o r d e r e d ) p - p a r t i t i o n of a n y set consisting of 2p distinct elements.

Lemma 7.1. Balancedness is hereditary. Theorem 7.2. Balancedness is 2-consistent. Proof. W e will show that each b i n a r y p a r t i t i o n which is n o t b a l a n c e d has a 2s u b p a r t i t i o n which is n o t balanced. Let rc be a b i n a r y p a r t i t i o n of N' which is not balanced. S u p p o s e N' = {al, ... , a 2 p l where al < a2 < .-- < a2~. T h e n the q u a n t i t y q(n) -= m i n { k = 1. . . . ,p: ak a n d a 2 o + l _ k are not in the s a m e p a r t of n}

(7.2)

is well-defined. Let j l a n d j2 be the (distinct) indices of the p a r t s of n that c o n t a i n qtrc) a n d 2p + I - q(n), respectively. T h e n there exist u, v • { q(n) + 1. . . . . 2p q(n)} such that njl =ltaq(nl,au} a n d gj2={a,.,a2p+l_q~n~l. Let S~.7~jlk._)7~j2={aq(n),au,a,., azp+ 1 -qIn)lJ . T h e n m i n ( S ) = aq~.) a n d m a x ( S ) = azp+ t q~n~are n o t in the same part of the 2 - s u b p a r t i t i o n {nj1,7~j2 } of 7"C;hence, this 2 - s u b p a r t i t i o n of n is n o t balanced.

Corollary 7.3. Balancedness is k-consistent Jor erery k .>1 2. Proof. T h e c o n c l u s i o n is i m m e d i a t e from L e m m a s 2.1, 7.1 a n d T h e o r e m 7.2.

[]

W e next s h o w t h a t b a l a n c e d n e s s is 2 - s o r t a b l e by verifying the sufficient c o n d i t i o n of a m o d i f i c a t i o n of T h e o r e m 2.4.

Theorem 7.4. Balancedness is 2-sortable. Proof. Here, we use a m o d i f i c a t i o n of T h e o r e m 2.4 where only b i n a r y p a r t i t i o n s are considered. Let N' = {al . . . . ,a2p} where a l < a2 < .-- < a2p. F o r each b i n a r y partition n of N ' which is n o t b a l a n c e d define q(n) by (7.2). S u p p o s e n is a b i n a r y p a r t i t i o n of N' which is not balanced. Let .Jl a n d .j:~ be the indices of the p a r t s of n t h a t c o n t a i n q(n) a n d 2p + 1 - q(n), respectively. In p a r t i c u l a r , there exist u, v e {q(n) + 1. . . . ,2p - q(n)) such that 7Zjl = {aqln) , a,, } a n d 7z.j2 = [av,a2p+l_qln) }. C o n s i d e r a b i n a r y p a r t i t i o n n' of N' which is a [ j ~ . j 2 ~ - B -

F.K. Hwang et al. / Discrete Mathematics 160 (1996) 1-23

18

resorting of ~z and does not satisfy B, where B stands for balancedness. As rc)l •rc)2 = {aqt~),a,,av,azp+ l-qt~)}, q(rc) < u < 2p + 1 - q(TQ, q(g) < v < 2p + 1 q(Tr) and {Tr}~,~)2 } is Z-balanced, {Tr}~,7r)2} = {{aq(,), azp+l -q(~, }, {a., a~}}. It follows that for k = 1. . . . . q(rt), ak and azp+l-k are in the same part of ~z', implying that q0z') > q(~z). It follows that the assumptions of the modified version of Theorem 2.4, are satisfied with J = {jl,Jz} and the partial order ~> defined on the set of binary partitions of N' which are not balanced by ~r ~> 7r' if and only if q(~) < q(~'). By the modification of Theorem 2.4, it follows that balancedness is 2-sortable. []

8. Structured optimal partitions In this section we consider opitmization problems over partitions and show how k-sortability can be used to establish the existence of optimal partitions having desired properties. L e t / / b e a class of partitions. A (numerical) objective function over 171 is a function F: H ~ R, i.e., a function F which associates each partition n with a real number F(n). A partition n* is called F-optimal over 11 if n* ~ 11 and F(rc*) >>.F(1t)

for every n e 11;

(8.1)

the class of such (optimal) partitions is denoted 11,. Important classes of partitions over which such optimization problems are considered include the class of all partitions, the classes of partitions of given size or given shape, and classes of partitions whose shape satisfies given constraints. Suppose 11 is a nonempty class of partitions and F is an objective function over 11. In order to prove that 17" satisfies a k-sortable property Q, it suffices to show that 11, is k-Q-sortable locally. As consecutiveness, nestedness and balancedness were shown to be 2-sortable, we have a framework for establishing the existence of optimal partitions that satisfy these properties. The next result develops a tool for establishing the existence of optimal partitions that satisfy a given k-sortable property by considering (smaller) optimization problems over k-partitions. We will need a few additional definitions. Suppose H is a class of p-partitions, n = {nl, ... ,•p} is a partition i n / / , J is a k-subset of {1. . . . . p} and jc_~ {1. . . . . p } \ J . The J-projection of 1I at 7~, denoted 11(7~,J), is (the set of kpartitions of (Jj~j 7rj) defined by 11(re, J) =- {(~')s: ~'~11 and (~')so = ~s°}. If F is an objective function over 11, we define the J-projection of F at ~, denoted F(g, J), as the real-valued function defined on H(g, J) by having F(rc, J)(z) = F(~z')

if z = (~z')s where 7~'~ 11 and (~z')jo = ~j,.

Of course, for each z e / / ( g , J) there is exactly one partition g' E 11 with z = (~')j and (g')so = ~jo; hence, F(rc, J) is well-defined.

F.K. Hwang et al./Discrete Mathematics 160 (1996) l 23

19

T h e o r e m 8.1. Let p and k be positive integers where p >~k, let 11 be a nonempty class of

p-partitions, let F be a (numerical) objective function over 17 and let Q be a k-sortable property over partitions. If for every 7z ~ II* and subset J of {1 . . . . ,p } consistinq of k elements, II(Tr, J)~t,.j~ satisfies Q, then l-IF satisfies Q. Proof. Let ~ be a partition in / / , of size p and let J be a subset of { 1. . . . ,P I with k elements where ~j does not satisfy Q. The assumptions of the theorem assure the existence of a partition z e//(rr, J)F(~.Jt * that satisfies Q. Let ~' be the p-partition with (rt')j~ = ~s~ and (rt')j = z. Like every partition in IIOz, J), 7z' is a J-resorting of~, and as (zr')j = r satisfies Q we have that ~z' is a J-Q-resorting of 7r. Also, the F(rr, J)-optimality of r implies that F(Tr') = F(rc, J)(r) >/F(Tr, J)(rcs) = F(rc), hence, the F-optimality of rt implies that F(~z) = F(Tr') and that 7t' is F-optimal. So, H~ contains the partition rr' which is a J-Q-resorting of ~z. This proves that H* is k-Q-sortable locally, hence, by the k-sortability of Q it is k-Q-sortable. In particular, L e m m a 2.2 implies that H* satisfies Q. [] We next consider optimization p r o b l e m s over partitions where the objective function has particular structure. We show that in this case the optimality requirement in T h e o r e m 8.1 can be simplified. Let N' be a subset N and let k be a positive integer. Suppose f : 2 N - , R, i.e., f is a real-valued function over the subsets of N. We say that the objective function F defined over all partitions of subsets of N is determined by f via summation if p

F(~) = ~ f0zfl

for a p-partition 7r = {~1 . . . . . 7~p}.

(8.2)

j=l

F o r a subset N' of N and positive integer p, let Fp(N') denote the set of all p-partitions of N'. We say that f-optimality over p-partitions Jacilitates Q if for every subset N' of N there exists a partition ~r that maximizes F over Fp(N') and satisfies Q. T h e next result shows that if we want to establish that f - o p t i m a l i t y over p-partitions facilitates a k-sortable p r o p e r t y for all p >~ k, it suffices to show that the property is facilitated by f - o p t i m a l i t y over k-partitions. T h e o r e m 8.2. Let f: 2 N ~ R and suppose the objective function F is determined by,[ via summation. Also, let k be a positive integer and let Q be a k-sortable property over partitions. Ill-optimality over k-partitions facilitates Q, then f-optimality over p-partitions facilitates Q for every p >~ k.

Proof. Suppose f - o p t i m a l i t y over k-partitions facilitates Q. Let N' be a subset of N, let p >~ k be a positive integer. We will prove that f-optimality over p-partitions facilitates Q by showing that [Fp(N')]* is k-Q-sortable locally. As Q is k-sortable, it will then follow that [F~(N')]* is k-Q-sortable, hence (by L e m m a 2.2) it must satisfy Q. In order to prove that [Fr(N')]* is k-Q-sortable locally, let ~ e [F,(N')3~ and let J be a subset of { 1. . . . . p} containing k elements where ~j does not satisfy Q. We will

F,K. H w a n g et al. / D i s c r e t e Mathematics 160 (1996) 1 23

20

show that [Fp(N')]~. contains a J-Q-resorting ~' of ~. Indeed, let N" _= ~)j~s ~j and consider the optimization p r o b l e m over Fk(N") with the objective function F. As f-optimality over k-partitions facilitates Q and as (obviously) Fk(N")~ O, there is a partition z e [Fk(N")]* that satisfies Q. In particular, as rcs~ Fk(N"),

f(vj) = F(r) >>-F(rrj) = ~ f(rcj). jeJ

jEJ

Let z' be the partition of N' with ~r) = ~j f o r j e J and ~r) = ~j f o r j e J¢ = {1, ... ,p}\J. Then p

p

F(~') = ~ f(rc'~)= 2 f(zj) + ~ f(rcj)>~ ~ f(~j)= F(7c). j= 1

jeJ

jeJ ~

j= 1

As ~' e Fv(N') and 7r is F-optimal over Fp(N'), we conclude that ~' is F-optimal over Fv(g' ) as well, i.e., 7c' e [Fv(N')]*. As ~) = 7rj for j e jc _ {1 . . . . . p}\J and ~rj = rs satisfies Q, we also have that 7r' is a J-Q-resorting of ~. [] T h e o r e m 8.2 is next modified by considering partitioning problems with shapeconstraints. F o r a subset N' of N, positive integer p and multiset v = {nl, ..., np} of positive integers with 52~=1 n / = IN' 1, let Fp. ~(N') denote the set of all p-partitions of N' with shape v. Given a function f : 2 N ~ R that determines F via summation, we say that f-shape optimality over p-partitions facilitates Q if for every subset N ' of N and every multiset v = {nl . . . . . np} of positive integers, there exists a p-partition with shape v that maximizes F over Fp.~(N') and satisfies Q. The next result modifies T h e o r e m 8.2 by considering f-shape-optimality, rather than f-optimality. The p r o o f is omitted as it is essentially identical to that of T h e o r e m 8.2. Theorem 8.3. Let f :

2 N ~ R and suppose the objective function F is determined by f via summation. Also, let k be a positive integer and let Q be a k-sortable property over partitions. If f-shape-optimality over k-partitions.facilitates Q, then f-shape-optimality over p-partitions facilitates Q for every p >~k. []

We note that T h e o r e m 8.3 can be extended to partitioning problems with shapeconstraints rather than shape-specification, e.g., when optimality is defined over partitions with a c o m m o n lower b o u n d on the cardinality of the parts. Above, we have considered optimization problems over partitions with objective functions that are determined by functions f : 2 N ---, R t h r o u g h summation. In general, f can be used to define objective functions over partitions t h r o u g h more general functions than summation, i.e., for any function h : R p ~ R it is possible to have the objective function F given by F(Tc) = h ( f ( ~ I), ... , f(Trp)) for each ordered partition ~ = (7q . . . . , ~p).

(8.3)

Of course, when, the parts of the partition are not assumed to be ordered, one will require that the function h be symmetric. One example of the m o r e general setup

F.K. Hwang et al./ Discrete Mathematics 160 (1996) 1 23

21

concerns partitioning problems where the objective function is the reliability of an assembled coherent system with series modules. In this case we have N = ~1. . . . . n I, {r/: i • N I are given positive numbers in [0, 1], f has representation f ( S ) = I~i~s ri for each subset S of N and h is given by

s~ ',(). 11 p

H ,1 xi,llI H lforal 011,

~ti: s,=O]

li: s, = 1 :

where .I: [0, 1} p --. {0, 1} is a m o n o t o n e (Boolean) function; see [-16]. This more general setup does not induce a natural objective function over subpartitions. Specifically, when some of the parts are fixed and subpartitions of the remaining set of elements are considered, the objective function over the subpartitions depends on the elements assigned to the fixed parts and is not just a function of the considered subpartitions. When h is the summation function, preferences over subpartitions are independent of the composition of the remaining parts and are representable by (the restriction of) summation-maximization over the parts of the subpartition. Hence, one can consider optimal subpartitions without referring to the remaining parts, implying that optimality is a property of subpartitions and not just of the partitions of which they are part. But, optimality is not a property of subpartitions in the more general case. Still, we emphasize that Theorem 8.1 is applicable in the more general context.

9. Concluding remarks While the basic ideas of inheritance, consistency and sortability have been seeded in the literature of optimal partitions, this paper represents the first attempt to extract these combinatorial notions from the 'optimality' context (see Section 8) and to give them a life of their own. We focused on the important roles these notions play as links between a global property and its local version. We also developed techniques to prove consistency and sortability. Finally, we showed how these combinatorial notions apply to optimization problems over partitions. Some of our findings about consistency and sortability are summarized in the following figure. In this figure Q(i,j) means that property Q is /-consistent and j-sortable and that there are no smaller indices for which these conclusions are true, respectively. Arrows represent implication, that is~ set-inclusion of the class of partitions satisfying the corresponding properties. Finally, the question marks represent the fact that we don't know if order-consecutiveness or full nestedness are k-sortable for any positive integer k, Still, we do know that order-consecutiveness is not k-sortable for k = 2 or k = 3 (by Theorem 2.3) and Examples 5.2 and 5.3) and that full nestedness is not k-sortable for k = 2 (by Example 6.1). As all the properties we consider are inherited by subpartitions, Q(i, j) implies that property Q is k-consistent for every integer k >~ i.

22

F.K. Hwang et al./ Discrete Mathematics 160 (1996) 1-23

[Consecutive(2,2) I

[ Fully nested (2,?) ]

L / I

l

I sost ,2>

I

I

The above figure suggests that neither consistency nor minimum sortability are monotone with respect to the partial order of inclusion. Sortability can be quite difficult to prove or disprove. Our basic tool to prove sortability is Theorem 2.4, which requires that if a local neighborhood is sorted into property Q (in whichever way), then the new partition must be lower in some partial order. This requirement can be easily weakened if a particular class of partitions is of concern. For example, if we are dealing with partitions having fixed shape, then it suffices to require decrease in the partial order just for those sortings which preserve the shape. We did not emphasize this finer sortability notion since it was not needed for the current paper, but, it could potentially be helpful in other cases.

References I-1] S. Anily and A. Federgruen, Structured partitioning problems, Oper. Res. 39 (1991) 130-149. 1-2] R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing (Holt, Rinchart and Winston, New York, 1975). [3] E.R. Barnes, A.J. Hoffman and U.G. Rothblum, On optimal partitions having disjoint convex and conic hulls, Math. Programming 54 (1991) 69-86. 1-4] E. Boros and P.L. Hammer, On clustering problems with connected optima in Euclidean spaces, Discrete Math. 75 (1989) 81 88. I-5] A.K. Chakravarty, J.B. Orlin and U.G. Rothblum, A partitioning problem with additive objective with an application to optimal inventory groupings for joint replenishment, Oper. Res. 30 (1982) 1018-1022. I-6] A.K. Chakravarty, J.B. Orlin and U.G. Rothblum, Consecutive optimizers for a partition problem with applications to optimal inventory groups for joint replenishment, Oper. Res. 33 (1985) 821 834. [7] R. Dorfman, The detection of defective members of large populations, Ann. Math. Statist. 14 (1943) 436-440. [8] M.C. Easton and C.K. Wong, The effect of a capacity constraint on the minimal cost of a partition, J. Assoc. Comput. Mach. 22 (1975) 441-449. 1-9] W.D. Fisher, On grouping for maximum homogeneity, J. Amer. Statist. Assoc. 53 (1958) 789 798. [10] M.R. Garey, F.K. Hwang and D.S. Johnson, Algorithms for a set partitioning problem arising in the design of multipurpose units, IEEE Trans. Comput. 26 (1977) 321-328. 1-11] C.Z. Gilstein, Optimal partitions of finite populations for Dorfman-type group testing, J. Statist. Plan. Infern. 12 (1985) 385 394.

F.K. Hwang et al./Discrete Mathematics 160 (1996) 1 23

23

[12] R.L. Graham, Bounds on multiprocessing timing anomalies, SIAM J. Appl. Math. 17 (1969) 416 429. [13] F.K. Hwang, A generalized binomial group testing problem, J. Amer. Statist. Assoc. 70 (19751 923 926. [14] F.K. Hwang, Optimal partitions, J. Optim. Theory Appl. 34 (1981) 1 10. [15] F.K. Hwang and C.L. Mallows, Enumerating nested and consecutive partitions (1993) J. Combin. Theory Set. A 70 (1995) 323 333. [16] F.K. Hwang and U.G. Rothblum, Optimality of monotone assemblies for coherent systems composed of series modules, Oper. Res. 42 (1994) 709 720. [17] F.K. Hwang, J. Sun and E.Y. Yao, Optimal set-partitioning~ SIAM J. Algebraic Discrete Methods 6(1985) 163 170.