Linear extension diameter of subposets of Boolean lattice induced by two levels

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Electronic Notes in Discrete Mathematics 38 (2011) 337–342 www.elsevier.com/locate/endm

Linear extension diameter of subposets of Boolean lattice induced by two levels Jiˇr´ı Fink

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Department of Applied Mathematics Faculty of Mathematics and Physics, Charles University in Prague

Petr Gregor Department of Theoretical Computer Science and Mathematical Logic Faculty of Mathematics and Physics, Charles University in Prague

Abstract The linear extension diameter of a finite poset P is the diameter of the graph on all linear extensions of P as vertices, two of them being adjacent whenever they differ in exactly one (adjacent) transposition. Recently, Felsner and Massow determined the linear extension diameter of the Boolean lattice Bn , and they posed a question of determining the linear extension diameter of a subposet of Bn induced by two levels. We solve the case of the 1st and kth level. The diametral pairs are obtained from minimal vertex covers of so called dependency graphs, a new concept which may be useful also for the general case. Keywords: Linear extension graph, boolean lattice.

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The Institute for Theoretical Computer Science (ITI) is supported by project 1M0545 of the Czech Ministry of Education. 2 fi[email protected]ff.cuni.cz 1571-0653/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2011.09.055

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Introduction

The linear extension graph G(P) of a finite poset P has all its linear extensions as vertices, two of them being adjacent whenever they differ in exactly one (adjacent) transposition. This graph was originally defined in [5] and many of its properties have been studied, see for example [4] for an overview. The linear extension diameter of P, denoted by led(P), is the diameter of G(P). A study of this parameter has been initiated in [3], where it has been also conjectured that the linear extension diameter of the Boolean lattice Bn is led(Bn ) = 22n−2 − (n + 1)2n−2 . Recently, Felsner and Massow [1] (see also [2]) proved this conjecture. They also posed a question of determining led(Bnj,k ) and the diametral pairs of G(Bnj,k ) for a subposet Bnj,k of the Boolean lattice Bn induced by two levels 1 ≤ j < k ≤ n. We solve the case j = 1 and 1 < k < n. n Clearly, for k = n trivially led(Bnj,n ) = (2j ) . Theorem 1.1 For every 1 < k ≤ n, n led(Bn1,k )

=

k

2



   n n +2 + − k+1 2

n−2  i=k i≡n (mod 2)

  i k

Moreover, we show that the diametral pairs are formed by every two linear extensions that reverse all pairs of atoms, all pairs of k-sets and certain pairs of an atom and a k-set that correspond to a minimum vertex cover of so called dependency graph, which is defined below.

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Preliminaries

The distance d(L1 , L2 ) in G(P) between two linear extensions L1 , L2 of a poset P is the number of pairs of elements of P that appear in L1 and L2 in a reversed order. Such pair is called a reversal (or a reversed pair ). The poset Bn1,k consists of all atoms and k-sets over [n], ordered by inclusion. We use letters S, T, . . . to denote subsets of [n] whereas u, v, . . . denotes the elements from [n]. For ease of notation, let us write subsets of [n] compactly; for example {u, v, w} as uvw. (Thus, uv represents the 2-set {u, v} whereas {u, v} represents the pair {{u}, {v}} of atoms.) For a permutation σ of [n] we write u <σ v if u is before v in σ; that is, −1 σ (u) < σ −1 (v). For a set S ⊆ [n] let maxσ (S) denote the maximum in S with respect to <σ . Furthermore, let σ be the reversed permutation and let inv(σ) be the number of inversions in σ. For example, for σ = 2341 we have

J. Fink, P. Gregor / Electronic Notes in Discrete Mathematics 38 (2011) 337–342

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3 <σ 1, σ = 1432, and inv(σ) = 3. Let L(σ) be the set of all linear extensions of Bn1,k with the order of atoms preserving the relation <σ . When looking for a diametral pair of linear extensions L1 , L2 in G(Bn1,k ), we may assume without loss of generality that L1 ∈ L(id) where id is the identity permutation. All other diametral pairs can be obtained by automorphisms of Bn1,k . Let σ be a fixed permutation of [n], L1 ∈ L(id), and L2 ∈ L(σ). The order in L1 and L2 is denoted by
free in L(σ) if u >σ v for every v ∈ S; otherwise, it is fixed in L(σ),

•

reversible if it is free in L(id) or in L(σ),

•

simple if it is free both in L(id) and in L(σ); otherwise it is nonsimple.

If {u, S} is free in L(σ), there exists L ∈ L(σ) such that S id v and u >σ v for every v ∈ S. Let s(σ) denote the number of simple pairs {S, u}. Clearly, the number of all reversible pairs  n is 2 k+1 − s(σ). Indeed, every (k + 1)-set T corresponds to reversible pairs {T \ {maxid (T )}, maxid (T )}, {T \ {maxσ (T )}, maxσ (T )}, which are distinct if and only if they are nonsimple. Finally, nr(L1 , L2 ) = nrk,k (L1 , L2 )+nr1,k (L1 , L2 )+nrs1,k (L1 , L2 ) is the number of not reversed pairs in L1 , L2 where •

nrk,k (L1 , L2 ) is the number of pairs of k-sets not reversed in L1 , L2 ,

•

nr1,k (L1 , L2 ) is the number of reversible nonsimple pairs {S, u} not reversed in L1 , L2 ,

•

nrs1,k (L1 , L2 ) is the number of reversible simple pairs {S, u} not reversed in L1 , L2 .

The following equality follows directly from the definitions. Proposition 2.1 For every permutation σ of [n], L1 ∈ L(id), L2 ∈ L(σ) it n  n  holds that d(L1 , L2 ) = (2k ) + 2 k+1 + inv(σ) − s(σ) − nr(L1 , L2 ).

J. Fink, P. Gregor / Electronic Notes in Discrete Mathematics 38 (2011) 337–342

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A12

A13

A23

A14

A24

A34

A: 12, 3 12, 4 12, 5 13, 4 13, 5 14, 5 23, 4 23, 5 24, 5 34, 5 C23

C24

C34

B: 32, 1 42, 1 43, 1 43, 2 52, 1 53, 1 53, 2 54, 1 54, 2 54, 3 B23 B24 B34 B25 B35 B45

Fig. 1. The dependency graph G(id) for n = 5 and k = 2. The empty sets AS , BS are not shown.

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Dependency graphs

Let σ be a fixed permutation of [n]. For a k-set S let AS and BSσ be the sets of all pairs {S, u} that are free in Lid , respectively in Lσ . That is, AS = {{S, u}; u >id v for every v ∈ S},

BSσ = {{S, u}; u >σ v for every v ∈ S}. (1) σ Note that AS and BS can be both empty as |AS | = n − maxid (S),

|BSσ | = n − σ −1 (maxσ (S)).

(2)

Let CSσ be the complete bipartite graph on AS ∪ BSσ . The index σ in BSσ and CSσ is omitted whenever it is clear from the context. Clearly, {S, u} is simple if and only if {S, u} ∈ AS ∩ BS . The edge between two copies of a simple {S, u} is called a simple edge.   Let A = S∈([n]) AS and B = S∈([n]) BS . The dependency graph G(σ) of k k  σ is a (bipartite) graph on A ∪ B defined by G(σ) = S∈([n]) CS . See Figure 1 k for an illustration. The edges of G(σ) have the following interpretation, called dependency. Proposition 3.1 Let σ be a permutation of [n], L1 ∈ L(id), and L2 ∈ L(σ). For every edge of G(σ) between {S, u} ∈ AS and {S, v} ∈ BS , if there is a kset T containing u and v such that {S, T } is reversed in L1 , L2 , then x
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Lemma 3.2 For every permutation σ of [n], L1 ∈ L(id), L2 ∈ L(σ), the dependency graph G(σ) has a vertex cover of size at most nr(L1 , L2 ) + s(σ). From Proposition 2.1 and Lemma 3.2 we obtain the following upper bound. Corollary 3.3 For every permutation σ of [n], L1 ∈ L(id), and L2 ∈ L(σ) it  n  n holds that d(L1 , L2 ) ≤ (2k ) + 2 k+1 + inv(σ) − α(G(σ)).

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Tight construction

The next lemma shows that the upper bound from Corollary 3.3 is attained by some pair of linear extensions L1 ∈ L(id), L2 ∈ L(σ). For a linear extension L of Bn1,k and u ∈ [n], the set of positions in L between the i-th atom u and the next atom is called the i-th slot. The last slot of L is the slot after the last atom in L. Lemma 4.1 For every permutation σ of [n], there exists L1 ∈ L(id), L2 ∈ L(σ) such that   n n d(L1 , L2 ) = k + 2 + inv(σ) − α(G(σ)). 2 k+1  Proof. By the definition of G(σ) we have α(G(σ)) = S∈([n]) min(|AS |, |BS |). k To construct the desired extensions L1 ∈ L(id), L2 ∈ L(σ), we first decide for each k-set S into which slot it is placed in L1 and in L2 . Our aim is to reverse all free pairs {S, u} in a larger component of CS . For this purpose, we put S into the leftmost slot possible in one extension and into the last slot of the other extension. Let i = maxid (S) = n − |AS |,

j = σ −1 (maxσ (S)) = n − |BS |.

•

If i ≤ j, we put S into the i-th slot in L1 and into the last slot in L2 ;

•

if i > j, we put S into the j-th slot in L2 and into the last slot in L1 .

Now, if |AS | ≥ |BS |, then every pair {S, u} ∈ AS is reversed since S
(3)

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It remains to show that by a proper ordering within the last slots of L1 , L2 , we reverse all pairs of k-sets. (i) If distinct k-sets are not together in a last slot of the same linear extension, then they are already reversed. (ii) Let Z be the set of all k-sets that are in the last slots of both linear extensions. First, we order Z at the beginning in the last slot of L1 , then we order Z in the reversed way at the beginning in the last slot of L2 . (iii) Finally, we order the remaining k-sets in the last slots of each extension in a reversed order as they appear in the other extension. Since nrk,k (L1 , L2 ) = 0 and nrs1,k (L1 , L2 ) = 0, it follows from Proposition 2.1 n  n  + inv(σ) − s(σ) − nr1,k (L1 , L2 ) so the statement that d(L1 , L2 ) = (2k ) + 2 k+1 follows from (3). 2 Note that the constructed linear extensions are not unique. By Corollary 3.3 and Lemma 4.1, for the value of led(Bn1,k ) it only remains to find the maximal value of inv(σ) − α(G(σ)) over all permutations σ of [n]. Lemma 4.2 inv(σ) − α(G(σ)) is maximized (only) for σ = id. From Corollary 3.3, Lemma 4.1 and Lemma 4.2 we obtain an exact value of the linear extension diameter of Bn1,k . n  n  n Corollary 4.3 led(Bn1,k ) = (2k ) + 2 k+1 + 2 − α(G(id)). n n Lemma 4.4 Let αnk = α(G(id)). For every n it holds n that αn = αn+1 = 0 k k and for every n ≥ k + 2 it holds that αn+2 = αn + k .

References [1] S. Felsner and M. Massow. Linear extension diameter of downset lattices of 2-dimensional posets. Elect. Notes in Disc. Math., 34:313–317, 2009. [2] S. Felsner and M. Massow. Linear extension diameter of downset lattices of two-dimensional posets. SIAM J. Discrete Math., 25(1):112–129, 2011. [3] S. Felsner and K. Reuter. The linear extension diameter of a poset. SIAM J. Discrete Math., 12:360–373, 1999. [4] M. Naatz. The graph of linear extensions revisited. SIAM J. Discrete Math., 13:354–369, 2000. [5] G. Pruesse and F. Ruskey. Generating the linear extensions of certain posets by transpositions. SIAM J. Discrete Math., 4:413–422, 1991.