- Email: [email protected]
Two Posets of Noncrossing Set Partitions
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Electronic Notes in Discrete Mathematics 61 (2017) 885–891 www.elsevier.com/locate/endm
Two Posets of Noncrossing Set Partitions Henri M¨ uhle 1 Institut f¨ ur Algebra Technische Universit¨ at Dresden 01069 Dresden, Germany.
Abstract In this manuscript we consider a subposet of the lattice of noncrossing set partitions of an n-element set under refinement order. This subposet is induced by those noncrossing set partitions, which do not contain the block {n − 1, n}, or which do not contain the singleton block {n} whenever 1 and n − 1 are in the same block. We prove that the resulting poset is in fact a supersolvable lattice, and we give a combinatorial proof for the value of its M¨obius function between least and greatest element by using Blass and Sagan’s theory of NBB bases. As a corollary we prove a conjecture by Bruce, Dougherty, Hlavacek, Kudo and Nicolas about the homotopy type of a certain subposet of our poset, which comes from parking functions with undesired positions. Keywords: noncrossing partition, supersolvable lattice, lexicographic shellability, NBB base, M¨obius function
1
Introduction
A set partition of [n] = {1, 2, . . . , n} is noncrossing if there are no indices i < j < k < l such that i, k and j, l belong to distinct blocks. Let us denote 1
E-mail: [email protected]
http://dx.doi.org/10.1016/j.endm.2017.07.050 1571-0653/© 2017 Elsevier B.V. All rights reserved.
886
H. Mühle / Electronic Notes in Discrete Mathematics 61 (2017) 885–891
the set of all noncrossing set partitions of [n] by NCn . We can partially order noncrossing set partitions by dual refinement, meaning that x ∈ NCn is smaller than y ∈ NCn if every block of x is contained in some block of y. Let us denote this partial order by ≤dref . The lattice (NCn , ≤dref ) of noncrossing set partitions is a remarkable poset with a rich combinatorial structure. It was introduced by Kreweras in the early 1970s [4], and has gained a lot of attention since then. It has, among other things, surprising ties to group theory, algebraic topology, representation theory, and free probability. See [5] and [9] for surveys on these lattices. In this abstract we focus on a particular subposet of (NCn , ≤dref ). Let x ∈ NCn and i, j ∈ [n]. We write i ∼x j if there is a block B ∈ x with i, j ∈ B. Consider the two sets Xn = x ∈ NCn | {n − 1, n} ∈ x , Yn = x ∈ NCn | 1 ∼x n − 1 and {n} ∈ x . For n ≥ 3 define PEn = NCn \ (Xn ∪ Yn ). It is the main purpose of this abstract to present some basic properties of the poset (PEn , ≤dref ), which is displayed in Figure 1 for n = 4. Recall for instance from [4, Corollaire 4.2] that 2n NCn = Cat(n) = 1 . n+1 n It is quickly verified that Xn = Cat(n − 2) = Yn . We thus obtain the following lemma. Lemma 1.1 ([3]) We have PE3 = 3 and for n ≥ 4 we have 2n − 4 5 9 PEn = Cat(n) − 2Cat(n − 2) = . + n−4 n+1 n−3
2
Main Results
Let us now state the main results of this manuscript. For the details of the proofs and any undefined notation we refer to the full version [8]. Theorem 2.1 For n ≥ 3 the poset (PEn , ≤dref ) is a supersolvable lattice. Let 0 denote the trivial set partition into singleton blocks, and let 1 denote the full partition into a single block. Let μ(PEn ,≤dref ) denote the M¨obius function of the lattice (PEn , ≤dref ).
H. Mühle / Electronic Notes in Discrete Mathematics 61 (2017) 885–891
887
1234
1|234
124|3
14|23
134|2
1|24|3
1|23|4
12|3|4
14|2|3
1|2|3|4 Fig. 1. The lattice (PE4 , ≤dref ).
Theorem 2.2 For n ≥ 3 we have μ(PEn ,≤dref ) (0, 1) = (−1) 2.1
2n − 5 . n n−4
n−1 4
Lattice Property and Supersolvability
Recall that a chain in a poset is a set of pairwise comparable elements. A chain is maximal if it cannot be extended to a larger chain. A poset is graded if all maximal chains have the same size. Finally, a poset is a lattice if for every two elements there exists a least upper bound (the join) and a greatest lower bound (the meet). Proposition 2.3 For n ≥ 3 the poset (PEn , ≤dref ) is a graded lattice. Proof. This is a straightforward computation. The meet of two elements of PEn agrees with their meet in (NCn , ≤dref ) unless it belongs to Xn . In that case their meet is obtained by splitting the block {n − 1, n} into two singleton blocks. Similarly, the join of two elements of PEn agrees with their join in (NCn , ≤dref ) unless it belongs to Yn . In that case their join is obtained by joining the block containing 1 and n − 1 with the singleton block {n}. 2 An element x ∈ PEn is left-modular if for all y ≤dref z the equality (y ∨ x) ∧ z = y ∨ (x ∧ z) holds. A chain is left-modular if it consists entirely of left-modular elements. For i ∈ [n] define xi to be the noncrossing set partition with the unique non-singleton block [i − 1] ∪ {n}. We thereby understand x1 = 0 and xn = 1.
888
H. Mühle / Electronic Notes in Discrete Mathematics 61 (2017) 885–891
Proposition 2.4 For n ≥ 3 the chain {x1 , x2 , . . . , xn } is a left-modular maximal chain in (PEn , ≤dref ). 2
Proof. This is a straightforward computation.
Now we can prove Theorem 2.1 by applying [7, Theorem 2], which states that a graded lattice is supersolvable if and only if it contains a left-modular maximal chain. (Let us at the same time take this as a definition of supersolvability. See the full version for the actual definition.) 2.2
M¨ obius Function
Let us for the moment take a general viewpoint, and consider an arbitrary posetP = (P, ≤). Its M¨ obius function is defined recursively by μP (x, x) = 1 and x≤z≤y μP (z, y) = 0 for x, y ∈ P with x < y. (It is zero otherwise.) If P is a lattice, then any element covering its least element ˆ0 is an atom. Let AP denote the set of atoms of P. Let be an arbitrary partial order on AP . A set X ⊆ AP is bounded below (or BB for short) if for every d ∈ X there exists a ∈ AP such that a d and a < X. A set X ⊆ AP is NBB if none of its nonempty subsets is BB. If X is NBB and X = x, then X is an NBB-base for x. The following result states that the NBB-bases for x ∈ P can be used to compute μP (ˆ0, x). Theorem 2.5 ([2, Theorem 1.1]) Let P = (P, ≤) be a finite lattice, and let be any partial order on AP . For x ∈ P we have μP (ˆ0, x) =
(−1)|X| , X
where the sum is over all NBB-bases for x with respect to . For our purposes, we consider the following order coming from the leftmodular chain of (PEn , ≤dref ) considered in Proposition 2.4. Let ai,j denote the noncrossing set partition whose only non-singleton block is {i, j}. We have An = A(NCn ,≤dref ) = ai,j | 1 ≤ i < j ≤ n , A¯n = A(PEn ,≤dref ) = An \ {a1,n−1 , an−1,n }.
and
Consider the partition of An given by Ai = a ∈ An | a ≤dref xi and a ≤dref xi−1
H. Mühle / Electronic Notes in Discrete Mathematics 61 (2017) 885–891
889
for i ∈ [n − 1], and let A¯i be the restriction of Ai to A¯n . It is a straightforward to verify that Ai = {aj,i | 1 ≤ j < i} ∪ {ai,n } for i ∈ [n − 1]. Define a partial order on An by setting a a if and only if a ∈ Ai and a ∈ Aj for i ≤ j. For X ⊆ An we can define a graph on [n] vertices as follows. Write the vertices from 1 through n on a horizontal line and connect the vertices i and j with an arc rising above this line if and only if ai,j ∈ X. Proposition 2.6 For n ≥ 3 a subset X ⊆ An is an NBB-base for 1 in (NCn , ≤dref ) if and only if X is a maximal chain in (An , ) and in its associated graph no two edges cross. Such a subset X is an NBB-base for 1 in (PEn , ≤dref ) if and only if it contains neither a1,n−1 nor an−1,n , and X \ {a1,n } does not join to 1 in (PEn , ≤dref ). To illustrate Proposition 2.6, consider n = 4. There are exactly two maximal chains in (A¯4 , ), namely X1 = {a1,4 , a1,2 , a2,3 } and X2 = {a1,4 , a2,4 , a2,3 }. Both have noncrossing diagrams, but we can verify in Figure 1 that a1,2 ∨a2,3 = 1. There is thus one NBB-base for 1 in (PE4 , ≤dref ). We can also verify directly in Figure 1 that μ(PE4 ,≤dref ) (0, 1) = −1 in accordance with Theorem 2.5. We conclude this section with the proof of Theorem 2.2. Proof of Theorem 2.2. The key observation is that any NBB-base for 1 in (NCn , ≤dref ) corresponds to a tree on n vertices with the following two properties: (i) it contains an edge between 1 and n, and (ii) the removal of this edge yields two trees on vertex sets [k] and {k + 1, k + 2, . . . , n} for some k ∈ [n − 1]. It is then easy to verify that there are exactly Cat(n − 1) such NBB-bases. Proposition 2.6 implies that the NBB-bases for 1 in (PEn , ≤dref ) are a particular subset of the NBB-bases for 1 in (NCn , ≤dref ). It is quickly verifed that there are Cat(n − 2)-many NBB-bases for 1 in (NCn , ≤dref ) that contain an−1,n . The remaining NBB-bases for 1 in (NCn , ≤dref ) that are not NBBbases for 1 in (PEn , ≤dref ) have the property that in the associated tree the only vertex adjacent to n is 1; consequently there are also Cat(n − 2) of them. The total number
of NBB-bases for 1 in (PEn , ≤dref ) is thus Cat(n − 1) − 4 2n−5 2Cat(n − 2) = n n−4 . Since every NBB-base for 1 has n − 1 elements, we conclude from Theorem 2.5 that n−1 4 2n − 5 . μ(PEn ,≤dref ) (0, 1) = (−1) n n−4 2
890
H. Mühle / Electronic Notes in Discrete Mathematics 61 (2017) 885–891
1234
1|234
124|3
14|23
134|2
1|24|3
1|23|4
12|3|4
14|2|3
1|2|3|4 Fig. 2. The poset (PE4 , ≤pchn ).
3
A Corollary
The motivation for the work presented in this extended abstract comes from [3], where particular subposets of (NCn , ≤dref ) induced by parking functions with certain forbidden values were considered. It is well known that the maximal chains of (NCn , ≤dref ) are in bijection with parking functions of length n − 1 [10], and thus any subset of parking functions generates a subposet of (NCn , ≤dref ) simply by taking the union of these chains. It was shown in [3] that the subposet induced by all parking functions not containing the value n − 1 has ground set PEn . Let us denote this poset by (PEn , ≤pchn ). By definition there are fewer order relations in ≤pchn than in ≤dref . Figure 2 displays (PE4 , ≤pchn ), which coincides with the poset in Figure 1 except for the missing edge between 1|23|4 and 1|234. Theorem C of [3] states that the M¨obius function of (PEn , ≤pchn ) vanishes between 0 and 1, and it was conjectured that these posets have a contractible order complex. We can use Theorem 2.1 to prove this conjecture. Recall for instance from [6, Theorem 1] that supersolvable lattices of rank n admit an edge-labeling with the property that the sequence of edge-labels along any maximal chain is a permutation of [n] and there is a unique maximal chain (in each interval) whose label sequence is increasing. Posets with such a labeling have the nice property that the order complex of their proper part is a wedge of (n − 2)-dimensional spheres, and the number of spheres is given by the absolute value of their M¨obius function between least and greatest element [1, Theorem 5.9]. Theorem 2.2 thus implies
that the order complex of 4 2n−5 the proper part of (PEn , ≤dref ) is a wedge of n n−4 -many (n−2)-dimensional
H. Mühle / Electronic Notes in Discrete Mathematics 61 (2017) 885–891
891
spheres. It turns out that (PEn , ≤pchn ) inherits the labeling coming from the supersolvability of (PEn , ≤dref ) [8, Proposition 4.4], and it retains the crucial property that there is a unique maximal chain (in each interval) whose label sequence is increasing. As a consequence, the order complex of its proper part is a wedge of zero (n − 2)-dimensional spheres, and thus contractible.
References [1] Bj¨orner, A. and Wachs, M. L., Shellable and Non-Pure Complexes and Posets I, Trans. Amer. Math. Soc. 348 (1996), pp. 1299–1327. [2] Blass, A. and Sagan, B. E., M¨ obius Functions of Lattices, Adv. Math. 127 (1997), pp. 94–123. [3] Bruce, M., Dougherty, M., Hlavacek, M., Kudo, R. and Nicolas, I., A Decomposition of Parking Functions by Undesired Spaces, Electron. J. Combin. 23 (2016), # P3.32. [4] Kreweras, G., Sur les partitions non crois´ees d’un cycle, Discrete Math. 1 (1972), pp. 333–350. [5] McCammond, J., Noncrossing Partitions in Surprising Locations, Amer. Math. Monthly 113 (2006), pp. 598–610. [6] McNamara, P., EL-Labelings, Supersolvability, and 0-Hecke Algebra Actions on Posets, J. Combin. Theory (Ser. A) 101 (2003), pp. 69–89. [7] McNamara, P. and Thomas, H., Poset Edge-Labellings and Left Modularity, Europ. J. Combin. 27 (2006), pp. 101–113. [8] M¨ uhle, H., Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces, arXiv:1701.02109 (2017). [9] Simion, R., Noncrossing Partitions, Discrete Math. 217 (2000), pp. 397–409. [10] Stanley, R. P., Parking Functions and Noncrossing Partitions, Elec. J. Combin. 4 (1997), # R20.
Electronic Notes in Discrete Mathematics 61 (2017) 885–891 www.elsevier.com/locate/endm
Two Posets of Noncrossing Set Partitions Henri M¨ uhle 1 Institut f¨ ur Algebra Technische Universit¨ at Dresden 01069 Dresden, Germany.
Abstract In this manuscript we consider a subposet of the lattice of noncrossing set partitions of an n-element set under refinement order. This subposet is induced by those noncrossing set partitions, which do not contain the block {n − 1, n}, or which do not contain the singleton block {n} whenever 1 and n − 1 are in the same block. We prove that the resulting poset is in fact a supersolvable lattice, and we give a combinatorial proof for the value of its M¨obius function between least and greatest element by using Blass and Sagan’s theory of NBB bases. As a corollary we prove a conjecture by Bruce, Dougherty, Hlavacek, Kudo and Nicolas about the homotopy type of a certain subposet of our poset, which comes from parking functions with undesired positions. Keywords: noncrossing partition, supersolvable lattice, lexicographic shellability, NBB base, M¨obius function
1
Introduction
A set partition of [n] = {1, 2, . . . , n} is noncrossing if there are no indices i < j < k < l such that i, k and j, l belong to distinct blocks. Let us denote 1
E-mail: [email protected]
http://dx.doi.org/10.1016/j.endm.2017.07.050 1571-0653/© 2017 Elsevier B.V. All rights reserved.
886
H. Mühle / Electronic Notes in Discrete Mathematics 61 (2017) 885–891
the set of all noncrossing set partitions of [n] by NCn . We can partially order noncrossing set partitions by dual refinement, meaning that x ∈ NCn is smaller than y ∈ NCn if every block of x is contained in some block of y. Let us denote this partial order by ≤dref . The lattice (NCn , ≤dref ) of noncrossing set partitions is a remarkable poset with a rich combinatorial structure. It was introduced by Kreweras in the early 1970s [4], and has gained a lot of attention since then. It has, among other things, surprising ties to group theory, algebraic topology, representation theory, and free probability. See [5] and [9] for surveys on these lattices. In this abstract we focus on a particular subposet of (NCn , ≤dref ). Let x ∈ NCn and i, j ∈ [n]. We write i ∼x j if there is a block B ∈ x with i, j ∈ B. Consider the two sets Xn = x ∈ NCn | {n − 1, n} ∈ x , Yn = x ∈ NCn | 1 ∼x n − 1 and {n} ∈ x . For n ≥ 3 define PEn = NCn \ (Xn ∪ Yn ). It is the main purpose of this abstract to present some basic properties of the poset (PEn , ≤dref ), which is displayed in Figure 1 for n = 4. Recall for instance from [4, Corollaire 4.2] that 2n NCn = Cat(n) = 1 . n+1 n It is quickly verified that Xn = Cat(n − 2) = Yn . We thus obtain the following lemma. Lemma 1.1 ([3]) We have PE3 = 3 and for n ≥ 4 we have 2n − 4 5 9 PEn = Cat(n) − 2Cat(n − 2) = . + n−4 n+1 n−3
2
Main Results
Let us now state the main results of this manuscript. For the details of the proofs and any undefined notation we refer to the full version [8]. Theorem 2.1 For n ≥ 3 the poset (PEn , ≤dref ) is a supersolvable lattice. Let 0 denote the trivial set partition into singleton blocks, and let 1 denote the full partition into a single block. Let μ(PEn ,≤dref ) denote the M¨obius function of the lattice (PEn , ≤dref ).
H. Mühle / Electronic Notes in Discrete Mathematics 61 (2017) 885–891
887
1234
1|234
124|3
14|23
134|2
1|24|3
1|23|4
12|3|4
14|2|3
1|2|3|4 Fig. 1. The lattice (PE4 , ≤dref ).
Theorem 2.2 For n ≥ 3 we have μ(PEn ,≤dref ) (0, 1) = (−1) 2.1
2n − 5 . n n−4
n−1 4
Lattice Property and Supersolvability
Recall that a chain in a poset is a set of pairwise comparable elements. A chain is maximal if it cannot be extended to a larger chain. A poset is graded if all maximal chains have the same size. Finally, a poset is a lattice if for every two elements there exists a least upper bound (the join) and a greatest lower bound (the meet). Proposition 2.3 For n ≥ 3 the poset (PEn , ≤dref ) is a graded lattice. Proof. This is a straightforward computation. The meet of two elements of PEn agrees with their meet in (NCn , ≤dref ) unless it belongs to Xn . In that case their meet is obtained by splitting the block {n − 1, n} into two singleton blocks. Similarly, the join of two elements of PEn agrees with their join in (NCn , ≤dref ) unless it belongs to Yn . In that case their join is obtained by joining the block containing 1 and n − 1 with the singleton block {n}. 2 An element x ∈ PEn is left-modular if for all y ≤dref z the equality (y ∨ x) ∧ z = y ∨ (x ∧ z) holds. A chain is left-modular if it consists entirely of left-modular elements. For i ∈ [n] define xi to be the noncrossing set partition with the unique non-singleton block [i − 1] ∪ {n}. We thereby understand x1 = 0 and xn = 1.
888
H. Mühle / Electronic Notes in Discrete Mathematics 61 (2017) 885–891
Proposition 2.4 For n ≥ 3 the chain {x1 , x2 , . . . , xn } is a left-modular maximal chain in (PEn , ≤dref ). 2
Proof. This is a straightforward computation.
Now we can prove Theorem 2.1 by applying [7, Theorem 2], which states that a graded lattice is supersolvable if and only if it contains a left-modular maximal chain. (Let us at the same time take this as a definition of supersolvability. See the full version for the actual definition.) 2.2
M¨ obius Function
Let us for the moment take a general viewpoint, and consider an arbitrary posetP = (P, ≤). Its M¨ obius function is defined recursively by μP (x, x) = 1 and x≤z≤y μP (z, y) = 0 for x, y ∈ P with x < y. (It is zero otherwise.) If P is a lattice, then any element covering its least element ˆ0 is an atom. Let AP denote the set of atoms of P. Let be an arbitrary partial order on AP . A set X ⊆ AP is bounded below (or BB for short) if for every d ∈ X there exists a ∈ AP such that a d and a < X. A set X ⊆ AP is NBB if none of its nonempty subsets is BB. If X is NBB and X = x, then X is an NBB-base for x. The following result states that the NBB-bases for x ∈ P can be used to compute μP (ˆ0, x). Theorem 2.5 ([2, Theorem 1.1]) Let P = (P, ≤) be a finite lattice, and let be any partial order on AP . For x ∈ P we have μP (ˆ0, x) =
(−1)|X| , X
where the sum is over all NBB-bases for x with respect to . For our purposes, we consider the following order coming from the leftmodular chain of (PEn , ≤dref ) considered in Proposition 2.4. Let ai,j denote the noncrossing set partition whose only non-singleton block is {i, j}. We have An = A(NCn ,≤dref ) = ai,j | 1 ≤ i < j ≤ n , A¯n = A(PEn ,≤dref ) = An \ {a1,n−1 , an−1,n }.
and
Consider the partition of An given by Ai = a ∈ An | a ≤dref xi and a ≤dref xi−1
H. Mühle / Electronic Notes in Discrete Mathematics 61 (2017) 885–891
889
for i ∈ [n − 1], and let A¯i be the restriction of Ai to A¯n . It is a straightforward to verify that Ai = {aj,i | 1 ≤ j < i} ∪ {ai,n } for i ∈ [n − 1]. Define a partial order on An by setting a a if and only if a ∈ Ai and a ∈ Aj for i ≤ j. For X ⊆ An we can define a graph on [n] vertices as follows. Write the vertices from 1 through n on a horizontal line and connect the vertices i and j with an arc rising above this line if and only if ai,j ∈ X. Proposition 2.6 For n ≥ 3 a subset X ⊆ An is an NBB-base for 1 in (NCn , ≤dref ) if and only if X is a maximal chain in (An , ) and in its associated graph no two edges cross. Such a subset X is an NBB-base for 1 in (PEn , ≤dref ) if and only if it contains neither a1,n−1 nor an−1,n , and X \ {a1,n } does not join to 1 in (PEn , ≤dref ). To illustrate Proposition 2.6, consider n = 4. There are exactly two maximal chains in (A¯4 , ), namely X1 = {a1,4 , a1,2 , a2,3 } and X2 = {a1,4 , a2,4 , a2,3 }. Both have noncrossing diagrams, but we can verify in Figure 1 that a1,2 ∨a2,3 = 1. There is thus one NBB-base for 1 in (PE4 , ≤dref ). We can also verify directly in Figure 1 that μ(PE4 ,≤dref ) (0, 1) = −1 in accordance with Theorem 2.5. We conclude this section with the proof of Theorem 2.2. Proof of Theorem 2.2. The key observation is that any NBB-base for 1 in (NCn , ≤dref ) corresponds to a tree on n vertices with the following two properties: (i) it contains an edge between 1 and n, and (ii) the removal of this edge yields two trees on vertex sets [k] and {k + 1, k + 2, . . . , n} for some k ∈ [n − 1]. It is then easy to verify that there are exactly Cat(n − 1) such NBB-bases. Proposition 2.6 implies that the NBB-bases for 1 in (PEn , ≤dref ) are a particular subset of the NBB-bases for 1 in (NCn , ≤dref ). It is quickly verifed that there are Cat(n − 2)-many NBB-bases for 1 in (NCn , ≤dref ) that contain an−1,n . The remaining NBB-bases for 1 in (NCn , ≤dref ) that are not NBBbases for 1 in (PEn , ≤dref ) have the property that in the associated tree the only vertex adjacent to n is 1; consequently there are also Cat(n − 2) of them. The total number
of NBB-bases for 1 in (PEn , ≤dref ) is thus Cat(n − 1) − 4 2n−5 2Cat(n − 2) = n n−4 . Since every NBB-base for 1 has n − 1 elements, we conclude from Theorem 2.5 that n−1 4 2n − 5 . μ(PEn ,≤dref ) (0, 1) = (−1) n n−4 2
890
H. Mühle / Electronic Notes in Discrete Mathematics 61 (2017) 885–891
1234
1|234
124|3
14|23
134|2
1|24|3
1|23|4
12|3|4
14|2|3
1|2|3|4 Fig. 2. The poset (PE4 , ≤pchn ).
3
A Corollary
The motivation for the work presented in this extended abstract comes from [3], where particular subposets of (NCn , ≤dref ) induced by parking functions with certain forbidden values were considered. It is well known that the maximal chains of (NCn , ≤dref ) are in bijection with parking functions of length n − 1 [10], and thus any subset of parking functions generates a subposet of (NCn , ≤dref ) simply by taking the union of these chains. It was shown in [3] that the subposet induced by all parking functions not containing the value n − 1 has ground set PEn . Let us denote this poset by (PEn , ≤pchn ). By definition there are fewer order relations in ≤pchn than in ≤dref . Figure 2 displays (PE4 , ≤pchn ), which coincides with the poset in Figure 1 except for the missing edge between 1|23|4 and 1|234. Theorem C of [3] states that the M¨obius function of (PEn , ≤pchn ) vanishes between 0 and 1, and it was conjectured that these posets have a contractible order complex. We can use Theorem 2.1 to prove this conjecture. Recall for instance from [6, Theorem 1] that supersolvable lattices of rank n admit an edge-labeling with the property that the sequence of edge-labels along any maximal chain is a permutation of [n] and there is a unique maximal chain (in each interval) whose label sequence is increasing. Posets with such a labeling have the nice property that the order complex of their proper part is a wedge of (n − 2)-dimensional spheres, and the number of spheres is given by the absolute value of their M¨obius function between least and greatest element [1, Theorem 5.9]. Theorem 2.2 thus implies
that the order complex of 4 2n−5 the proper part of (PEn , ≤dref ) is a wedge of n n−4 -many (n−2)-dimensional
H. Mühle / Electronic Notes in Discrete Mathematics 61 (2017) 885–891
891
spheres. It turns out that (PEn , ≤pchn ) inherits the labeling coming from the supersolvability of (PEn , ≤dref ) [8, Proposition 4.4], and it retains the crucial property that there is a unique maximal chain (in each interval) whose label sequence is increasing. As a consequence, the order complex of its proper part is a wedge of zero (n − 2)-dimensional spheres, and thus contractible.
References [1] Bj¨orner, A. and Wachs, M. L., Shellable and Non-Pure Complexes and Posets I, Trans. Amer. Math. Soc. 348 (1996), pp. 1299–1327. [2] Blass, A. and Sagan, B. E., M¨ obius Functions of Lattices, Adv. Math. 127 (1997), pp. 94–123. [3] Bruce, M., Dougherty, M., Hlavacek, M., Kudo, R. and Nicolas, I., A Decomposition of Parking Functions by Undesired Spaces, Electron. J. Combin. 23 (2016), # P3.32. [4] Kreweras, G., Sur les partitions non crois´ees d’un cycle, Discrete Math. 1 (1972), pp. 333–350. [5] McCammond, J., Noncrossing Partitions in Surprising Locations, Amer. Math. Monthly 113 (2006), pp. 598–610. [6] McNamara, P., EL-Labelings, Supersolvability, and 0-Hecke Algebra Actions on Posets, J. Combin. Theory (Ser. A) 101 (2003), pp. 69–89. [7] McNamara, P. and Thomas, H., Poset Edge-Labellings and Left Modularity, Europ. J. Combin. 27 (2006), pp. 101–113. [8] M¨ uhle, H., Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces, arXiv:1701.02109 (2017). [9] Simion, R., Noncrossing Partitions, Discrete Math. 217 (2000), pp. 397–409. [10] Stanley, R. P., Parking Functions and Noncrossing Partitions, Elec. J. Combin. 4 (1997), # R20.