A Decomposition of Parking Functions by Undesired Spaces

Bruce, Melody; Dougherty, Michael; Hlavacek, Max; Kudo, Ryo; Nicolas, Ian

doi:10.37236/5940

Cited by 4 publications

(9 citation statements)

References 6 publications

(8 reference statements)

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“…Now we consider a subposet of (PE n , ≤ dref ) that was introduced in [4]. To that end recall that a function f : [n] → [n] is a parking function if for all k ∈ [n] the cardinality of f −1 [k] is at least k. It is a classical result that the number of parking functions of length n is (n + 1) n−1 [6, Proposition 2.6.1].…”

Section: A Subposet Of (Pe N ≤ Dref )mentioning

confidence: 99%

“…Figure 6 shows the poset (PE 4 , ≤ pchn ). This poset was extensively studied in [4]. For our purposes the next statement is the most relevant.…”

Section: A Subposet Of (Pe N ≤ Dref )mentioning

confidence: 99%

“…The maximal chains of (NC n , ≤ dref ) are naturally in bijection with parking functions of length n − 1, see [18]. This connection was used in [4] to define a subposet of (NC n , ≤ dref ) as follows. Fix some k ≤ n and take the set of all parking functions which do not have k in their image, but every value larger than k. Let us consider the poset (PE n,k , ≤ pchn ), which is the subposet of (NC n , ≤ dref ) determined by the maximal chains corresponding to these parking functions.…”

Section: Introductionmentioning

confidence: 99%

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Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces

Mühle

2018

rev.colomb.mat

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Consider the noncrossing set partitions of an n-element set which either 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. In this article we study the subposet of the noncrossing partition lattice induced by these elements, and show that it is a supersolvable lattice, and therefore lexicographically shellable. We give a combinatorial model for the NBB bases of this lattice and derive an explicit formula for the value of its Möbius function between least and greatest element. This work is motivated by a recent article by M. Bruce, M. Dougherty, M. Hlavacek, R. Kudo, and I. Nicolas, in which they introduce a subposet of the noncrossing partition lattice that is determined by parking functions with certain forbidden entries. In particular, they conjecture that the resulting poset always has a contractible order complex. We prove this conjecture by embedding their poset into ours, and showing that it inherits the lexicographic shellability.2010 Mathematics Subject Classification. 05A18, 06A07.

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Section: A Subposet Of (Pe N ≤ Dref )mentioning

confidence: 99%

“…Figure 6 shows the poset (PE 4 , ≤ pchn ). This poset was extensively studied in [4]. For our purposes the next statement is the most relevant.…”

Section: A Subposet Of (Pe N ≤ Dref )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces

Mühle

2018

rev.colomb.mat

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“…For example, in [13] Edelman introduced the k-divisible noncrossing partition lattice, the subposet of N C n where each partition has all of its block sizes divisible by k. Later, Armstrong [2] introduced and studied the k-divisible noncrossing partition lattice for each finite Coxeter group. More recently, motivated by the connection between the noncrossing partition lattice and parking functions, Bruce, Dougherty, Hlavacek, Kudo, and Nicolas [12] introduced a subposet of the noncrossing partition lattice obtained by removing chains that corresponded to parking functions with certain restrictions. To solve a conjecture put forward in that article, Mühle [23] defined two new subposets of the noncrossing partition lattice obtained by removing partitions which do not contain certain blocks.…”

Section: Introductionmentioning

confidence: 99%

The Noncrossing Bond Poset of a Graph

Farmer

Hallam

Smyth

2020

Electron. J. Combin.

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The partition lattice and noncrossing partition lattice are well studied objects in combinatorics. Given a graph $G$ on vertex set $\{1,2,\dots, n\}$, its bond lattice, $L_G$, is the subposet of the partition lattice formed by restricting to the partitions whose blocks induce connected subgraphs of $G$. In this article, we introduce a natural noncrossing analogue of the bond lattice, the noncrossing bond poset, $NC_G$, obtained by restricting to the noncrossing partitions of $L_G$. Both the noncrossing partition lattice and the bond lattice have many nice combinatorial properties. We show that, for several families of graphs, the noncrossing bond poset also exhibits these properties. We present simple necessary and sufficient conditions on the graph to ensure the noncrossing bond poset is a lattice. Additionally, for several families of graphs, we give combinatorial descriptions of the Möbius function and characteristic polynomial of the noncrossing bond poset. These descriptions are in terms of a noncrossing analogue of non-broken circuit (NBC) sets of the graphs and can be thought of as a noncrossing version of Whitney's NBC theorem for the chromatic polynomial. We also consider the shellability and supersolvability of the noncrossing bond poset, providing sufficient conditions for both. We end with some open problems.

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“…For example, in [13] Edelman introduced the k-divisible noncrossing partition lattice, the subposet of NC n where each partition has all of its block sizes divisible by k. Later, Armstrong [2] introduced and studied the k-divisible noncrossing partition lattice for each finite Coxeter group. More recently, motivated by the connection between the noncrossing partition lattice and parking functions, Bruce, Dougherty, Hlavacek, Kudo, and Nicolas [12] introduced a subposet of the noncrossing partition lattice obtained by removing chains that corresponded to parking functions with certain restrictions. To solve a conjecture put forward in that article, Mühle [23] defined two new subposets of the noncrossing partition lattice obtained by removing partitions which do not contain certain blocks.…”

Section: Introductionmentioning

confidence: 99%

The Noncrossing Bond Poset of a Graph

Farmer,

Hallam,

Smyth

2020

Preprint

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The partition lattice and noncrossing partition lattice are well studied objects in combinatorics. Given a graph G on vertex set {1, 2, . . . , n}, its bond lattice, L G , is the subposet of the partition lattice formed by restricting to the partitions whose blocks induce connected subgraphs of G. In this article, we introduce a natural noncrossing analogue of the bond lattice, the noncrossing bond poset, N C G , obtained by restricting to the noncrossing partitions of L G .Both the noncrossing partition lattice and the bond lattice have many nice combinatorial properties. We show that, for several families of graphs, the noncrossing bond poset also exhibits these properties. We present simple necessary and sufficient conditions on the graph to ensure the noncrossing bond poset is a lattice. Additionally, for several families of graphs, we give combinatorial descriptions of the Möbius function and characteristic polynomial of the noncrossing bond poset. These descriptions are in terms of a noncrossing analogue of non-broken circuit (NBC) sets of the graphs and can be thought of as a noncrossing version of Whitney's NBC theorem for the chromatic polynomial. We also consider the shellability and supersolvability of the noncrossing bond poset, providing sufficient conditions for both. We end with some open problems.

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A Decomposition of Parking Functions by Undesired Spaces

Cited by 4 publications

References 6 publications

Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces

Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces

The Noncrossing Bond Poset of a Graph

The Noncrossing Bond Poset of a Graph

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