Some properties of the parking function poset (original) (raw)

The topology of restricted partition posets

Journal of Algebraic Combinatorics, 2013

For each composition c we show that the order complex of the poset of pointed set partitions Π • c is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition c. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module S B where B is a border strip associated to the composition. We also study the filter of pointed set partitions generated by a knapsack integer partition and show the analogous results on homotopy type and action on the top homology.

Semi-pointed partition posets

Discrete Mathematics & Theoretical Computer Science, 2015

Some problems arising from partition poset homology

The Mathematical Legacy of Richard P. Stanley, 2016

We describe some open problems related to homology representations of subposets of the partition lattice, beginning with questions first raised in Stanley's work on group actions on posets.

Semi-pointed partition posets Bérénice Delcroix

2019

We present here a family of posets which generalizes both partition and pointed partition posets. After a short description of these new posets, we show that they are Cohen-Macaulay, compute their Moebius numbers and their characteristic polynomials. The characteristic polynomials are obtained using a combinatorial interpretation of the incidence Hopf algebra associated to these posets. Résumé. Nous introduisons ici une famille de posets qui généralise à la fois les poset de partitions et les posets de partitions pointées. Après une description rapide de ces nouveaux posets, nous montrons qu’ils sont Cohen-Macaulay et nous calculons leurs nombres de Moebius et leurs polynômes caractéristiques. Ces derniers sont obtenus grâce à une interprétation combinatoire de l’algèbre de Hopf d’incidence associée à ces posets.

Generalized parking functions, descent numbers, and chain polytopes of ribbon posets

Advances in Applied Mathematics, 2010

We consider the inversion enumerator In(q), which counts labeled trees or, equivalently, parking functions. This polynomial has a natural extension to generalized parking functions. Substituting q = −1 into this generalized polynomial produces the number of permutations with a certain descent set. In the classical case, this result implies the formula In(−1) = En, the number of alternating permutations. We give a combinatorial proof of these formulas based on the involution principle. We also give a geometric interpretation of these identities in terms of volumes of generalized chain polytopes of ribbon posets. The volume of such a polytope is given by a sum over generalized parking functions, which is similar to an expression for the volume of the parking function polytope of Pitman and Stanley. q b 1 +b 2 +···+bn−n = q ( n 2 ) · I n (q −1 ), where P n is the set of all parking functions of length n. Cayley's formula states that |T n | = |P n | = (n + 1) n−1 , hence I n (1) = (n + 1) n−1 .

On Combinatorics of Partition Posets Mathematics Subject Classification: 05E11P

We give basic notions concerning partially ordered sets (posets) using the notations of R.P.Stanely [7]and [10]. We discuss pointed partitions, pointed compositions and knapsack partitions of integers and sets, and we give an overview of the Möbius function of restricted partitions mainly based on the works of Richard Ehrenborg and Margaret A. Readdy. In addition, we refer to [8],[3],and [9] for some concepts to provide a clear summary of the major topics we set out in this project.

On the topology of two partition posets with forbidden block sizes

Journal of Pure and Applied Algebra, 2001

We study two subposets of the partition lattice obtained by restricting block sizes. The ÿrst consists of set partitions of {1; : : : ; n} with block size at most k, for k ≤ n − 2. We show that the order complex has the homotopy type of a wedge of spheres, in the cases 2k + 2 ≥ n and n = 3k + 2. For 2k + 2 ¿ n, the posets in fact have the same Sn−1-homotopy type as the order complex of n−1, and the Sn-homology representation is the "tree representation" of Robinson and Whitehouse. We present similar results for the subposet of n in which a unique block size k ≥ 3 is forbidden. For 2k ≥ n, the order complex has the homotopy type of a wedge of (n − 4)-spheres. The homology representation of Sn can be simply described in terms of the Whitehouse lifting of the homology representation of n−1.

Combinatorics of Topological Posets:\ Homotopy complementation formulas

arXiv (Cornell University), 1998

We show that the well known homotopy complementation formula of Björner and Walker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated on some classes of topological posets including the Grassmannian and configuration posets, G n (R) and exp n (X) which were introduced and studied by V. Vassiliev. Among other applications we present a reasonably complete description, in terms of more standard spaces, of homology types of configuration posets exp n (S m) which leads to a negative answer to a question of Vassilev raised at the workshop "Geometric Combinatorics" (MSRI, February 1997).

Semi-pointed partition posets and species

Journal of Algebraic Combinatorics, 2016

We define semi-pointed partition posets, which are a generalisation of partition posets and show that they are Cohen-Macaulay. We then use multichains to compute the dimension and the character for the action of the symmetric groups on their homology. We finally study the associated incidence Hopf algebra, which is similar to the Faà di Bruno Hopf algebra.

Some properties of the parking function poset (original) (raw)

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