We study exact and asymptotic enumerative aspects of the Hilbert series of the cohomology ring of the moduli space of stable pointed curves of genus zero. This manifold is related to the WDVV ...(Witten–Dijkgraaf–Verlinde–Verlinde) equations of string theory.
The notion of the negative $q$-binomial was recently introduced by Fu, Reiner, Stanton and Thiem. Mirroring the negative $q$-binomial, we show the classical $q$ -Stirling numbers of the second kind ...can be expressed as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in $q$ and $(1+q)$. We extend this enumerative result via a decomposition of the Stirling poset, as well as a homological version of Stembridge’s $q=-1$ phenomenon. A parallel enumerative, poset theoretic and homological study for the $q$-Stirling numbers of the first kind is done beginning with de Médicis and Leroux’s rook placement formulation. Letting $t=1+q$ we give a bijective combinatorial argument à la Viennot showing the $(q; t)$-Stirling numbers of the first and second kind are orthogonal.
La notion de la $q$-binomial négative était introduite par Fu, Reiner, Stanton et Thiem. Réfléchissant la $q$-binomial négative, nous démontrons que les classiques $q$-nombres de Stirling de deuxième espèce peuvent être exprimés comme une paire de statistiques sur un sous-ensemble des mots de croissance restreinte. Les expressions résultantes sont les polynômes en $q$ et $1+q$. Nous étendons ce résultat énumératif via une décomposition du poset de Stirling, ainsi que d’une version homologique du $q=-1$ phénomène de Stembridge. Un parallèle énumératif, poset théorique et étude homologique des $q$-nombres de Stirling de première espèce se fait en commençant par la formulation du placement des tours par suite des auteurs de Médicis et Leroux. On laisse $t=1+q$ et on donne les arguments combinatoires et bijectifs à la Viennot qui démontrent que les $(q;t)$-nombres de Stirling de première et deuxième espèces sont orthogonaux.
Pizza and 2-Structures Ehrenborg, Richard; Morel, Sophie; Readdy, Margaret
Discrete & computational geometry,
12/2023, Volume:
70, Issue:
4
Journal Article
Peer reviewed
Open access
Let
H
be a Coxeter hyperplane arrangement in
n
-dimensional Euclidean space. Assume that the negative of the identity map belongs to the associated Coxeter group
W
. Furthermore assume that the ...arrangement is not of type
A
1
n
. Let
K
be a measurable subset of the Euclidean space with finite volume which is stable by the Coxeter group
W
and let
a
be a point such that
K
contains the convex hull of the orbit of the point
a
under the group
W
. In a previous article the authors proved the generalized pizza theorem: that the alternating sum over the chambers
T
of
H
of the volumes of the intersections
T
∩
(
K
+
a
)
is zero. In this paper we give a dissection proof of this result. In fact, we lift the identity to an abstract dissection group to obtain a similar identity that replaces the volume by any valuation that is invariant under affine isometries. This includes the cases of all intrinsic volumes. Apart from basic geometry, the main ingredient is a theorem of the authors where we relate the alternating sum of the values of certain valuations over the chambers of a Coxeter arrangement to similar alternating sums for simpler subarrangements called 2-structures introduced by Herb to study discrete series characters of real reduced groups.
q-Stirling numbers: A new view Cai, Yue; Readdy, Margaret A.
Advances in applied mathematics,
20/May , Volume:
86
Journal Article
Peer reviewed
Open access
We show the classical q-Stirling numbers of the second kind can be expressed compactly as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in q ...and 1+q. We extend this enumerative result via a decomposition of a new poset Π(n,k) which we call the Stirling poset of the second kind. Its rank generating function is the q-Stirling number Sqn,k. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done. Letting t=1+q we give a bijective argument showing the (q,t)-Stirling numbers of the first and second kinds are orthogonal.
Balanced and Bruhat Graphs Ehrenborg, Richard; Readdy, Margaret
Annals of combinatorics,
09/2020, Volume:
24, Issue:
3
Journal Article
Peer reviewed
We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets ...having an
R
-labeling, imply the existence of the (non-homogeneous)
c
d
-index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian posets, we show an analogue of Alexander duality for bounded balanced digraphs. For Bruhat graphs of Coxeter groups, an important family of balanced graphs, our theory gives elementary proofs of the existence of the complete
c
d
-index and its properties. We also introduce the rising and falling quasisymmetric functions of a labeled acyclic digraph and show they are Hopf algebra homomorphisms mapping balanced digraphs to the Stembridge peak algebra. We conjecture non-negativity of the
c
d
-index for acyclic digraphs having a balanced linear edge labeling.
The purpose of this paper is to compute the Möbius function of filters in the partition lattice formed by restricting to partitions by type. The Möbius function is determined in terms of the descent ...set statistics on permutations and the Möbius function of filters in the lattice of integer compositions. When the underlying integer partition is a knapsack partition, the Möbius function on integer compositions is determined by a topological argument. In this proof the permutahedron makes a cameo appearance.
Euler flag enumeration of Whitney stratified spaces Ehrenborg, Richard; Goresky, Mark; Readdy, Margaret
Discrete mathematics and theoretical computer science,
01/2013, Volume:
DMTCS Proceedings vol. AS,..., Issue:
Proceedings
Journal Article
Peer reviewed
Open access
We show the $\mathrm{cd}$-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi-graded poset. This is a poset endowed with an order-preserving ...rank function and a weighted zeta function. This allows us to generalize the classical notion of Eulerianness, and obtain a $\mathrm{cd}$-index in the quasi-graded poset arena. We also extend the semi-suspension operation to that of embedding a complex in the boundary of a higher dimensional ball and study the shelling components of the simplex.
Nous montrons le $\mathrm{cd}$-index existe pour les manifolds de Whitney stratifiées en élargissant la notion d’un poset gradué à celle qu'un poset quasi-gradué. C’est un poset doté d'une fonction de rang que préservant l’ordre du poset et une fonction de zêta pondérée. Ceci nous permet de généraliser la notion classique de “Eulerianness”, et obtenir un $\mathrm{cd}$-index dans l’arène des posets quasi-gradués. Nous tenons également à l’opération de semi-suspension pour que d’intégrer une complexe dans la frontière d’une balle de dimension supérieur et étudions les composants shelling d’un simplex.
This article is concerned with the constants that appear in Harish-Chandra's character formula for stable discrete series of real reductive groups, although it does not require any knowledge about ...real reductive groups or discrete series. In Harish-Chandra's work the only information we have about these constants is that they are uniquely determined by an inductive property. Later Goresky--Kottwitz--MacPherson and Herb gave different formulas for these constants. In this article we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longer make sense), and give a direct proof that the two formulas agree. We actually prove a slightly more general identity that also implies the combinatorial identity underlying the discrete series character identities of Morel. We deduce this identity from a general abstract theorem giving a way to calculate the alternating sum of the values of a valuation on the chambers of a Coxeter arrangement. We also introduce a ring structure on the set of valuations on polyhedral cones in Euclidean space with values in a fixed ring. This gives a theoretical framework for the valuation appearing in Appendix A of the Goresky--Kottwitz--MacPherson paper. In Appendix B we extend the notion of 2-structures (due to Herb) to pseudo-root systems.
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