Related papers: Poset homology of Rees products, and $q$-Eulerian …
The Rees product of partially ordered sets was introduced by Bj\"orner and Welker. Using the theory of lexicographic shellability, Linusson, Shareshian and Wachs proved formulas, of significance in the theory of gamma-positivity, for the…
Segre products of posets were defined by Bj\"orner and Welker (2005). We investigate the homology representations of the $t$-fold Segre power $B_n^{(t)}$ of the Boolean lattice $B_n$. The direct product $\sym_n^{\times t}$ of the symmetric…
We determine how the flag f-vector of any graded poset changes under the Rees product with the chain, and more generally, any t-ary tree. As a corollary, the M\"obius function of the Rees product of any graded poset with the chain, and more…
We use the theory of lexicographic shellability to provide various examples in which the rank of the homology of a Rees product of two partially ordered sets enumerates some set of combinatorial objects, perhaps according to some natural…
Polyhedral products were defined by Bahri, Bendersky, Cohen and Gitler, to be spaces obtained as unions of certain product spaces indexed by the simplices of an abstract simplicial complex. In this paper we give a very general homotopy…
We define a homology theory for a certain class of posets equipped with a representation. We show that when restricted to Boolean lattices this homology is isomorphic to the homology of the "cube" complex defined by Khovanov.
Given a symmetric monoidal category $C$ with product $\sqcup$, where the neutral element for the product is an initial object, we consider the poset of $\sqcup$-complemented subobjects of a given object $X$. When this poset has finite…
In this note we introduce a new class of refined Eulerian polynomials defined by $$A_n(p,q)=\sum_{\pi\in\mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)},$$ where ${\rm odes}(\pi)$ and ${\rm edes}(\pi)$ enumerate the number of descents…
We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric functions}, which have the property of specializing to enumerators for the joint distribution of the permutation statistics, major index and excedance…
This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and…
We introduce the short toric polynomial associated to a graded Eulerian poset. This polynomial contains the same information as the two toric polynomials introduced by Stanley, but allows different algebraic manipulations. The intertwined…
This paper studies the poset of eigenspaces of elements of an imprimitive unitary reflection group, for a fixed eigenvalue, ordered by the reverse of inclusion. The study of this poset is suggested by the eigenspace theory of Springer and…
Via duality of Hopf algebras, there is a direct association between peak quasisymmetric functions and enumeration of chains in Eulerian posets. We study this association explicitly, showing that the notion of $\cd$-index, long studied in…
Given a finite poset, Greene introduced a rational function obtained by summing certain rational functions over the linear extensions of the poset. This function has interesting interpretations, and for certain families of posets, it…
Anders Bjorner characterized which finite graded partially ordered sets arise as the posets of closure relations on cells of a finite, regular CW complex. His characterization of these "CW posets" required each open interval $(\hat{0},u)$…
In this paper, we first give formulas for the order polynomial $\Omega (\Pw; t)$ and the Eulerian polynomial $e(\Pw; \lambda)$ of a finite labeled poset $(P, \omega)$ using the adjacency matrix of what we call the $\omega$-graph of $(P,…
This paper introduces and develops M\"obius homology, a homology theory for representations of finite posets into abelian categories. Although the connection between poset topology and M\"obius functions is classical, we go further by…
Let $P$ be a finite partially ordered set. In a recent series of works, Proudfoot introduced the notion of $Z$-polynomials associated with $P$-kernels, providing a unified framework for various intersection cohomology Poincar\'e polynomials…
We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine $h$-vector of balanced semi-Eulerian complexes…
The eigenspace theory of unitary reflection groups, initiated by Springer and Lehrer, suggests that the following object is worthy of study: the poset of eigenspaces of elements of a unitary reflection group, for a fixed eigenvalue, ordered…