Margaret M. Bayer, Gabor Hetyei
The closed cone of flag vectors of Eulerian partially ordered sets is studied. It is completely determined up through rank seven. Half-Eulerian posets are defined. Certain limit posets of Billera and Hetyei are half-Eulerian; they give rise to extreme rays of the cone for Eulerian posets. A new family of linear inequalities valid for flag vectors of Eulerian posets is given.
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A partially ordered set is r-thick if every nonempty open interval contains at least r elements. This paper studies the flag vectors of graded, r-thick posets and shows the smallest convex cone containing them is isomorphic to the cone of flag vectors of all graded posets. It also defines a k-analogue of the Mobius function and k-Eulerian posets, which are 2k-thick. Several characterizations of k-Eulerian posets are given. The generalized Dehn-Sommerville equations are proved for flag vectors of k-Eulerian posets. A new inequality is proved to be valid and sharp for rank 8 Eulerian posets.
We prove that the group algebra of the hyperoctahedral group contains a subalgebra corresponding to the flag descent number of Adin, Brenti, and Roichman. This algebra is in fact the span of the basis elements of the type A and type B Eulerian descent algebras. We describe a set of orthogonal idempotents which spans the flag descent algebra and prove that it contains the type A Eulerian descent algebra as a two-sided ideal. Using a new colored analogue of Stanley's $P$-partitions, we prove the existence of a colored Eulerian descent algebra which is a subalgebra of the Mantaci-Reutenauer algeb…
We investigate the $α$-colored Eulerian polynomials and a notion of descents introduced in a recent paper of Hedmark and show that such polynomials can be computed as a polynomial encoding descents computed over a quotient of the wreath product $\mathbb{Z}_α\wr\mathfrak{S}_n$. Moreover, we consider the flag descent statistic computed over this same quotient and find that the flag Eulerian polynomial remains palindromic. We prove that the flag descent polynomial is palindromic over this same quotient by giving a combinatorial proof that the flag descent statistic is symmetrically distributed ov…
In this paper, we characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems, which generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including flag descent polynomials for hyperoctahedral group, flag ascent-plateau polynomials for Stirling permutations, up-down run polynomials for symmetric group and alternating run polynomials for hyperoctahedral group. As applications, we derive some properties of associated enumerative polynomials. In particular, we find that both the ascent-plateau polynomials and left ascent-plate…
Recently, Hyatt introduced some colored Eulerian quasisymmetric function to study the joint distribution of excedance number and major index on colored permutation groups. We show how Hyatt's generating function formula for the fixed point colored Eulerian quasisymmetric functions can be deduced from the Decrease value theorem of Foata and Han. Using this generating function formula, we prove two symmetric function generalizations of the Chung-Graham-Knuth symmetrical Eulerian identity for some flag Eulerian quasisymmetric functions, which are specialized to the flag excedance numbers. Combina…
A graded partially ordered set is Eulerian if every interval has the same number of elements of even rank and of odd rank. Face lattices of convex polytopes are Eulerian. For Eulerian partially ordered sets, the flag vector can be encoded efficiently in the cd-index. The cd-index of a polytope has all positive entries. An important open problem is to give the broadest natural class of Eulerian posets having nonnegative cd-index. This paper completely determines which entries of the cd-index are nonnegative for all Eulerian posets. It also shows that there are no other lower or upper bounds on …
By considering Eulerian numbers and ordered Stirling numbers of the second and third kinds over a multiset, we generalize identities of Eulerian numbers and Stirling numbers of the second and third kinds and provide $q$-analogs of these generalizations. Using these generalizations, we also compute Eulerian numbers and ordered Stirling numbers of the second kind over a multiset.
It is known that the normalized volume of standard hypersimplices (defined as some slices of the unit hypercube) are the Eulerian numbers. More generally, a recent conjecture of Stanley relates the Ehrhart series of hypersimplices with descents and excedences in permutations. This conjecture was proved by Nan Li, who also gave a generalization to colored permutations. In this article, we give another generalization to colored permutations, using the flag statistics introduced by Foata and Han. We obtain in particular a new proof of Stanley's conjecture, and some combinatorial identities relati…
Normal ordered grammars
引入正规序文法理论以推广正规序问题，为 Eulerian 多项式及二阶 Eulerian 多项式等提供文法解释，并揭示其生成函数与 Catalan 数的联系。
The classical Dehn--Sommerville relations assert that the $h$-vector of an Eulerian simplicial complex is symmetric. We establish three generalizations of the Dehn--Sommerville relations: one for the $h$-vectors of pure simplicial complexes, another one for the flag $h$-vectors of balanced simplicial complexes and graded posets, and yet another one for the toric $h$-vectors of graded posets with restricted singularities. In all of these cases, we express any failure of symmetry in terms of "errors coming from the links." For simplicial complexes, this further extends Klee's semi-Eulerian relat…
Well-Quasi-Ordering Eulerian Digraphs: Bounded Carving Width
本文证明了具有受限刻痕宽度的欧拉图类在强浸入下是良quis序的，并提出了一种新的强浸入概念。同时指出无界度数的欧拉图类在强浸入下不满足良quis序，但弱浸入下可能满足。
Stellahedral geometry of partially ordered sets
本文提出偏序集的星状变换，保持欧拉性等性质，并证明其与环面h-多项式联系，从而得出增广Chow多项式的正性结果，并否定两个开放问题。
The flag vector contains all the face incidence data of a polytope, and in the poset setting, the chain enumerative data. It is a classical result due to Bayer and Klapper that for face lattices of polytopes, and more generally, Eulerian graded posets, the flag vector can be written as a cd-index, a non-commutative polynomial which removes all the linear redundancies among the flag vector entries. This result holds for regular CW complexes. We relax the regularity condition to show the cd-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi…
Well-Quasi-Ordering Eulerian Digraphs Embeddable in Surfaces by Strong Immersion
本文证明可嵌入任意曲面的最大度至多 4 的欧拉有向图在强浸入下良拟序，并给出圆图、算法与 Erdős-Pósa 性质等应用。
The P-Eulerian polynomial counts the linear extensions of a labeled partially ordered set, P, by their number of descents. It is known that the P-Eulerian polynomials are real-rooted for various classes of posets P. The purpose of this paper is to extend these results to polynomials in several variables. To this end we study multivariate extensions of P-Eulerian polynomials and prove that for certain posets these polynomials are stable, i.e., non-vanishing whenever all variables are in the upper half-plane of the complex plane. A natural setting for our proofs is the Malvenuto-Reutenauer algeb…
This is a survey of the $cd$-index of Eulerian partially ordered sets. The $cd$-index is an encoding of the numbers of chains, specified by ranks, in the poset. It is the most efficient such encoding, incorporating all the affine relations on the flag numbers of Eulerian posets. Eulerian posets include the face posets of regular CW spheres (in particular, of convex polytopes), intervals in the Bruhat order on Coxeter groups, and the lattices of regions of oriented matroids. The paper discusses inequalities on the $cd$-index, connections with other combinatorial parameters, computation, and alg…
Eulerian posets and $Z$-polynomials
该研究解决了Proudfoot提出的问题，证明欧拉偏序集的$Z$-多项式与对应区间偏序集的环面$h$-多项式一致，还得到周多项式与维罗纳变换的关联结果。
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) ${\bf cd}$-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 balanced digraphs. For Bruhat graphs of Coxeter groups, an important family of balanced graphs, our theory gives elementary proofs of the existe…
Using Grothendieck's "functor of points" approach to algebraic geometry, we define a new infinite-dimensional algebro-geometric flag space as a $k$-functor (for $k$ a ring) which maps a $k$-algebra $R$ to the set of certain well-ordered chains of submodules of an infinite rank free $R$-module. This generalizes the well known construction of a $k$-functor that is represented by the classical (i.e. finite-dimensional) full flag scheme. We prove that as in the finite-dimensional case, there is an action of a general linear group on our flag space, that the stabilizer of the standard flag is the s…
The number of flags in a complete fan, or more generally in an Eulerian poset, is encoded in the cd-index. We prove non-negativity of the cd-index for complete fans, regular CW-spheres and Gorenstein* posets.
A chain poset, by definition, consists of chains of ordered elements in a poset. We study the chain posets associated to two posets: the Boolean algebra and the poset of isotropic flags. We prove that, in both cases, the chain posets satisfy the strong Sperner property and are rank-log concave.
A colored complex of type $\mathbf{a} = (a_1, \dots, a_n)$ is a simplicial complex $Δ$ on a vertex set $V$, together with an ordered partition $(V_1, \dots, V_n)$ of $V$, such that every face $F$ of $Δ$ satisfies $|F \cap V_i| \leq a_i$. For each $\mathbf{b} = (b_1, \dots, b_n) \leq \mathbf{a}$, let $f_{\mathbf{b}}$ be the number of faces $F$ of $Δ$ such that $|F \cap V_i| = b_i$. The array of integers $\{f_{\mathbf{b}}\}_{\mathbf{b} \leq \mathbf{a}}$ is called the fine $f$-vector of $Δ$, and it is a refinement of the $f$-vector of $Δ$. In this paper, we generalize the notion of Macaulay repre…
For any $n > 0$ and $0 \leq m < n$, let $P_{n,m}$ be the poset of projective equivalence classes of $\{-,0,+\}$-vectors of length $n$ with sign variation bounded by $m$, ordered by reverse inclusion of the positions of zeros. Let $Δ_{n,m}$ be the order complex of $P_{n,m}$. A previous result from the third author shows that $Δ_{n,m}$ is Cohen-Macaulay over $\mathbb{Q}$ whenever $m$ is even or $m = n-1$. Hence, it follows that the $h$-vector of $Δ_{n,m}$ consists of nonnegative entries. Our main result states that $Δ_{n,m}$ is partitionable and we give an interpretation of the $h$-vector when $…
Chow functions for partially ordered sets
本文提出偏序集上的Chow函数理论，作为KLS理论的平行框架，在不依赖Hard Lefschetz定理的情况下证明单峰性与正性结果，并揭示多面体面格、拟阵Chow环及Bruhat区间中多个猜想之间的统一联系。
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