An Extension of Stanley's Symmetric Acyclicity Theorem to Signed Graphs

A bivariate chromatic polynomial for signed graphs

We study Dohmen--Pönitz--Tittmann's bivariate chromatic polynomial $c_Γ(k,l)$ which counts all $(k+l)$-colorings of a graph $Γ$ such that adjacent vertices get different colors if they are $\le k$. Our first contribution is an extension of $c_Γ(k,l)$ to signed graphs, for which we obtain an inclusion--exclusion formula and several special evaluations giving rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is to derive combinatorial reciprocity theorems for $c_Γ(k,l)$ and its signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking chromatic polynom…

带符号图的色拟对称函数

Chromatic quasisymmetric functions for signed graphs

本文将Stanley的色对称函数及Shareshian-Wachs-Ellzey的拟对称细化推广到带符号图，引入带符号色拟对称不变量并研究其结构性质，进而定义和研究了带符号拟对称函数代数SQSym。

Barycentric subdivisions and derangement polynomials for the even-signed permutation groups

The derangement polynomial for the symmetric group enumerates derangements by the number of excedances. It can be interpreted as the local $h$-polynomial, in the sense of Stanley, of the barycentric subdivision of the simplex. Motivated by this interpretation, we define a derangement polynomial for the even-signed permutation group. The coefficients of this polynomial are nonnegative, symmetric and unimodal. We show that they enumerate derangements in the even-signed permutation group according to a notion of excedance, which is analogous to the one introduced by Brenti for signed permutations…

带符号图的组合Hopf代数

The combinatorial Hopf algebra of signed graphs

本研究定义了带符号图的组合Hopf代数，利用其对极给出相关不变量组合互易结果的新证明，还研究了两类图的双变量多项式不变量，拓展了组合Hopf代数在带符号图领域的应用。

Signed graphs and the freeness of the Weyl subarrangements of type $B_{\ell}$

A Weyl arrangement is the hyperplane arrangement defined by a root system. Arnold and Saito proved that every Weyl arrangement is free. The Weyl subarrangements of type $A_{\ell}$ are represented by simple graphs. Stanley gave a characterization of freeness for this type of arrangements in terms of thier graph. In addition, The Weyl subarrangements of type $B_{\ell}$ can be represented by signed graphs. A characterization of freeness for them is not known. However, characterizations of freeness for a few restricted classes are known. For instance, Edelman and Reiner characterized the freeness …

Flow polytopes of signed graphs and the Kostant partition function

We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide entirely combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As a fascinating special family of flow polytopes, we study the Chan-Robbins-Yuen polytopes. Motivated by the beautiful volume formula $\prod_{k=1}^{n-2} Cat(k)$ for the type $A_n…

Bialgebras for Stanley symmetric functions

We construct a non-commutative, non-cocommutative, graded bialgebra $\mathbfΠ$ with a basis indexed by the permutations in all finite symmetric groups. Unlike the formally similar Malvenuto-Poirier-Reutenauer Hopf algebra, this bialgebra does not have finite graded dimension. After giving formulas for the product and coproduct, we show that there is a natural morphism from $\mathbfΠ$ to the algebra of quasi-symmetric functions, under which the image of a permutation is its associated Stanley symmetric function. As an application, we use this morphism to derive some new enumerative identities. …

Signed-eliminable graphs and free multiplicities on the braid arrangement

We define specific multiplicities on the braid arrangement by using edge-bicolored graphs. To consider their freeness, we introduce the notion of bicolor-eliminable graphs as a generalization of Stanley's classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of the deformation of the braid arrangement in terms of directed graphs.

Triangular Ladders $P_{d,2}$ are $e$-positive

In 1995 Stanley conjectured that the chromatic symmetric functions of the graphs $P_{d,2}$, which we call triangular ladders, were $e$-positive. In this paper we confirm this conjecture, which is also an unsolved case of the celebrated $(3+1)$-free conjecture. Our method is to follow the generalization of the chromatic symmetric functions by Gebhard and Sagan to symmetric functions in non-commuting variables. These functions satisfy a deletion-contraction property unlike the chromatic symmetric function in commuting variables. We do this by proving a new signed combinatorial formula for \emph{…

Tutte Polynomials for Directed Graphs

The Tutte polynomial is a fundamental invariant of graphs. In this article, we define and study a generalization of the Tutte polynomial for directed graphs, that we name B-polynomial. The B-polynomial has three variables, but when specialized to the case of graphs (that is, digraphs where arcs come in pairs with opposite directions), one of the variables becomes redundant and the B-polynomial is equivalent to the Tutte polynomial. We explore various properties, expansions, specializations, and generalizations of the B-polynomial, and try to answer the following questions: 1. what properties o…

无环多胞形与环多胞形

Acyclotopes and Tocyclotopes

研究将 zonotopes 相关结论推广到其他根系，计算无环多胞形的 Ehrhart 多项式；还将拟阵对偶概念拓展到 zonotopes，构造环多胞形并计算其 Ehrhart 多项式，对一般整数矩阵应用对偶构造得到格 Gale zonotope 并描述其 Ehrhart 多项式。