Alan Stapledon
We present a new geometric proof of Stanley's monotonicity theorem for lattice polytopes, using an interpretation of $δ$-polynomials of lattice polytopes in terms of orbifold Chow rings.
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We give new proofs of three theorems of Stanley on generating functions for the integer points in rational cones. The first, Stanley's reciprocity theorem, relates the rational generating functions for the integer points in a cone K and for those in its interior. The second, Stanley's Positivity theorem asserts that the generating function of the Ehrhart quasipolynomial of a rational polytope P can be written as a rational function with nonnegative numerator. The third, Stanley's Monotonicity Theorem, asserts that if a polytope Q contains P, then each coefficient in the numerator for Q is at l…
Kazhdan-Lusztig-Stanley polynomials are a combinatorial generalization of Kazhdan-Lusztig polynomials of for Coxeter groups that include g-polynomials of polytopes and Kazhdan-Lusztig polynomials of matroids. In the cases of Weyl groups, rational polytopes, and realizable matroids, one can count points over finite fields on flag varieties, toric varieties, or reciprocal planes to obtain cohomological interpretations of these polynomials. We survey these results and unite them under a single geometric framework.
A geometrization of Stanley--Reisner theory
本文提出Stanley--Reisner对应的几何解释，将其推广至概形，并用一个元素的域进行诠释。
Stanley has studied a symmetric function generalization X_G of the chromatic polynomial of a graph G. The innocent-looking Stanley-Stembridge Poset Chain Conjecture states that the expansion of X_G in terms of elementary symmetric functions has nonnegative coefficients if G is a clawfree incomparability graph. Here we give a combinatorial interpretation of these coefficients by combining Gasharov's work on the conjecture with Egecioglu and Remmel's combinatorial interpretation of the inverse Kostka matrix. This gives a new proof of a partial nonnegativity result of Stanley. As an interesting b…
A proof of the Stanley--Stembridge conjecture
本文通过概率解释证明了单位间隔图的色准对称函数展开式系数，并由此推导出斯坦利-斯蒂姆布里奇猜想的成立。
Luck and magic for Pitman-Stanley polytopes and parking functions
本文研究Pitman-Stanley多面体的Ehrhart理论，证明其Ehrhart多项式具有强正性现象，进而推导出其h*-多项式实根性。通过解读系数于magic基中，关联到修改停车协议中的幸运汽车数。
Stanley in his paper [Stanley, Richard P.: Acyclic orientations of graphs In: Discrete Mathematics 5 (1973), Nr. 2, S. 171-178.] provided interpretations of the chromatic polynomial when it is substituted with negative integers. Greene and Zaslavsky interpreted the coefficients of the chromatic polynomial in [Greene, Curtis ; Zaslavsky, Thomas: On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs. In: Transactions of the American Mathematical Society 280 (1983), jan, Nr. 1, S. 97-97.]. We shall develop an invo…
This article is dedicated to the study of positivity phenomena for the chromatic symmetric function of a graph with respect to various bases of symmetric functions. We give a new proof of Gasharov's theorem on the Schur-positivity of the chromatic symmetric function of a $(3 + 1)$-free poset. We present a combinatorial interpretation of the Schur-coefficients in terms of planar networks. Compared to Gasharov's proof, it gives a clearer visual illustration of the cancellation procedures and is quite similar in spirit to the proof of monomial positivity of Schur functions via the Lindstrom-Gesse…
Dual Schubert polynomials via a Cauchy identity
本文通过组合证明建立了Postnikov-Stanley对偶舒伯特多项式公式与经典柯西恒等式的等价性，从而为Huang和Pylyavskyy的插入算法提供了自然解释。
Decompositions of Ehrhart $h^*$-polynomials for rational polytopes
本文为有理多胞形的 Ehrhart $h^*$-多项式提供了两种分解公式，推广了 Betke-McMullen 定理与 Stapledon 结果，并据此给出了 Stanley 单调性定理的新证明及 Stanley 和 Hibi 不等式的有理扩展。
In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the results of Stanley and Elder from a fixed integer to an array of subsequent integers, and propose an analogue of Ramanujan's congruence relations for the `number of parts' function instead of the partition function. We also deduce the generating function for the `number of parts', and relate the technical results with their graphical interpretations through a …
Combinatorial interpretation of the coefficients of the order polynomial of fence posets
研究针对围栏偏序集定义新统计量blP，为其序多项式系数提供组合解释，还借此解释舒伯特拟阵基多面体埃尔哈特多项式系数，建立序多面体埃尔哈特多项式线性系数下界，并猜想该统计量对其他偏序集的推广。
The Rayleigh monotonicity is a principle from the theory of electrical networks. Its combinatorial interpretation says for each two edges of a graph G, that the presence of one of them in a random spanning tree of G is negatively correlated with the presence of the other edge. In this paper we give a self-contained (inductive) proof of Rayleigh monotonicity for graphs.
Egecioglu and Remmel gave an interpretation for the entries of the inverse Kostka matrix K^{-1} in terms of special rim-hook tableaux. They were able to use this interpretation to give a combinatorial proof that KK^{-1}=I but were unable to do the same for the equation K^{-1}K=I. We define a sign-reversing involution on rooted special rim-hook tableaux which can be used to prove that the last column of this second product is correct. In addition, following a suggestion of Chow we combine our involution with a result of Gasharov to give a combinatorial proof of a special case of the (3+1)-free …
We introduce the notion of combinatorial positivity of translation-invariant valuations on convex polytopes that extends the nonnegativity of Ehrhart h*-vectors. We give a surprisingly simple characterization of combinatorially positive valuations that implies Stanley's nonnegativity and monotonicity of h*-vectors and generalizes work of Beck et al. (2010) from solid-angle polynomials to all translation-invariant simple valuations. For general polytopes, this yields a new characterization of the volume as the unique combinatorially positive valuation up to scaling. For lattice polytopes our re…
For a simplicial complex $Δ$, the graded Betti number $β_{i,j}(k[Δ])$ of the Stanley-Reisner ring $k[Δ]$ over a field $k$ has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if $Δ$ is the boundary complex of a $d$-dimensional stacked polytope with $n$ vertices for $d\geq3$, then $β_{k-1,k}(k[Δ])=(k-1)\binom{n-d}{k}$. We prove this combinatorially.
Lattice point enumeration of polytopes associated to integer compositions
本文证明了Chapoton关于整数组合关联多面体h向量的猜想，并给出了gamma向量非负性及h*多项式组合解释。
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