The median Genocchi numbers, Q-analogues and continued fractions

Cycles of even-odd drop permutations and continued fractions of Genocchi numbers

Recently, Lazar and Wachs (arXiv:1910.07651) showed that the (median) Genocchi numbers play a fundamental role in the study of the homogenized Linial arrangement and obtained two new permutation models (called D-permutations and E-permutations) for (median) Genocchi numbers. They further conjecture that the distributions of cycle numbers over the two models are equal. In a follow-up, Eu et al. (arXiv:2103.09130) further proved the gamma-positivity of the descent polynomials of even-odd descent permutations, which are in bijection with E-permutations by Foata's fundamental transformation. This …

Torus fixed points in Schubert varieties and normalized median Genocchi numbers

We give a new proof for the fact that the number of torus fixed points for the degenerated flag variety is equal to the normalized median Genocchi number, using the identification with a certain Schubert variety. We further study the torus fixed points for the symplectic degenerated flag variety and develop a combinatorial model, symplectic Dellac configurations, so parametrize them. The number of these symplectic fixed points is conjectured to be the median Euler number.

Alternation acyclic tournaments

We define a tournament to be alternation acyclic if it does not contain a cycle in which descents and ascents alternate. Using a result by Athanasiadis on hyperplane arrangements, we show that these tournaments are counted by the median Genocchi numbers. By establishing a bijection with objects defined by Dumont, we show that alternation acyclic tournaments in which at least one ascent begins at each vertex, except for the largest one, are counted by the Genocchi numbers of the first kind. Unexpected consequences of our results include a pair of ordinary generating function formulas for the Ge…

彩色交错三角形与Genocchi中位数

Colored Interlacing Triangles and Genocchi Medians

本文研究了固定深度N=2和任意颜色数n的彩色交错三角形，证明其与Dumont错位的双射关系，并引入q-变形，揭示了Genocchi中位数的q-类比。

On the Homogenized Linial Arrangement: Intersection Lattice and Genocchi Numbers

Hetyei recently introduced a hyperplane arrangement (called the homogenized Linial arrangement) and used the finite field method of Athanasiadis to show that its number of regions is a median Genocchi number. These numbers count a class of permutations known as Dumont derangements. Here, we take a different approach, which makes direct use of Zaslavsky's formula relating the intersection lattice of this arrangement to the number of regions. We refine Hetyei's result by obtaining a combinatorial interpretation of the Möbius function of this lattice in terms of variants of the Dumont permutation…

奇偶模式遇上Genocchi数，I：四种标号与三个双射

Parity patterns meet Genocchi numbers, I: four labelings and three bijections

本文通过四种奇偶限制排列标号超平面排列区域，建立双射并推导交织两类Genocchi数的类Seidel三角形，统一了Hetyei与Lazar-Wachs的计数结果。

使用 Foata--Zeilberger 双射及其变体的 Laguerre 有向图解释的连分数

Continued fractions using a Laguerre digraph interpretation of the Foata--Zeilberger bijection and its variants

本文用 Laguerre 有向图解释 Foata--Zeilberger 双射以计算排列中的环，从而得到了枚举排列和 D-排列的多变量多项式的 Jacobi 型、Thron 型和 Stieltjes 型连分数，并证明了 Sokal、Zeng 等人提出的连分数猜想。

Combinatorial interpretations of the Kreweras triangle in terms of subset tuples

We show how the combinatorial interpretation of the normalized median Genocchi numbers in terms of multiset tuples, defined by Hetyei in his study of the alternation acyclic tournaments, is bijectively equivalent to previous models like the normalized Dumont permutations or the Dellac configurations, and we extend the interpretation to the Kreweras triangle.

Generalized Dumont-Foata polynomials and alternative tableaux

Dumont and Foata introduced in 1976 a three-variable symmetric refinement of Genocchi numbers, which satisfies a simple recurrence relation. A six-variable generalization with many similar properties was later considered by Dumont. They generalize a lot of known integer sequences, and their ordinary generating function can be expanded as a Jacobi continued fraction. We give here a new combinatorial interpretation of the six-variable polynomials in terms of the alternative tableaux introduced by Viennot. A powerful tool to enumerate alternative tableaux is the so-called "matrix Ansatz", and usi…

基于连分数组合学的循环筛分现象

Cyclic sieving phenomena via combinatorics of continued fractions

本文利用连分数恒等式研究排列、集合划分、完美匹配和D-排列上的循环筛分现象，重新证明并推广Corteel对合相关结果，证明两个猜想，并构造新的Genocchi-Corteel对合。

Quiver Grassmannians and degenerate flag varieties

Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by the second named author. This leads to the consideration of a class of Grassmannians of subrepresentations of the direct sum of a projective and an injective representation of a Dynkin quiver. It is proven that these are (typically singular) irreducible normal local complete intersection varieties, which admit a group action with finitely many orbits, and a cell…