Gamma-positivity for a Refinement of Median Genocchi Numbers

简并旗型品种和Genocchi数中位数

Degenerate flag varieties and the median Genocchi numbers

研究了$A$型旗叶型水稻品种$\Fl_\la$的$\bG_a^M$退化$\Fl^a_\la$。我们明确地描述这些退化作为Grassmanians的产品中的变种。我们构建胞腔剖分的$\Fl^a_\la$，并表明，对于完整的标志，细胞的数量等于归一化的中位数Genocchi数$h_n$。这导致了数字$h_n$的一个新的组合定义。我们还计算了庞加莱多项式的完全退化的标志品种通过一个自然统计的集合上的Dellac的配置，类似的长度统计的集合上的排列。因此，我们得到一个自然的$q$版本的归一化中位数Genocchi数。

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…

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 …

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…

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。

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。

Parity considerations for drops in cycles on $\{1,2,\ldots,n\}$

In 2019, A. Lazar and M. L. Wachs conjectured that the number of cycles on $[2n]$ with only even-odd drops equals the $n$-th Genocchi number. In this paper, we restrict our attention to a subset of cycles on $[n]$ that in all drops in the cycle, the latter entry is odd. We deduce two bivariate generating functions for such a subset of cycles with an extra variable introduced to count the number of odd-odd and even-odd drops, respectively. One of the generating function identities confirms Lazar and Wachs' conjecture, while the other identity implies that the number of cycles on $[2n-1]$ with o…

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.