The median Genocchi numbers, Q-analogues and continued fractions

TL;DR

This paper explores the combinatorial and geometric properties of median Genocchi numbers, linking them to degenerate flag varieties and continued fractions, and introduces a q-analogue consistent with previous definitions.

Contribution

It reviews a geometric approach to median Genocchi numbers and establishes a continued fraction representation for their Poincaré polynomial generating functions.

Findings

Poincaré polynomials match the q-analogue of median Genocchi numbers.

Generating functions can be expressed as simple continued fractions.

Provides geometric interpretation of median Genocchi numbers.

Abstract

The goal of this paper is twofold. First, we review the recently developed geometric approach to the combinatorics of the median Genocchi numbers. The Genocchi numbers appear in this context as Euler characteristics of the degenerate flag varieties. Second, we prove that the generating function of the Poincar\' e polynomials of the degenerate flag varieties can be written as a simple continued fraction. As an application we prove that the Poincar\' e polynomials coincide with the $q$-version of the normalized median Genocchi numbers introduced by Han and Zeng.