Generalized Dellac configurations

TL;DR

This paper explores two generalizations of Dellac configurations, establishing a connection with generalized Dumont permutations and introducing configurations with boundaries, along with recurrence relations for their Poincaré polynomials.

Contribution

It introduces a new generalization called Dellac configurations with boundaries and links generalized Dellac configurations to Dumont permutations.

Findings

Established correspondence between generalized Dellac configurations and Dumont permutations.

Introduced Dellac configurations with boundaries and derived recurrence relations.

Provided combinatorial insights into the structure of these configurations.

Abstract

We study combinatorics of two generalizations of Dellac configurations. First, we establish a correspondence between a generalized Dellac configuration with three parameters and a generalized Dumont permutations. Secondly, by relaxing conditions on Dellac configurations, we introduce a generalization which we call Dellac configurations with boundaries. We show several recurrence relations for the Poincar\'e polynomials of Dellac configurations with boundaries.