Cardinality of $\ell_1$-Segments and Genocchi Numbers

TL;DR

This paper establishes a connection between Genocchi numbers and the size of specific segments in permutation spaces measured by the -distance, supported by experimental evidence of maximality.

Contribution

It reveals that Genocchi numbers count the elements in certain -distance segments in permutation spaces, a novel combinatorial interpretation.

Findings

Genocchi numbers correspond to segment cardinalities

Segments studied tend to have maximal size

Experimental data supports maximality conjecture

Abstract

We prove that the Genocchi numbers of first and second kind give the cardinality of certain segments in permutation spaces, with respect to the $\ell_1$-distance. Experimental data suggests that those segments have maximal cardinality among all segments in the corresponding spaces.