Lonesum and $\Gamma$-free $0$-$1$ fillings of Ferrers shapes

TL;DR

This paper establishes a combinatorial equivalence between two classes of 0-1 fillings of Ferrers shapes, introduces new bijections, and connects these to Genocchi numbers and alternating acyclic tournaments.

Contribution

It introduces a new bijection between Callan sequences and Dumont-like permutations, and provides a combinatorial interpretation of Genocchi numbers.

Findings

Lonesum and $\Gamma$-free fillings are equinumerous.

New bijection between Callan sequences and Dumont-like permutations.

Genocchi numbers interpreted via Callan sequences.

Abstract

We show that $\Gamma$-free fillings and lonesum fillings of Ferrers shapes are equinumerous by applying a previously defined bijection on matrices for this more general case and by constructing a new bijection between Callan sequences and Dumont-like permutations. As an application, we give a new combinatorial interpretation of Genocchi numbers in terms of Callan sequences. Further, we recover some of Hetyei's results on alternating acyclic tournaments. Finally, we present an interesting result in the case of certain other special shapes.