Bijective enumerations of $\Gamma$-free 0-1 matrices

TL;DR

This paper introduces a bijection between certain 0-1 matrices avoiding a specific pattern and Callan sequences, leading to new generating functions and combinatorial proofs related to poly-Bernoulli numbers.

Contribution

It constructs a novel bijection linking $ ext{Gamma}$-free matrices to Callan sequences, enabling new enumerative and combinatorial insights.

Findings

Derived the generating function for $ ext{Gamma}$-free matrices.

Provided a bijective proof connecting non-ambiguous forests and permutation pairs.

Established a combinatorial correspondence involving poly-Bernoulli numbers.

Abstract

We construct a new bijection between the set of $n\times k$ $0$-$1$ matrices with no three $1$'s forming a $\Gamma$ configuration and the set of $(n,k)$-Callan sequences, a simple structure counted by poly-Bernoulli numbers. We give two applications of this result: We derive the generating function of $\Gamma$-free matrices, and we give a new bijective proof for an elegant result of Aval et al. that states that the number of complete non-ambiguous forests with $n$ leaves is equal to the number of pairs of permutations of $\{1,\dots,n\}$ with no common rise.