Bijective enumerations of $\mathrm{\Gamma }$-free 0-1 matrices

We construct a new bijection between the set of $n\times k$ $0$-$1$ matrices with no three $1$’s forming a $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$-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.