Bijective enumerations of Γ-free 0-1 matrices

We construct a new bijection between the set of n×k 0-1 matrices with no three 1's forming a Γ 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 Γ-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,…,n} with no common rise.