  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.
