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arXiv · 1707.06899

Bijective enumerations of $Γ$-free 0-1 matrices

Beáta Bényi·Gábor V. Nagy
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Abstract

We construct a new bijection between the set of $n\times k$ $0$-$1$ 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 $\{1,\dots,n\}$ with no common rise.

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Beáta Bényi, Gábor V. Nagy. 2017-07-21. Bijective enumerations of $Γ$-free 0-1 matrices. https://doi.org/10.1016/j.aam.2017.12.002

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