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arXiv · 1102.5568

Counting (3+1) - Avoiding permutations

M. D. Atkinson·Bruce E. Sagan·Vincent Vatter
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Abstract

A poset is {\it $(\3+\1)$-free} if it contains no induced subposet isomorphic to the disjoint union of a 3-element chain and a 1-element chain. These posets are of interest because of their connection with interval orders and their appearance in the $(\3+\1)$-free Conjecture of Stanley and Stembridge. The dimension 2 posets $P$ are exactly the ones which have an associated permutation $\pi$ where $i\prec j$ in $P$ if and only if $i<j$ as integers and $i$ comes before $j$ in the one-line notation of $\pi$. So we say that a permutation $\pi$ is {\it $(\3+\1)$-free} or {\it $(\3+\1)$-avoiding} if its poset is $(\3+\1)$-free. This is equivalent to $\pi$ avoiding the permutations 2341 and 4123 in the language of pattern avoidance. We give a complete structural characterization of such permutations. This permits us to find their generating function.

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M. D. Atkinson, Bruce E. Sagan, Vincent Vatter. 2011-02-28. Counting (3+1) - Avoiding permutations. https://arxiv.org/abs/1102.5568

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