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arXiv · 2303.13115

Blockwise simple permutations

Eli Bagno·Estrella Eisenberg·Shulamit Reches·Moriah Sigron
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Abstract

A permutation is called {\it {block-wise simple}} if it contains no interval of the form $p_1\oplus p_2$ or $p_1 \ominus p_2$. We present this new set of permutations and explore some of its combinatorial properties. We present a generating function for this set, as well as a recursive formula for counting block-wise simple permutations. Following Tenner, who founded the notion of interval posets, we characterize and count the interval posets corresponding to block-wise simple permutations. We also present a bijection between these interval posets and certain tiling's of the $n$-gon. Finally, we prove that the bi-variate distribution of the descent and inverse descent numbers are gamma-positive, provided the correctness of our recent conjecture on simple permutations.

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Eli Bagno, Estrella Eisenberg, Shulamit Reches, Moriah Sigron. 2023-03-23. Blockwise simple permutations. https://arxiv.org/abs/2303.13115

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