Blockwise simple permutations

2023/03/23 by Bagno, Eli, Eisenberg, Estrella, Reches, Shulamit +1
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2303.13115

Abstract

A permutation is called \it block-wise simple if it contains no interval of the form p₁⊕ p₂ or p₁ \ominus p₂. 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.

Related

• The enumeration of simple permutations
• Simple permutations poset
• Interval posets for permutations
• The interval posets of permutations seen from the decomposition tree perspective
• Intervals of Permutations with a Fixed Number of Descents are Shellable