Michael Albert ; Mathilde Bouvel

Operators of equivalent sorting power and related Wilfequivalences
dmtcs:2333 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2013,
DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)

https://doi.org/10.46298/dmtcs.2333
Operators of equivalent sorting power and related Wilfequivalences
Authors: Michael Albert ^{1}; Mathilde Bouvel ^{2}
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Michael Albert;Mathilde Bouvel
1 Department of Computer Science, University of Otago
2 Laboratoire Bordelais de Recherche en Informatique
We study sorting operators $\textrm{A}$ on permutations that are obtained composing Knuth's stack sorting operator \textrmS and the reverse operator $\textrm{R}$, as many times as desired. For any such operator $\textrm{A}$, we provide a bijection between the set of permutations sorted by $\textrm{S} \circ \textrm{A}$ and the set of those sorted by $\textrm{S} \circ \textrm{R} \circ \textrm{A}$, proving that these sets are enumerated by the same sequence, but also that many classical permutation statistics are equidistributed across these two sets. The description of this family of bijections is based on an apparently novel bijection between the set of permutations avoiding the pattern $231$ and the set of those avoiding $132$ which preserves many permutation statistics. We also present other properties of this bijection, in particular for finding families of Wilfequivalent permutation classes.