The Sorting Index and Permutation Codes
William Y. C. Chen, George Z. Gong, Jeremy J. F. Guo

TL;DR
This paper explores the sorting index in permutation groups, establishes bijections with known permutation codes, and proves equidistribution of various permutation statistics across types A, B, and D.
Contribution
It provides a combinatorial proof connecting the sorting index with Foata-Han codes and extends these results to signed permutations and Coxeter groups of types B and D.
Findings
The bijection of Foata and Han maps (inv, rl-min) to (sor, cyc).
Distribution of (inv_B, Lmap_B, Rmil_B) and (sor_B, Lmap_B, Cyc_B) over signed permutations.
Equidistribution of six pairs of set-valued statistics over signed permutations.
Abstract
In the combinatorial study of the coefficients of a bivariate polynomial that generalizes both the length and the reflection length generating functions for finite Coxeter groups, Petersen introduced a new Mahonian statistic , called the sorting index. Petersen proved that the pairs of statistics and have the same joint distribution over the symmetric group, and asked for a combinatorial proof of this fact. In answer to the question of Petersen, we observe a connection between the sorting index and the B-code of a permutation defined by Foata and Han, and we show that the bijection of Foata and Han serves the purpose of mapping to . We also give a type analogue of the Foata-Han bijection, and we derive the quidistribution of and over…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algorithms and Data Compression · semigroups and automata theory
