Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm

TL;DR

This paper introduces a new probability measure on Young diagrams linked to the Hecke insertion algorithm, explores its properties, and investigates its typical shape, connecting combinatorics, probability, and algebraic geometry.

Contribution

It defines the Plancherel-Hecke measure, interprets the Hecke algorithm as an exact sampling method, and studies its symmetry and limiting shape, extending classical combinatorial theorems.

Findings

Hecke algorithm provides polynomial-time exact sampling for the measure.

Symmetry property of the Hecke algorithm related to subsequences of words.

Conjecture on the limit shape of the measure, analogous to classical longest increasing subsequence results.

Abstract

We define and study the Plancherel-Hecke probability measure on Young diagrams; the Hecke algorithm of [Buch-Kresch-Shimozono-Tamvakis-Yong '06] is interpreted as a polynomial-time exact sampling algorithm for this measure. Using the results of [Thomas-Yong '07] on jeu de taquin for increasing tableaux, a symmetry property of the Hecke algorithm is proved, in terms of longest strictly increasing/decreasing subsequences of words. This parallels classical theorems of [Schensted '61] and of [Knuth '70], respectively, on the Schensted and Robinson-Schensted-Knuth algorithms. We investigate, and conjecture about, the limit typical shape of the measure, in analogy with work of [Vershik-Kerov '77], [Logan-Shepp '77] and others on the ``longest increasing subsequence problem'' for permutations. We also include a related extension of [Aldous-Diaconis '99] on patience sorting. Together, these results provide a new rationale for the study of increasing tableau combinatorics, distinct from the original algebraic-geometric ones concerning K-theoretic Schubert calculus.