Partition into heapable sequences, heap tableaux and a multiset extension of Hammersley's process

TL;DR

This paper explores the partitioning of integer sequences into heapable subsequences, introduces a new sorting method, connects it to a particle system, and extends classical combinatorial structures with new inequalities and correspondences.

Contribution

It introduces a heap-based extension of patience sorting, links the minimal heapable subsequences to a multiset Hammersley's process, and extends Young tableaux and Robinson-Schensted correspondence.

Findings

The minimal number of heapable subsequences can be computed via an extended patience sorting.

Expected value of the process on random permutations scales as (1+√5)/2 * ln(n).

New heap-based Young tableaux and related inequalities are established.

Abstract

We investigate partitioning of integer sequences into heapable subsequences (previously defined and established by Mitzenmacher et al). We show that an extension of patience sorting computes the decomposition into a minimal number of heapable subsequences (MHS). We connect this parameter to an interactive particle system, a multiset extension of Hammersley's process, and investigate its expected value on a random permutation. In contrast with the (well studied) case of the longest increasing subsequence, we bring experimental evidence that the correct asymptotic scaling is $\frac{1+\sqrt{5}}{2}\cdot \ln(n)$. Finally we give a heap-based extension of Young tableaux, prove a hook inequality and an extension of the Robinson-Schensted correspondence.