Partition into Heapable Sequences, Heap Tableaux and a Multiset Extension of Hammersley’s Process

Abstract

We investigate partitioning of integer sequences into heapable subsequences (previously defined and established by Mitzenmacher et al.[BHMZ11]). 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 isFinally we give a heap-based extension of Young tableaux, prove a hook inequality and an extension of the Robinson-Schensted correspondence.

The Language (and Series) of Hammersley-Type Processes

“…Thus it is natural to call the Hammersley-Ulam problem for heapable sequences the investigation of the scaling behavior of the number of classes of the partition of a random permutation into a minimal number of (max) heapable subsequences. This was the approach we took in [9]. Unlike the case of LIS, for heapable subsequences the relevant parameter (denoted in [9] by M HS k (π)) scales logarithmically, and the following conjecture was proposed:…”

“…The intuition for Conjecture 19 relies on the extension from the LIS problem to heapable sequences of a correspondence between LIS and an interactive particle system [1] called the Hammersley-Aldous-Diaconis (shortly, Hammersley or HAD) process. The validity of correspondence was noted, for heapable sequences, in [9]. The generalized process was further investigated in [4], where it was called the Hammersley tree process.…”

“…Experimental evidence presented in [10] seems to confirm the accuracy of this heuristic description. A proof of the existence of constants c 0 , c 1 , c 2 was attempted in [9] based on subadditivity (Fekete's lemma). However, part of the proof in [9] is incorrect.…”

“…While it could perhaps be fixed using more sophisticated tools (e.g. the subadditive ergodic theorem [19]) than those in [9], an alternate approach involves analyzing the asymptotic behavior of process HAD k using (noncommutative) power series ( [18,6]).…”

Almost-sure asymptotics for the number of heaps inside a random sequence

“…In [2], Byers et al proposed variations on this problem where the question of finding increasing subsequences in a permutation is replaced by that of finding heapable subsequences. Subsequently, Istrate and Bonchis [3] introduced a modification of the classical patience sorting algorithm called heap sorting algorithm which now computes the minimal number of binary heaps required to partition {1, . .…”