The monoids of the patience sorting algorithm

The left patience sorting ([Formula: see text][Formula: see text]PS) monoid, also known in the literature as the Bell monoid, and the right patient sorting ([Formula: see text]PS) monoid are introduced by defining certain congruences on words. Such congruences are constructed using insertion algorithms based on the concept of decreasing subsequences. Presentations for these monoids are given. Each finite-rank [Formula: see text]PS monoid is shown to have polynomial growth and to satisfy a nontrivial identity (dependent on its rank), while the infinite rank [Formula: see text]PS monoid does not satisfy any nontrivial identity. Each [Formula: see text][Formula: see text]PS monoid of finite rank has exponential growth and does not satisfy any nontrivial identity. The complexity of the insertion algorithms is discussed. [Formula: see text]PS monoids of finite rank are shown to be automatic and to have recursive complete presentations. When the rank is [Formula: see text] or [Formula: see text], they are also biautomatic. [Formula: see text][Formula: see text]PS monoids of finite rank are shown to have finite complete presentations and to be biautomatic.

Identities in Plactic, Hypoplactic, Sylvester, Baxter, and Related Monoids

This paper considers whether non-trivial identities are satisfied by certain `plactic-like' monoids that, like the plactic monoid, are closely connected to combinatorics. New results show that the hypoplactic, sylvester, Baxter, stalactic, and taiga monoids satisfy identities, and indeed give shortest identities satisfied by these monoids. The existing state of knowledge is discussed for the plactic monoid and left and right patience sorting monoids.

A Cost Optimal Parallel Algorithm for Patience Sorting

In this paper, we consider a parallel algorithm for the patience sorting. The problem is not known to be in the class NC or P-complete. We propose two algorithms for the patience sorting of n distinct integers. The first algorithm runs in [Formula: see text] time using p processors on the EREW PRAM, where m is the number of decreasing subsequences in a solution of the patience sorting. The second algorithm runs in [Formula: see text] time using p processors on the EREW PRAM. If [Formula: see text] is satisfied, the second algorithm becomes cost optimal.