Random $k$-noncrossing partitions

TL;DR

This paper presents polynomial time algorithms for uniformly generating random $k$-noncrossing and 2-regular, $k$-noncrossing partitions using bijections and Markov process interpretations.

Contribution

It introduces the first polynomial time algorithms for uniform sampling of these complex combinatorial structures based on bijections and Markov processes.

Findings

Algorithms run in polynomial time

Uniform sampling of $k$-noncrossing partitions achieved

Markov process interpretation enables efficient sampling

Abstract

In this paper, we introduce polynomial time algorithms that generate random $k$-noncrossing partitions and 2-regular, $k$-noncrossing partitions with uniform probability. A $k$-noncrossing partition does not contain any $k$ mutually crossing arcs in its canonical representation and is 2-regular if the latter does not contain arcs of the form $(i,i+1)$. Using a bijection of Chen {\it et al.} \cite{Chen,Reidys:08tan}, we interpret $k$-noncrossing partitions and 2-regular, $k$-noncrossing partitions as restricted generalized vacillating tableaux. Furthermore, we interpret the tableaux as sampling paths of a Markov-processes over shapes and derive their transition probabilities.