Random 3-noncrossing partitions

TL;DR

This paper presents polynomial-time algorithms for uniformly generating 3-noncrossing partitions and 2-regular, 3-noncrossing partitions, using bijections and Markov-process interpretations to facilitate sampling.

Contribution

It introduces efficient algorithms for sampling complex combinatorial structures by leveraging bijections and Markov-process models, extending previous theoretical frameworks.

Findings

Algorithms run in polynomial time

Uniform sampling of complex partitions achieved

Markov-process models derived for sampling paths

Abstract

In this paper, we introduce polynomial time algorithms that generate random 3-noncrossing partitions and 2-regular, 3-noncrossing partitions with uniform probability. A 3-noncrossing partition does not contain any three 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 3-noncrossing partitions and 2-regular, 3-noncrossing partitions as restricted generalized vacillating tableaux. Furthermore, we interpret the tableaux as sampling paths of Markov-processes over shapes and derive their transition probabilities.