On partitions avoiding 3-crossings

TL;DR

This paper studies partitions avoiding 3-crossings, proving their counting sequence is P-recursive and providing an explicit recurrence, while conjecturing that this property does not extend to higher k-crossings.

Contribution

It establishes that 3-noncrossing partitions are counted by a P-recursive sequence and provides an explicit recurrence relation.

Findings

Counting sequence for 3-noncrossing partitions is P-recursive.

Explicit linear recurrence relation is derived.

Conjecture that k-noncrossing partitions for k≥4 are not P-recursive.

Abstract

A partition on $[n]$ has a crossing if there exists $i\_1<i\_2<j\_1<j\_2$ such that $i\_1$ and $j\_1$ are in the same block, $i\_2$ and $j\_2$ are in the same block, but $i\_1$ and $i\_2$ are not in the same block. Recently, Chen et al. refined this classical notion by introducing $k$-crossings, for any integer $k$. In this new terminology, a classical crossing is a 2-crossing. The number of partitions of $[n]$ avoiding 2-crossings is well-known to be the $n$th Catalan number $C\_n={{2n}\choose n}/(n+1)$. This raises the question of counting $k$-noncrossing partitions for $k\ge 3$. We prove that the sequence counting 3-noncrossing partitions is P-recursive, that is, satisfies a linear recurrence relation with polynomial coefficients. We give explicitly such a recursion. However, we conjecture that $k$-noncrossing partitions are not P-recursive, for $k\ge 4$.