Front representation of set partitions

TL;DR

This paper introduces the front representation of set partitions, providing recurrence relations for specific pattern-avoiding partitions and analyzing symmetric distributions of crossings and nestings.

Contribution

It defines the front representation of set partitions and derives new recurrence relations for pattern-avoiding and noncrossing partitions, extending understanding of their combinatorial properties.

Findings

Derived recurrence relations for 12...k12-avoiding partitions.

Established recurrence relations for k-distant noncrossing partitions.

Proved symmetric distribution properties of crossings and nestings in front representation.

Abstract

Let $\pi$ be a set partition of $[n]=\{1,2,...,n\}$. The standard representation of $\pi$ is the graph on the vertex set $[n]$ whose edges are the pairs $(i,j)$ of integers with $i<j$ in the same block which does not contain any integer between $i$ and $j$. The front representation of $\pi$ is the graph on the vertex set $[n]$ whose edges are the pairs $(i,j)$ of integers with $i<j$ in the same block whose smallest integer is $i$. Using the front representation, we find a recurrence relation for the number of $12... k12$-avoiding partitions for $k\geq2$. Similarly, we find a recurrence relation for the number of $k$-distant noncrossing partitions for $k=2,3$. We also prove that the front representation has several joint symmetric distributions for crossings and nestings as the standard representation does.