On the lattice of weighted partitions

TL;DR

This paper introduces the lattice of weighted partitions, explores its combinatorial properties, and provides explicit labelings, M"obius function, and connections to binary trees, extending classical partition lattice theory.

Contribution

It defines the lattice of weighted partitions, establishes its shellability via an explicit EL-labeling, and links its structure to binary trees and Stirling transforms.

Findings

The lattice of weighted partitions is shellable.

The total number of weighted partitions is given by Stirling transforms.

A bijection exists between maximal decreasing chains and labeled binary trees.

Abstract

We introduce and study the lattice of generalized partitions, called weighted partitions. This lattice possesses similar properties of the lattice of partitions. By use of the pictorial representation of a weighted partition, the total number is given by the successive Stirling transforms of the Stirling number of the second kind. We construct an explicit $EL$-labeling on the lattice, which implies this lattice is $EL$-shellable and hence shellable. We compute the M\"obius function and the characteristic polynomial by use of a pictorial representation of a maximal decreasing chain. Further, a maximal decreasing chain is shown to be bijective to a labeled rooted complete binary tree.