Actions and identities on set partitions

TL;DR

This paper introduces a group action on labeled set partitions, providing new combinatorial proofs for identities related to Narayana polynomials and exploring supercharacter theories of types B and D.

Contribution

It defines a novel group action on labeled set partitions and uses it to prove combinatorial identities and enumerate supercharacter theories.

Findings

New combinatorial proofs of Coker's identity for Narayana polynomial

Establishment of identities for type B analogues

Enumerative results for supercharacter theories of types B and D

Abstract

A labeled set partition is a partition of a set of integers whose arcs are labeled by nonzero elements of an abelian group $A$. Inspired by the action of the linear characters of the unitriangular group on its supercharacters, we define a group action of $A^n$ on the set of $A$-labeled partitions of an $(n+1)$-set. By investigating the orbit decomposition of various families of set partitions under this action, we derive new combinatorial proofs of Coker's identity for the Narayana polynomial and its type B analogue, and establish a number of other related identities. In return, we also prove some enumerative results concerning Andr\'e and Neto's supercharacter theories of type B and D.