On the combinatorics of several integrable hierarchies

TL;DR

This paper reveals that the combinatorial properties of specific set partitions are governed by generating functions connected to well-known integrable hierarchies like Burgers, Ibragimov--Shabat, and Korteweg--de Vries.

Contribution

It establishes a novel link between combinatorics of set partitions and integrable hierarchies through generating functions.

Findings

Set partition statistics are described by integrable hierarchy-related generating functions.

Connections between combinatorics and integrable systems are demonstrated.

Provides a new perspective on the structure of set partitions.

Abstract

We demonstrate that statistics of certain classes of set partitions is described by generating functions related to the Burgers, Ibragimov--Shabat and Korteweg--de Vries integrable hierarchies.