On the Bivariate Characteristic Polynomial of the Shuffle Lattice

TL;DR

This paper derives an explicit formula for the bivariate characteristic polynomial of the shuffle lattice, a structure modeling DNA mutation, and explores its relation to other combinatorial invariants.

Contribution

It proves a conjectured formula for the $M$-triangle of the shuffle lattice and relates it to the $H$-triangle, advancing understanding of its combinatorial properties.

Findings

Explicit formula for the $M$-triangle of the shuffle lattice

Relation established between the $M$-triangle and the $H$-triangle

Confirmation of conjecture by McConville and Mühle (2022)

Abstract

The shuffle lattice was introduced by Greene in 1988 as an idealized model for DNA mutation, when he revealed remarkable combinatorial properties of this structure. In this paper, we prove an explicit formula for the $M$-triangle of the shuffle lattice, a bivariate refinement of the characteristic polynomial, as conjectured by McConville and M\"uhle in 2022, and find a relation between the $M$-triangle and the $H$-triangle, a bivariate refinement of the rank generating function.