On the Poset and Asymptotics of Tesler Matrices

Jason O'Neill

doi:10.37236/6877

Abstract

Tesler matrices are certain integral matrices counted by the Kostant partition function and have appeared recently in Haglund's study of diagonal harmonics. In 2014, Drew Armstrong defined a poset on such matrices and conjectured that the characteristic polynomial of this poset is a power of $q-1$. We use a method of Hallam and Sagan to prove a stronger version of this conjecture for posets of a certain class of generalized Tesler matrices. We also study bounds for the number of Tesler matrices and how they compare to the number of parking functions, the dimension of the space of diagonal harmonics.

Highlights

Tesler matrices were introduced by Glenn Tesler to study Macdonald polynomials
In (1), the Hilbert series is over the space DHn which has dimension (n + 1)n−1
We show that certain powers of (q − 1) divide the characteristic polynomial of the Tesler poset corresponding to a hook sum vector with either a trailing or a leading binary word

Summary

Introduction

Tesler matrices were introduced by Glenn Tesler to study Macdonald polynomials. They have been recently studied due to their relationship with diagonal harmonics and Haglund proved in [9] that the bigraded Hilbert series for the space of diagonal harmonics, denoted DHn, is the sum over Tesler matrices of a bivariate weight. We prove another such result that was initially conjectured by Armstrong in [1] by using a different alternating sum He defines a poset on the set of Tesler matrices which we will denote as P (1n) and refer to as the Tesler poset. The method that we use in this paper extends to the larger class of generalized Tesler matrices with binary hook sums and settles Armstrong’s conjecture with a simple calculation. In the last two sections, we will explore asymptotics and other enumerative questions regarding generalized Tesler matrices and explore the significance of settling Conjecture 1.2 in respect to the asymptotics of Tesler matrices

Background

The Tesler Poset

Application of Hallam-Sagan to the Tesler Poset

Armstrong polynomial

Understanding Different Hook Sum Vectors