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Valuative Invariants of Catalan Matroids

Yiming Chen, Yao Li, Ming Yao

TL;DR

We address the problem of computing valuative invariants for $(a,b)$-Catalan matroids by expressing their polytopes as convex combinations of direct sums of uniform matroids. The authors establish that Catalan matroids are non-extremal and lie in the interior of the relevant convex hull, enabling a universal formula: for any valuative invariant $f$, $f(C_n^{a,b})=\sum_{\lambda\vdash n} \frac{1}{z_{\lambda}} f(U_{\lambda}^{a,b})$. This reduction to uniform matroids yields explicit, nonnegative expressions for Ehrhart polynomials and a suite of invariants (Tutte, KL, inverse KL, $Z$, Whitney) via the same partition-sum; explicit small-n data are included. The results provide a unifying framework that proves Ehrhart positivity for $(a,b)$-Catalan matroids and reveals structural tiling and interior-position phenomena in matroid polytope geometry, with broad implications for evaluating valuative invariants across matroid classes.

Abstract

We decompose the indicator function of each $(a, b)$-Catalan matroid polytope as a weighted sum of indicator function of matroid polytopes that correspond to direct sums of uniform matroids. Catalan matroids lie in the interior of the convex hull of direct sums of uniform matroids in the polytope of all matroids introduced by Ferroni and Fink. Moreover, we describe combinatorially the coefficients of the convex combination of direct sums of uniform matroid corresponding to an $(a,b)$-Catalan matroid. In particular, this allows us to derive explicit formulas for arbitrary valuative invariants of $(a, b)$-Catalan matroids. Among other applications, we prove that $(a,b)$-Catalan matroids are Ehrhart positive, and we find formulas for the Kazhdan--Lusztig invariants of these matroids.

Valuative Invariants of Catalan Matroids

TL;DR

We address the problem of computing valuative invariants for -Catalan matroids by expressing their polytopes as convex combinations of direct sums of uniform matroids. The authors establish that Catalan matroids are non-extremal and lie in the interior of the relevant convex hull, enabling a universal formula: for any valuative invariant , . This reduction to uniform matroids yields explicit, nonnegative expressions for Ehrhart polynomials and a suite of invariants (Tutte, KL, inverse KL, , Whitney) via the same partition-sum; explicit small-n data are included. The results provide a unifying framework that proves Ehrhart positivity for -Catalan matroids and reveals structural tiling and interior-position phenomena in matroid polytope geometry, with broad implications for evaluating valuative invariants across matroid classes.

Abstract

We decompose the indicator function of each -Catalan matroid polytope as a weighted sum of indicator function of matroid polytopes that correspond to direct sums of uniform matroids. Catalan matroids lie in the interior of the convex hull of direct sums of uniform matroids in the polytope of all matroids introduced by Ferroni and Fink. Moreover, we describe combinatorially the coefficients of the convex combination of direct sums of uniform matroid corresponding to an -Catalan matroid. In particular, this allows us to derive explicit formulas for arbitrary valuative invariants of -Catalan matroids. Among other applications, we prove that -Catalan matroids are Ehrhart positive, and we find formulas for the Kazhdan--Lusztig invariants of these matroids.

Paper Structure

This paper contains 11 sections, 19 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Interior Position of Catalan Matroids
  4. Valuative Invariants of Catalan Matroids
  5. The Ehrhart Polynomials
  6. The Tutte Polynomials
  7. The Kazhdan-Lusztig polynomials
  8. The KL polynomials
  9. The inverse KL polynomial
  10. $Z$-polynomial
  11. Whitney polynomial

Key Result

Theorem 1.2

Catalan matroids are non-extremal matroids. Moreover, the coordinate vector $p_{[C_{n}]} \in \Omega_{n, 2 n}$ of the Catalan matroid $C_{n}$ lies in the interior of the convex hull of the coordinate vectors of direct sum of uniform matroids.

Figures (1)

  • Figure 2.1: The filling $\mathcal{F}_{S}(w)$ for $w=67185$.

Theorems & Definitions (28)

  • Definition 1.1: F-allmatroids, Definition 4.2
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1: FL1
  • Lemma 2.2
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Proposition 3.5
  • ...and 18 more