A new formula for intersection numbers

TL;DR

This paper introduces a novel closed-form formula for calculating Witten--Kontsevich intersection numbers, simplifying computations by avoiding recursion and involving sums over partitions with factorials, double factorials, and Kostka numbers.

Contribution

It presents a new explicit formula for intersection numbers that does not rely on recursive methods or solving equations, and proves a related conjecture about generating polynomial coefficients.

Findings

The formula simplifies the computation of intersection numbers.

Proves a conjecture about vanishing coefficients in generating polynomials.

Provides a new combinatorial approach involving partitions and Kostka numbers.

Abstract

We propose a new formula to compute Witten--Kontsevich intersection numbers. It is a closed formula, not involving recursion neither solving equations. It only involves sums over partitions of products of factorials, double factorials and Kostka numbers (numbers of semi-standard tableau of given shape and weight) with bounded weights. As an application, we prove a conjecture of [ELO21] stating that the generating polynomials of the intersection numbers expressed in the basis of elementary symmetric polynomials have an unexpected vanishing of their coefficients.