Logarithms of iteration matrices, and proof of a conjecture by Shadrin and Zvonkine

TL;DR

This paper proves a conjecture linking matrix entries in Hurwitz number studies to rational sequences, using iteration matrices and their logarithms as key tools.

Contribution

It introduces a novel proof of a conjecture by employing iteration matrices and their logarithms, connecting matrix entries to rational sequences in Hurwitz number analysis.

Findings

Confirmed the conjecture relating matrix entries to rational numbers.

Demonstrated the utility of iteration matrices and their logarithms in combinatorial proofs.

Provided a new methodological approach for studying Hurwitz numbers.

Abstract

A proof for a conjecture by Shadrin and Zvonkine, relating the entries of a matrix arising in the study of Hurwitz numbers to a certain sequence of rational numbers, is given. The main tools used are iteration matrices of formal power series and their (matrix) logarithms.