A combinatorial proof of the non-vanishing of Hankel determinants of the Thue--Morse sequence

TL;DR

This paper provides a new combinatorial proof confirming that Hankel determinants of the Thue--Morse sequence are always nonzero, and extends similar results to other classical sequences.

Contribution

It introduces a purely combinatorial proof for the non-vanishing of Hankel determinants of the Thue--Morse sequence and re-proves related results for other sequences.

Findings

Hankel determinants of the Thue--Morse sequence are always nonzero

A new combinatorial proof method is developed

Non-vanishing results are extended to other classical sequences

Abstract

In 1998, Allouche, Peyri\`ere, Wen and Wen established that the Hankel determinants associated with the Thue--Morse sequence on $\{-1, 1\}$ are always nonzero. Their proof depends on a set of sixteen recurrence relations. We present an alternative, purely combinatorial proof of the same result. We also re-prove a recent result of Coons on the non-vanishing of the Hankel determinants associated to two other classical integer sequences.