A Combinatorial Proof of the Non-Vanishing of Hankel Determinants of the Thue-Morse Sequence

Abstract

In 1998, Allouche, Peyrière, 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.