Proof of a conjecture of Stanley about Stern's array

2019/01/18 by Speyer, David E
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1901.06301

Abstract

Stanley, building on work of Stern, defined an array of numbers by the recurrence s(n, 2k) = s(n-1, k), s(n, 2k+1) = s(n-1, k) + s(n-1, k+1). Stanley showed that, for each positive integer r, the sequence s_n^r:= ∑_k s(n,k)^r obeys a homogeneous linear recurrence in n of length r/2+O(1). Numerical evidence, however, suggested that s_n^r obeys shorter recurrences, of length r/3+O(1). We prove Stanley's conjecture.

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