Congruences for Fishburn numbers modulo prime powers

Abstract

The Fishburn numbers ξ(n) are defined by the formal power series [Formula: see text] Recently, Andrews and Sellers discovered congruences of the form ξ(pm + j) ≡ 0 modulo p, valid for all m ≥ 0. These congruences have then been complemented and generalized to the case of r-Fishburn numbers by Garvan. In this note, we answer a question of Andrews and Sellers regarding an extension of these congruences to the case of prime powers. We show that, under a certain condition, all these congruences indeed extend to hold modulo prime powers.