Congruences for Fishburn numbers modulo prime powers

Abstract

The Fishburn numbers $\xi (n)$ are defined by the formal power series \[ \sum_{n \geq 0} \xi (n) q^n = \sum_{n \geq 0} \prod_{j = 1}^n (1 - (1 - q)^j). \] Recently, G. Andrews and J. Sellers discovered congruences of the form $\xi (p m + j) \equiv 0$ modulo $p$, valid for all $m \geq 0$. These congruences have then been complemented and generalized to the case of $r$-Fishburn numbers by F. 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.