Congruences for Fishburn numbers modulo prime powers

TL;DR

This paper extends known congruences for Fishburn numbers from prime moduli to prime power moduli, under certain conditions, enhancing understanding of their arithmetic properties.

Contribution

It generalizes existing congruences for Fishburn numbers to prime powers, answering an open question and broadening the scope of their modular properties.

Findings

Congruences for Fishburn numbers extend to prime powers under specific conditions.

The results deepen understanding of Fishburn numbers' arithmetic structure.

The work confirms conjectures about modular behavior of Fishburn numbers at 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.