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# Talk: Congruences for Fishburn numbers modulo prime powers (JMM)

Congruences for Fishburn numbers modulo prime powers (JMM)

Date: 2015/01/11
Occasion: AMS Joint Meetings 2015, Special Session on Partitions, q-Series, and Modular Forms
Place: San Antonio

## Abstract

The Fishburn numbers $$\xi (n)$$ are defined by the formal power series \begin{equation*} \sum_{n \geq 0} \xi (n) q^n = \sum_{n \geq 0} \prod_{j = 1}^n (1 - (1 - q)^j) . \end{equation*} 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 talk, 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.