Congruences for Fishburn numbers modulo prime powers

Congruences for Fishburn numbers modulo prime powers
Armin Straub — International Journal of Number Theory — Volume 11, Number 5, 2015, Pages 1679-1690


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.


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    author = {Armin Straub},
    title = {Congruences for {F}ishburn numbers modulo prime powers},
    journal = {International Journal of Number Theory},
    year = {2015},
    volume = {11},
    number = {5},
    pages = {1679--1690},
    doi = {10.1142/S1793042115400175},