Journal of Integer Sequences, Vol. 24 (2021), Article 21.6.3

Mex-Related Partition Functions of Andrews and Newman

Rupam Barman and Ajit Singh
Department of Mathematics
Indian Institute of Technology


In a recent paper, Andrews and Newman introduced certain families of partition functions using the minimal excludant or "mex" function. In this article, we study two of the families of functions Andrews and Newman introduced, namely pt,t(n) and p2t,t(n). We establish identities connecting the ordinary partition function p(n) to pt,t(n) and p2t,t(n) for all t ≥ 1. Using these identities, we prove that Ramanujan's famous congruences for p(n) are also satisfied by pt,t(n) and p2t,t(n) for infinitely many values of t.

Very recently, da Silva and Sellers provided complete parity characterizations of p1,1(n) and p3,3(n). We prove that pt,t(n) ≡ C4t,t(n) (mod 2) for all n ≥ 0 and t ≥ 1, where C4t,t(n) is Andrews' singular overpartition function. Using this congruence, the parity characterization of p_{1,1}(n) given by da Silva and Sellers follows from that of C4,1(n).

We also give elementary proofs of certain congruences already proved by da Silva and Sellers.

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Received February 22 2021; revised versions received February 23 2021; May 17 2021; May 18 2021. Published in Journal of Integer Sequences, May 18 2021.

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