2020/09/24 by Rupam Barman, Ajit Singh, Barman, Rupam +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.2009.11605
9 pages
arxiv created 2020/09/24 · arxiv updated 2020/09/25
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 the 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 provide complete parity characterizations of p1,1(n) and p3,3(n). We prove that pt,t(n)≡ C4t,t(n) \pmod2 for all n≥ 0 and t≥ 1, where C4t,t(n) is the Andrews' singular overpartition function. Using this congruence, the parity characterization of p1,1(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.