ON A CONJECTURE FOR A REFINEMENT OF THE SUM OF MINIMAL EXCLUDANTS

Abstract

Abstract In 2019, Andrews and Newman [‘Partitions and the minimal excludant’, Ann. Comb.23(2) (2019), 249–254] introduced the arithmetic function $\sigma \textrm {mex}(n)$ , which denotes the sum of minimal excludants over all the partitions of n. Baruah et al. [‘A refinement of a result of Andrews and Newman on the sum of minimal excludants’, Ramanujan J., to appear] showed that the sum of minimal excludants over all the partitions of n is the same as the number of partition pairs of n into distinct parts. They proved three congruences modulo $4$ and $8$ for two functions appearing in this refinement and conjectured two further congruences modulo $8$ and $16$ . We confirm these two conjectures by using q-series manipulations and modular forms.