Estimates of Two Partition Theoretic Functions

A partition say π = (a sub 1 /sub ,a sub 2 /sub ,...,a sub k /sub ) of a positive integer n is said to be a modulo -- m partition of n if a sub i /sub = a sub j /sub (mod m) ∀i, j, where m is a positive integer greater than 1. Let R sub m /sub (n,k) be the number of modulo-m partitions of n with exactly k parts. In this article, we show that: R sub m /sub (n,2) ~ n/2m when gcd(m, 2) = 1 and R sub m /sub (n, 3) ~ n²/12m² when gcd(m, 3) = 1. Estimate for R sub m /sub (n, k) is conjectured.