Proof of a conjecture of Baruah and Sarmah on generalized Frobenius partitions with 6 colors

Abstract

In his 1984 AMS Memoir, Andrews introduced the k-colored generalized Frobenius partition function cϕk(n), which denotes the number of generalized Frobenius partitions of n with k colors. Recently, Baruah and Sarmah found the generating function for cϕ6(n). They also established 2- and 3-dissections of the generating function for cϕ6(n) and proved that cϕ6(2n+1)≡0(mod4), cϕ6(3n+1)≡cϕ6(3n+2)≡0(mod9). Furthermore, they conjectured that cϕ6(3n+2)≡0(mod27) for n≥0. In this paper, we confirm this conjecture by employing the generating function for cϕ6(3n+2) given by Baruah and Sarmah and the (p,k)-parametrization of theta functions due to Alaca, Alaca and Williams. Moreover, we conjecture that for n≥0, cϕ6(9n+7)≡0(mod27) and cϕ6(27n+16)≡0(mod35).