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Let spt(n) denote the total number of appearances of smallest parts in thepartitions of n. Recently, Andrews showed how spt(n) is related to the secondrank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and13. We prove a generalization of these congruences using known relationsbetween rank and crank moments. We obtain explicit Ramanujan-type congruencesfor spt(n) mod p for p = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Onoproved that Dyson's rank function has infinitely many Ramanujan-typecongruences. Their proof is non-constructive and utilizes the theory of weakMaass forms. We construct two explicit nontrivial examples mod 11 usingelementary congruences between rank moments and half-integer weight Heckeeigenforms.

CONGRUENCES FOR ANDREWS' SMALLEST PARTS PARTITION FUNCTION AND NEW CONGRUENCES FOR DYSON'S RANK