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3-REGULAR PARTITIONS AND A MODULAR K3 SURFACE

In classical representation theory, k-regular partitions of n label irreducible kmodular representations of the symmetric group Sn when k is prime [8]. More recently, such partitions have been studied for their arithmetic properties in connection with the theory of modular forms and Galois representations [1, 6, 10, 11, 12]. Although one may presumably use the ideas from [1, 10] to study the k-regular partitions modulo any prime, more focus has been placed on the most straightforward case, the p-adic behavior of p-regular partitions. For example, we have Theorem 1 (Gordon-Ono [6]). If S(p, j, a) denotes the set of natural numbers n such that bpj (n) is not divisible by p, then S(p, j, a) has arithmetic density 0. In general there is no elementary characterization of the sets S(p, j, a), but in the best cases we do have simple congruential formulas for bk(n). For example, the classical expansions