The Andrews–Sellers family of partition congruences

Abstract

In 1994, James Sellers conjectured an infinite family of Ramanujan type congruences for 2-colored Frobenius partitions introduced by George E. Andrews. These congruences arise modulo powers of 5. In 2002 Dennis Eichhorn and Sellers were able to settle the conjecture for powers up to 4. In this article, we prove Sellers’ conjecture for all powers of 5. In addition, we discuss why the Andrews–Sellers family is significantly different from classical congruences modulo powers of primes.

A family of congruences for Rogers–Ramanujan subpartitions

“…A proof of this can be found in [9,Section 3]. The value of this equation becomes immediate when we consider the following theorem:…”

“…Our method to resolve the complications associated with this new family of congruences is based largely on the methods developed by Paule and Radu to prove the Andrews-Sellers conjecture [9]. These methods themselves are related to the techniques developed by Atkin to prove Ramanujan's congruence family for p(m) over powers of 11 [2].…”

On certain weighted 7-colored partitions

“…(Here (4.1) can be viewed as mod 5 congruences like (2.18).) On the other hand, one readily sees that c 4 (n) = p n Theorem 4.3 (Corollary 2.12, [18]). For all n ≥ 0 and α ≥ 1, cφ 2 (5 α n + λ α ) ≡ 0 (mod 5 α ),…”

Mock theta functions and weakly holomorphic modular forms modulo 2 and 3

“…Andrews [6, Corollary 10•1 and Theorem 10•2] proved that cφ 2 (2n + 1) ≡ 0 (mod 2) and cφ 2 (5n + 3) (mod 5), and many congruence properties of these symbols have since been investigated (see, for example, [10,22,28]). Corollary 1•4 shows that there is no linear congruence of the form cφ 2 (mn + t) modulo 3 with odd m (it is likely that an adaptation of these methods can be used to remove the restriction on m in this case).…”