Congruences for 2t-Core Partition Functions

Parity results for broken k-diamond partitions and (2k+1)-cores

“…Numerous congruence properties are known for t-cores, although few such results are known modulo 2. Such parity results can be found in [7], [6], [10], [8], [2], [4]. In all of these papers, the value of t which was considered was even; in this paper, we provide a new set of parity results for t-cores wherein t is odd.…”

Ramanujan type congruences for quotients of level 7 Klein forms

“…A detailed proof of the modular properties of more general quotients subsuming the cases considered here appear in [7]. The required holomorphicity follows from (2.1) and Lemma 2.1 (2) and (3). The Fourier expansions T r (τ ) may be used to deduce the components of v j are linearly independent over C. Finally, Parts (1) and (2) follow from the coincidence of the number of components of v j (resp.…”

“…These are called Ramanujan type congruences because of their analogy to congruences for the classical partition function [19], and motivate a number of studies on Ramanujan type congruences for t-core partitions and other relevant counting functions (see, e.g., [2,3,4,5,6]). On the other hand, by the definition of a t (n) and simple manipulations of infinite products, the generating function may be written (1.2) ∞ n=0 a 7 (n)q n = (q 7 ; q 7 ) 7 ∞ (q; q) ∞ = (q 7 ; q 7 ) 6 ∞ (q, q 6 ; q 7 ) ∞ (q 2 , q 5 ; q 7 ) ∞ (q 3 , q 4 ; q 7 ) ∞ , where (a; q) n = ∞ j=0…”

Arithmetical properties of the number of t-core partitions

“…Some arithmetical properties of a(t; n) have also been investigated when t=4, 8, 16 and 2 n (see [3,11,12,15,22]). …”

“…, p s . For the special case, M. Boylan [3] gave a corollary of this proposition which asserts that s ≤ m if f (z) ∈ S 12m (mod 2). Therefore, for any 4 t −1 3 distinct odd primes p 1 , p 2 , .…”