Divisibility and Distribution of Partitions into Distinct Parts

Congruences for (3,11)-regular bipartitions modulo 11

“…The arithmetic of -regular partition functions has received a great deal of attention (see, for example, [1,2,5,10,[12][13][14][15]20,22,[24][25][26][27]30]). Recently, Xia and Yao [31] established several infinite families of congruences modulo 2 for b 9 (n).…”

Number theoretic properties of generating functions related to Dyson’s rank for partitions into distinct parts

“…Several infinite families of congruence identities have also been shown for Q. (See [13], [14], [17], [18].) In fact, it was shown in [14] that for any prime p, there exist positive integers a and b such that Q(an + b) ≡ 0 (mod p) for all positive integers n.…”

The Number of Partitions Into Distinct Parts Modulo Powers of 5

“…We give details only for the function f 1 (z); the remaining cases are analogous. There is a simple criterion for deciding when an η-product is a holomorphic modular form (see, for example, [11,Theorem 4]). We find that f 1 (z) ∈ M 4 (Γ 0 (1152), χ 2 ), where χ 2 denotes the Kronecker symbol for Q( √ 2).…”

“…Recent works [1,7,11] p-adically relating certain values of Q(n) to Fourier coefficients of holomorphic modular forms guarantee that for any modulus M coprime to 3, there are indeed infinitely many independent congruences of the form Q(an + b) ≡ 0 (mod M). However, it was believed that to identify examples explicitly in this theory would require a case-by-case computation of the action of the Hecke operators T (p) (see Proposition 5 for the definition) on the relevant forms.…”