Congruences for two restricted overpartitions

Congruences for overpartitions with restricted odd differences

“…3, we prove congruences modulo 6 for t(n), while in Sects. 4-7, we prove congruences modulo 2, 4 for t (3) (n) and t (7) (n), congruences modulo 4 and 5 for t (4) (n) and congruences modulo 3, 6, and 12 for t (8)…”

“…Now assume that (6.7) holds for some α ≥ 0. Substituting (2.13) into (6.7), we deduce that n≥0 t (7) 8 from both sides of (6.11), dividing Using (7.8), (7.7) can be reduces to n≥0 t (8)…”

“…Applying (2.8) in (7.9), we find that n≥0 t (8) Equating the coefficients of q 2n+1 , dividing throughout by q and then replacing q 2 by q in (7.10), we obtain n≥0 t (8) (8n + 7)q n ≡ 3 f Invoking (7.11) and (7.12), we find that n≥0 t (8) (8n + 7)q n ≡ 3 f 3 3 f 18 f 6 (mod 12), (7.13) which yields the desired results (7.2) and (7.3).…”

“…where t (k) (n) denotes the number of overpartitions of n where (i) consecutive parts differ by a multiple of (k + 1) unless the larger of the two is overlined, and (ii) the smallest part is overlined unless it is divisible by k + 1. Chern et al [8] studied Ramanujan-type congruences for the partition functions t(n) and s(n).…”

Infinite Families of Congruences for Overpartitions With Restricted Odd Differences

“…it can also be deduced by a simple combinatorial argument using an elementary analysis of Ferrers diagrams. Subsequently, arithmetic properties of t(n) have received attention and congruences modulo small powers of 2 and 3 have been found in [6,9,12]. For example, for n ≥ 0,…”