Hirakjyoti Das’s research while affiliated with Tezpur University and other places

In the eleventh paper in the series on MacMahons partition analysis, Andrews and Paule [1] introduced the k elongated partition diamonds. Recently, they [2] revisited the topic. Let $d_k(n)$ count the partitions obtained by adding the links of the k elongated plane partition diamonds of length n. Andrews and Paule [2] obtained several generating functions and congruences for $d_1(n)$, $d_2(n)$, and $d_3(n)$. They also posed some conjectures, among which the most difficult one was recently proved by Smoot [11]. Da Silva, Hirschhorn, and Sellers [5] further found many congruences modulo certain primes for $d_k(n)$ whereas Li and Yee [8] studied the combinatorics of Schmidt type partitions, which can be viewed as partition diamonds. In this article, we give elementary proofs of the remaining conjectures of Andrews and Paule [2], extend some individual congruences found by Andrews and Paule [2] and da Silva, Hirschhorn, and Sellers [5] to their respective families as well as find new families of congruences for $d_k(n)$, present a refinement in an existence result for congruences of $d_k(n)$ found by da Silva, Hirschhorn, and Sellers [5], and prove some new individual as well as a few families of congruences modulo 5, 7, 8, 11, 13, 16, 17, 19, 23, 25, 32, 49, 64 and 128.

We show that the series expansions of certain q-products have matching coefficients with their reciprocals. Several of the results are associated to Ra-manujan's continued fractions. For example, let R(q) denote the Rogers-Ramanujan continued fraction having the well-known q-product repesentation R(q) = (q; q 5) ∞ (q 4 ; q 5) ∞ (q 2 ; q 5) ∞ (q 3 ; q 5) ∞. If ∞ n=0 α(n)q n = 1 R 5 (q) = ∞ n=0 α (n)q n −1 , ∞ n=0 β(n)q n = R(q) R (q 16) = ∞ n=0 β (n)q n −1 , then α(5n + r) = −α (5n + r − 2) r ∈ {3, 4}, β(10n + r) = −β (10n + r − 6) r ∈ {7, 9}.

Recently, Chan and Wang studied the fractional partition function and found several infinite classes of congruences satisfied by the corresponding coefficients. In this paper, we find new families of congruences modulo powers of primes using the Rogers-Ramanujan continued fraction and some dissection formulae of certain q-products. We also find analogous congruences in the coefficients of the fractional powers of the generating function for the 2-color partition function.

Recently, Chan and Wang (Fractional powers of the generating function for the partition function. Acta Arith. 187(1), 59--80 (2019)) studied the fractional powers of the generating function for the partition function and found several congruences satisfied by the corresponding coefficients. In this paper, we find some new families of congruences modulo powers of primes. We also find analogous results for the coefficients of the fractional powers of the generating function for the 2-color partition function.

Relations involving the Rogers-Ramanujan continued fractions R(q), $R(q^3 ),$ and $R(q^4)$ are used to find new generating functions and congruences modulo 5 and 25 for 3-core, 4-core, 4-regular, and colored partition functions.