Arithmetic properties for 7-regular partition triples

Topology Based on New Generalized Closed Sets

“…Definition 2.3. [5] Let (X, ) be a topological space and A  X. We define the sg-closure of A (briefly, sg-cl(A)) to be the intersection of all sg-closed sets containing A.…”

“…In 1963, Levine [1][2] was introduced by the notions of semi-open sets and g-closed sets are investigated its fundamental properties also. Since the advent of the notion of semiopen sets, many mathematicians worked on such sets and also introduced some other notions such as [3][4][5][6][7][8][9][10][11][12][13][14][15].…”

Generating functions and congruences for 9-regular and 27-regular partitions in 3 colours

“…Recently, Gireesh and Naika [GN19] found an infinite family of congruences for b 3;3 (n) whereas Saikia and Boruah [SB17] obtained some infinite families of arithmetic identities and congruences modulo 2, 4, 5, and 9 for b ;3 (n) when ∈ {2, 3, 4, 5}. Very recently, Chern, Tang, and Xia [CTX20] found an infinite family of congruences for b 7;3 (n) modulo powers of 7. In this paper, we find some generating functions for b 9;3 (n) and b 27;3 (n) and use those to prove new infinite families of congruences.…”