Congruences for bipartitions with odd parts distinct

Abstract

Hirschhorn and Sellers studied arithmetic properties of the number of partitions with odd parts distinct. In another direction, Hammond and Lewis investigated arithmetic properties of the number of bipartitions. In this paper, we consider the number of bipartitions with odd parts distinct. Let this number be denoted by pod −2 (n). We obtain two Ramanujan-type identities for pod −2 (n), which imply that pod −2 (2n + 1) is even and pod −2 (3n + 2) is divisible by 3. Furthermore, we show that for any α ≥ 1 and n ≥ 0, pod −2 (3 2α+1 n + 23×3 2α −7 8 ) is a multiple of 3 and pod −2 (5 α+1 n + 11×5 α +1 4 ) is divisible by 5. We also find combinatorial interpretations for the two congruences modulo 2 and 3.

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

“…A bipartition (λ, μ) of n is a pair of partitions (λ, μ) such that the sum of all of the parts equals n. Recently, the arithmetic properties of bipartitions with certain restrictions on each partition have received a great deal of attention (see, for example, [4,[6][7][8][9][17][18][19][20][21][22]28]). …”

Congruences modulo 9 for bipartitions with designated summands

“…There are numerous remarkable results on arithmetic properties for p −2 (n) (see [2,11,12,20]). Recently, arithmetic properties for bipartitions with certain restrictions on each partition have drawn a great deal of interest (see [5,7,9,10]).…”

Comment on “Some Congruence Properties of a Restricted Bipartition Function cN(n)”

“…This function has some beautiful arithmetic properties (see, e.g., [2]). Recently, there is more and more research on the bipartitions with certain restrictions on each partition, for example, [3][4][5][6][7]. In this short review, we consider the number of bipartitions ( , ) with the restriction that each part of is divisible by .…”