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 is a multiple of 3 and pod_-2 is divisible by 5. We also find combinatorial interpretations for the two congruences modulo 2 and 3.

  AMS Classification:  05A17, 11P83

  Keywords:  partition, bipartition, congruence, birank

  Download:   pdf