In this paper, using the idea of Alford, Granville and Pomerance in [1] (or van Emde Boas and Kruyswijk [6]), we obtain an upper bound for the Davenport Constant of an Abelian group G in terms of the number of repetitions of the group elements in any given sequence. In particular, our result implies, [Formula: see text] where n is the exponent of G and k ≥ 0 denotes the number of distinct elements of G that are repeated at least twice in the given sequence.

In this work, we study the function B s,t (n), which counts the number of (s, t)-regular bipartitions of n. Recently, many authors proved infinite families of congruences modulo 11 for B 3,11 (n), modulo 3 for B 3,s (n) and modulo 5 for B 5,s (n). Very recently, Kathiravan proved several infinite families of congruences modulo 11, 13 and 17 for B 5,11 (n), B 5,13 (n) and B 81,17 (n). In this paper, we will prove infinite families of congruences modulo 5 for B 2,15 (n), modulo 11 for B 7,11 (n), modulo 11 for B 27,11 (n) and modulo 17 for B 243,17 (n) .

International audience A partition of n is l-regular if none of its parts is divisible by l. Let b l (n) denote the number of l-regular partitions of n. In this paper, using theta function identities due to Ramanujan, we establish some new infinite families of congruences for b l (n) modulo l, where l = 17, 23, and for b 65 (n) modulo 13.