BIASES in INTEGER PARTITIONS

We show that there are biases in the number of appearances of the parts in two residue classes in the set of ordinary partitions. More precisely, let pj,k,m(n) be the number of partitions of n such that there are more parts congruent to j modulo m than parts congruent to k modulo m for m ≥2. We prove that p1,0,m(n) is in general larger than p0,1,m(n). We also obtain asymptotic formulas for p1,0,m(n) and p0,1,m(n) for m ≥ 2.