Arithmetic properties of summands of partitions II

Let d is an element of N, d >= 2. We prove that a positive proportion of partitions of an integer n satisfies the following : for all 1 <= a < b <= d, the number of the parts congruent to a (mod d) is greater than the number of the parts congruent to b (mod d). We also show that for almost all partitions the rate of the number of square free parts is 6/pi(2) (1 + o(1)).