Restricted bipartitions

We consider bipartitions subject to certain restrictions and show that be(m,n)⩾bo(m,n), where be(m,n) (respectively, bo(m,n)) denotes the number of these partitions with an even (respectively, odd) number of even parts. Our principal tool is a lemma concerning the non-negativity of the coefficients of a certain rational function. As another corollary of this lemma, we deduce an inequality between the rank-counting numbers, N(r,m,n).