Let S_2(n,k) denote the 2-associated Stirling number of the second kind for n objects and k blocks, with n being at least two. That is, we partition n labeled objects into k unlabeled blocks such that each block has at least two objects in it, and S_2(n,k) is the number of those partitions. I’d like to ask if the following is known: q(n,k) is strictly decreasing in k, where q(n,k):=S_2(n,k)/S(n,k), and S is the Stirling number of the second kind. Many thanks for the help!

Janos