Enumeration of involutions by descents and symmetric matrices

Let InIn denote the set of all involutions of {1,2,…,n}{1,2,…,n}. We establish a connection between the number I(n,k)I(n,k) of involutions in InIn with kk descents and the number T(n,k)T(n,k) of k×kk×k symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to nn.