Eulerian numbers, Newcomb's problem and representations of symmetric groups

AbstractConsider a set of n positive integers consisting of μ1 1's, μ2 2's,…, μr r's. If the integer in the ith place in an arrangement σ of this set is σ(i), and a non-rise in σ is defined as σ(i+1)⩽σ(i), a problem that suggests itself is the determination of the number of arrangements σ with k non-rises. When each μi is unity, the problem is that of finding the number A(n, k) of permutations of distinct integers 1, 2,…, n with k descents, a descent being defined as σ(i+1)