Some Identities Related to the Second-Order Eulerian Numbers

The Stirling permutations were introduced by Gessel and Stanley [6]. For some related results on this subject, we refer to [1, 4, 7, 9]. Let Qn be the multiset {1, 1, 2, 2, . . . , n, n}. A Stirling permutation of order n is a permutation of Qn such that for each 1 ≤ m ≤ n, the elements lying between two occurrences of m are greater than m. The second-order Eulerian numbers Cn,k count the Stirling permutations of order n with k decents, which satisfy the recurrence relation: Cn,k = kCn−1,k + (2n− k)Cn−1,k−1 (1.1) with C1,1 = 1 and C1,0 = 0. By exhibiting a bijection between the set of partitions of [n +m] with m blocks and the bar permutations on the elements of Qn with m bars, Gessel and Stanley [6] proved that ∞