New Eulerian numbers of type \(D\). (English) Zbl 1382.05004
Summary: We introduce a new array of type \(D\) Eulerian numbers, different from that studied by F. Brenti [Eur. J. Comb. 15, No. 5, 417–441 (1994; Zbl 0809.05012)], C.-O. Chow [ibid. 24, No. 4, 391–408 (2003; Zbl 1033.05003)] and M. Hyatt [“Recurrences for Eulerian polynomials of type B and type D”, Preprint, arXiv:1404.3110]. We find in particular the recurrence relation, Worpitzky formula and the generating function. We also find the probability distributions whose moments are Eulerian polynomials of type \(A\), \(B\) and \(D\).
MSC:
| 05A05 | Permutations, words, matrices |
| 05A15 | Exact enumeration problems, generating functions |
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
Software:
OEISOnline Encyclopedia of Integer Sequences:
Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).Type D Eulerian triangle.
Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.
Eulerian numbers of type D, the primary type.
Eulerian numbers of type D, the complementary type.
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