Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers. (English) Zbl 07923886
Summary: A D-permutation is a permutation of \([2n]\) satisfying \(2k - 1 \leq \sigma(2k - 1)\) and \(2k \geq \sigma(2k)\) for all \(k\); they provide a combinatorial model for the Genocchi and median Genocchi numbers. We find Stieltjes-type and Thron-type continued fractions for some multivariate polynomials that enumerate D-permutations with respect to a very large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings.
MSC:
| 05A05 | Permutations, words, matrices |
| 05A15 | Exact enumeration problems, generating functions |
| 05A19 | Combinatorial identities, bijective combinatorics |
| 05A30 | \(q\)-calculus and related topics |
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 30B70 | Continued fractions; complex-analytic aspects |
Keywords:
Genocchi numbers; median Genocchi numbers; D-permutation; D-semiderangement; D-derangement; D-cycle; continued fraction; J-fraction; S-fraction; T-fraction; Dyck path; almost-Dyck path; Schröder pathReferences:
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