Cycles of even-odd drop permutations and continued fractions of Genocchi numbers. (English) Zbl 1517.05012
Summary: Recently A. Lazar and M. L. Wachs [Comb. Theory 2, No. 1, Paper No. 2, 34 p. (2022; Zbl 1502.52019)] proved two new permutation models, called D-permutations and E-permutations, for Genocchi and median Genocchi numbers. In a follow-up, S.-P. Eu et al. [Electron. J. Comb. 29, No. 2, Research Paper P2.15, 23 p. (2022; Zbl 1487.05011)] studied the even-odd descent permutations, which are in bijection with E-permutations. We generalize Eu et al.’s descent polynomials with eight statistics and obtain an explicit J-fraction formula for their ordinary generaing function. The J-fraction permits us to confirm two conjectures of Lazar-Wachs [loc. cit.] about cycles of D and E permutations and obtain a \((p,q)\)-analogue of Eu et al.’s gamma-formula [loc. cit.]. Moreover, the \((p,q)\) gamma-coefficients have the same factorization flavor as the gamma-coefficients of P. Brändén’s \((p,q)\)-Eulerian polynomials [in: Handbook of enumerative combinatorics. Boca Raton, FL: CRC Press. 437–483 (2015; Zbl 1327.05051)].
MSC:
| 05A05 | Permutations, words, matrices |
| 11B75 | Other combinatorial number theory |
| 11A55 | Continued fractions |
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |
Keywords:
even-odd drop permutations; even-odd descent permutations; cycles; descents; drops; Genocchi numbers; median Genocchi numbers; normalized median Genocchi numbersReferences:
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