Moscow Journal of Combinatorics and Number Theory, vol. 2, issue 3, 2012, pp. 34-84, [pp. 232-282].

Dominique Foata, Guo-Niu Han


Abstract. The derivative polynomials introduced by Knuth and Buckholtz in their calculations of the tangent and secant numbers are extended to a multivariable q-environment. The n-th q-derivatives of the classical q-tangent tan_q(u) and of the two q-secants sec_q(u), Sec_q(u) are given two polynomial expressions in tan_q(u), sec_q(u), Sec_q(u), indexed by triplets of integers for the first class, and compositions of integers for the second. The functional relation between those two classes is fully given by means of combinatorial techniques. Moreover, those polynomials are proved to be generating functions for so-called t-permutations by multivariable statistics. By giving special values to those polynomials we recover classical $q$-polynomials such as the Carlitz q-Eulerian polynomials and the (t,q)-tangent and -secant analogs recently introduced. They also provide q-analogs for the Springer numbers. Finally, the t-compositions used in this paper furnish a combinatorial interpretation to one of the Fibonacci triangles.

foata at unistra dot fr, guoniu dot han at unistra dot fr

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