THE t q ANALOGS OF SECANT AND TANGENT NUMBERS

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THE (t, q)-ANALOGS OF SECANT AND TANGENT NUMBERS Dominique Foata Institut Lothaire, 1 rue Murner F-67000 Strasbourg, France Guo-Niu Han I.R.M.A., Universite de Strasbourg et CNRS 7 rue Rene-Descartes, F-67084 Strasbourg, France Submitted: August 6, 2010; Accepted: April 12, 2011; Published: May 1, 2011 To Doron Zeilberger, with our warmest regards, on the occasion of his sixtieth birthday. Abstract. The secant and tangent numbers are given (t, q)-analogs with an explicit com- binatorial interpretation. This extends, both analytically and combinatorially, the classical evaluations of the Eulerian and Roselle polynomials at t = ?1. 1. Introduction As is well-known (see, e.g., [Ni23, p. 177-178], [Co74, p. 258-259]), the coefficients T2n+1 of the Taylor expansion of tanu, namely tanu = ∑ n≥0 u2n+1 (2n+ 1)!T2n+1(1.1) = u1!1 + u3 3! 2 + u5 5! 16 + u7 7! 272 + u9 9! 7936 + u11 11! 353792 + · · · are positive integral coefficients, usually called tangent numbers, while the secant numbers E2n, also positive and integral, make their appearances in the Taylor expansion of sec u: secu

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Taylor Alternating permutation

THE ( t q ) -ANALOGS OF SECANT AND TANGENT NUMBERS Dominique Foata Institut Lothaire, 1 rue Murner F-67000 Strasbourg, France foata@unistra.fr Guo-Niu Han I.R.M.A.,Universite´deStrasbourgetCNRS 7rueRen´e-Descartes,F-67084Strasbourg,France guoniu.han@unistra.fr Submitted: August 6, 2010; Accepted: April 12, 2011; Published: May 1, 2011

To Doron Zeilberger, with our warmest regards, on the occasion of his sixtieth birthday. Abstract . The secant and tangent numbers are given ( t q )-analogs with an explicit com-binatorial interpretation. This extends, both analytically and combinatorially, the classical evaluations of the Eulerian and Roselle polynomials at t = − 1. 1. Introduction As is well-known (see, e.g., [Ni23, p. 177-178], [Co74, p. 258-259]), the coeﬃcients T 2 n +1 of the Taylor expansion of tan u , namely 2 1  1) tan u = X u n +1 ( n ≥ 0 (2 n + 1)! T 2 n +1 u 3 =1 u !1+3!2+ u 5! 5 16 + u 7! 7 272 + u 9! 9 7936 + 1 u 1 1! 1 353792 +    are positive integral coeﬃcients, usually called tangent numbers , while the secant numbers E 2 n , also positive and integral, make their appearances in the Taylor expansion of sec u : (1  2) sec u =co1s u = 1 + n X ≥ 1 ( u 2 2 n n )! E 2 n u 2 u 4 u 6 u 8 u 10 = 1 + 2! 1 + 4! 5 + 6! 61 + 8! 1385 + 10! 50521 +    Key words and phrases. q -secant numbers, q -tangent numbers, ( t q )-secant numbers, ( t q )-tangent numbers, alternating permutations, pix, inverse major index, lec-statistic, inversion number, excedance number. Mathematics Subject Classiﬁcations. 05A15, 05A30, 33B10 the electronic journal of combinatorics 12 (2011), #R00 1

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