Doubloons and q-secant numbers
نویسندگان
چکیده
Based on the evaluation at t = −1 of the generating polynomial for the hyperoctahedral group by the number of descents, an observation recently made by Hirzebruch, a new q-secant number is derived by working with the Chow-Gessel q-polynomial involving the flag major index. Using the doubloon combinatorial model we show that this new q-secant number is a polynomial with positive integral coefficients, a property apparently hard to prove by analytical methods.
منابع مشابه
Author manuscript, published in "Quarterly Journal of Mathematics (2009) 17 pages" Doubloons and new q-tangent numbers
We introduce new q-tangent numbers based on the Carlitz q-analog of the Eulerian polynomial and the so-called doubloon combinatorial set-up. Those new q-tangent numbers are polynomials with positive integral coefficients. They are divisible by products of binomials of the form 1+ q, the quotients being q-analogs of the reduced tangent numbers having an explicit combinatorial interpretation.
متن کاملTHE q-TANGENT AND q-SECANT NUMBERS VIA BASIC EULERIAN POLYNOMIALS
The classical identity that relates Eulerian polynomials to tangent numbers together with the parallel result dealing with secant numbers is given a q-extension, both analytically and combinatorially. The analytic proof is based on a recent result by Shareshian and Wachs and the combinatorial one on the geometry of alternating permutations.
متن کاملCOMBINATORICS OF GEOMETRICALLY DISTRIBUTED RANDOM VARIABLES: NEW q-TANGENT AND q-SECANT NUMBERS
Up-down permutations are counted by tangent (respectively, secant) numbers. Considering words instead, where the letters are produced by independent geometric distributions, there are several ways of introducing this concept; in the limit they all coincide with the classical version. In this way, we get some new q-tangent and q-secant functions. Some of them also have nice continued fraction ex...
متن کاملThe doubloon polynomial triangle
The doubloon polynomials are generating functions for a class of combinatorial objects called normalized doubloons by the compressed major index. They provide a refinement of the q-tangent numbers and also involve two major specializations: the Poupard triangle and the Catalan triangle.
متن کاملThe (t,q)-Analogs of Secant and Tangent Numbers
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.
متن کامل