A combinatorial proof of a bibasic trigonometric identity

Abstract

The bibasic trigonometric functions , recently introduced by Foata and Han, give rise to the p , q - tangent numbers and the p , q - secant numbers . Foata and Han proposed a combinatorial interpretation of these bibasic coefficients as enumerations of alternating permutations by the bi-statistic ( inv 1 , inv 2 ) . Under this interpretation, the symmetry of the bibasic trigonometric functions yields a combinatorial identity. A combinatorial proof of the identity is desired. For permutations of even order, this has already been given by Foata and Han. Here we give a proof for permutations of odd order.