Octabasic Laguerre polynomials and permutation statistics
A set of orthogonal polynomials is studied which depends on ``eight independent \(q\)'s.'' If all ``eight \(q\)'s'' are set equal to 1, these orthogonal polynomials specialize to Laguerre polynomials. Using Viennot's combinatorial theory of orthogonal polynomials, it is shown that the moments for these polynomials can be expressed combinatorially, in several different ways, as certain generating functions for permutations. This opens the path for a study of these polynomials in terms of statistics for various combinatorial objects. It is this what the remainder of the paper is devoted to. Specializations of the octabasic orthogonal polynomials and their combinatorics are systematically studied. These specializations include numerous classical families of orthogonal polynomials (besides Laguerre polynomials) and related \(q\)-analogues. The combinatorics involves permutations, of course, set partitions, noncrossing partitions, involutions, etc., and various statistics on these objects. Particularly attractive and useful are the many links that are worked out, again in a systematic manner, between the various combinatorial objects.
scientific article; zbMATH DE number 908397
scientific article