Effective Scalar Products for D-finite Symmetric Functions

Details Effective Scalar Products for D-finite Symmetric Functions Effective Scalar Products for D-finite Symmetric Functions

2003-10-09

arxiv.org/abs/math/0310132v2

Journal of Combinatorial Theory Series A, Volume 112 , Issue 1 (October 2005), Pages: 1 - 43

Mathematics

Combinatorics

51 pages, full paper version of FPSAC 02 extended abstract; v2: corrections from original submission, improved clarity; now fo

Scientific paper

10.1016/j.jcta.2005.01.001

Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as sub-series and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are D-finite. We extend Gessel's work by providing algorithms that compute differential equations these generating functions satisfy in the case they are given as a scalar product of symmetric functions in Gessel's class. Examples of applications to k-regular graphs and Young tableaux with repeated entries are given. Asymptotic estimates are a natural application of our method, which we illustrate on the same model of Young tableaux. We also derive a seemingly new formula for the Kronecker product of the sum of Schur functions with itself.

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