The combinatorics of associated Hermite polynomials
Abstract
Description
We develop a combinatorial model of the associated Hermite polynomials and their moments, and prove their orthogonality with a sign-reversing involution. We find combinatorial interpretations of the moments as complete matchings, connected complete matchings, oscillating tableaux, and rooted maps and show weight-preserving bijections between these objects. Several identities, linearization formulas, the moment generating function, and a second combinatorial model are also derived.
[v1]: 18 pages, 16 figures; presented at FPSAC 2007 [v2]: Some minor errors fixed (thanks Bill Chen, Jang Soo Kim) and text rearranged and cleaned up; no real content changes [v3]: fixed typos, to appear in European J. Combinatorics
[v1]: 18 pages, 16 figures; presented at FPSAC 2007 [v2]: Some minor errors fixed (thanks Bill Chen, Jang Soo Kim) and text rearranged and cleaned up; no real content changes [v3]: fixed typos, to appear in European J. Combinatorics