15 research outputs found
Dyck paths and pattern-avoiding matchings
How many matchings on the vertex set V={1,2,...,2n} avoid a given
configuration of three edges? Chen, Deng and Du have shown that the number of
matchings that avoid three nesting edges is equal to the number of matchings
avoiding three pairwise crossing edges. In this paper, we consider other
forbidden configurations of size three. We present a bijection between
matchings avoiding three crossing edges and matchings avoiding an edge nested
below two crossing edges. This bijection uses non-crossing pairs of Dyck paths
of length 2n as an intermediate step.
Apart from that, we give a bijection that maps matchings avoiding two nested
edges crossed by a third edge onto the matchings avoiding all configurations
from an infinite family, which contains the configuration consisting of three
crossing edges. We use this bijection to show that for matchings of size n>3,
it is easier to avoid three crossing edges than to avoid two nested edges
crossed by a third edge.
In this updated version of this paper, we add new references to papers that
have obtained analogous results in a different context.Comment: 18 pages, 4 figures, important references adde
The combinatorics of associated Hermite polynomials
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.Comment: [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.
Combinatoric