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View CartSIAM J. Discrete Math. 20, pp. 960-976 (17 pages)
On Identities Concerning the Numbers of Crossings and Nestings of Two Edges in Matchings
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Add to FacebookLet $M,N$ be two matchings on $[2n]$ (possibly $M=N$). For an integer $l\ge 0$, let ${\cal T}(M,l)$ be the set of those matchings on $[2n+2l]$ which can be obtained from $M$ by successively adding $l$ times the first edge, and similarly for ${\cal T}(N,l)$. Let $s,t\in\{cr,ne\}$, where $cr$ is the statistic of the number of crossings in a matching and $ne$ is the statistic of the number of nestings (possibly $s=t$). We prove that if the statistics $s$ and $t$ coincide, respectively, on the sets of matchings ${\cal T}(M,l)$ and ${\cal T}(N,l)$ for $l=0,1$, then they coincide on these sets for every $l\ge 0$; similar identities hold for the joint statistic of $cr$ and $ne$. These results are instances of a general identity for crossings and nestings weighted by elements from an abelian group.
© 2006 Society for Industrial and Applied Mathematics
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Received February 25, 2005
Accepted June 05, 2006
Published online December 11, 2006
Accepted June 05, 2006
Published online December 11, 2006
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