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Nestings of Matchings and Permutations and North Steps in PDSAWs

Abstract : We present a simple bijective proof of the fact that matchings of $[2n]$ with N nestings are equinumerous to $\textit{partially directed self avoiding walks}$ confined to the symmetric wedge defined by $y= \pm x$, with $n$ east steps and $N$ north steps. A very similar construction connects permutations with $N$ nestings and $\textit{PDSAWs}$ remaining below the $x$-axis, again with $N$ north steps. Furthermore, both bijections transport several combinatorially meaningful parameters.
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Martin Rubey. Nestings of Matchings and Permutations and North Steps in PDSAWs. 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.691-704. ⟨hal-01185145⟩

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