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Bijections for lattice paths between two boundaries

Abstract : We prove that on the set of lattice paths with steps $N=(0,1)$ and $E=(1,0)$ that lie between two boundaries $B$ and $T$, the two statistics `number of $E$ steps shared with $B$' and `number of $E$ steps shared with $T$' have a symmetric joint distribution. We give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps $S=(0,-1)$ at prescribed $x$-coordinates. We also show that a similar equidistribution result for other path statistics follows from the fact that the Tutte polynomial of a matroid is independent of the order of its ground set. Finally, we extend the two theorems to $k$-tuples of paths between two boundaries, and we give some applications to Dyck paths, generalizing a result of Deutsch, and to pattern-avoiding permutations.
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Sergi Elizalde, Martin Rubey. Bijections for lattice paths between two boundaries. 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), 2012, Nagoya, Japan. pp.827-838. ⟨hal-01283135⟩

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