Matchings Avoiding Partial Patterns and Lattice Paths

Vít Jelínek, Nelson Y. Li, Toufik Mansour, Sherry H. F. Yan

Abstract


In this paper, we consider matchings avoiding partial patterns $1123$ and $1132$. We give a bijection between $1123$-avoiding matchings with $n$ edges and nonnegative lattice paths from $(0,2)$ to $(2n,0)$. As a consequence, the refined enumeration of $1123$-avoiding matchings can be reduced to the enumeration of certain lattice paths. Another result of this paper is a bijection between $1132$-avoiding matchings with $n$ edges and lattice paths from $(0,0)$ to $(2n,0)$ starting with an up step, which may go under the $x$-axis.


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