Matchings avoiding partial patterns and lattice paths (Q870005)

Summary: 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.