Generating Functions for Domino Matchings in the $2\times k$ Game of Memory

Abstract

When all the elements of the multiset {1, 1, 2, 2, 3, 3, . . . , k, k} are placed in the cells of a 2 × k rectangular array, in how many configurations are exactly v of the pairs directly over top one another, and exactly h directly beside one another -thus forming 1 × 2 dominoes? We consider the sum of matching numbers over the graphs obtained by deleting h horizontal and v vertical vertex pairs from the 2 × k grid graph in all possible ways, providing a generating function for these aggregate matching polynomials. We use this result to derive a formal generating function enumerating the domino matchings, making connections with linear chord diagrams.