King and domino polynomials for polyomino graphs

Abstract

For the purposes of treating several enumeration problems of lattice dynamics, king and domino polynomials are defined for a chessboard, polyomino, or square lattice of arbitrary size and shape. These polynomials are shown to be closely related to the partition function of the dimer statistics, the number of Kekulé structures, or maximum matching number. Several recursion formulas are found. Interpretation of these newly proposed quantities is given, and the possibility of extending them to the important physical models is discussed.