The rook polynomials of almost symmetric Arf numerical semigroups

In this study, we aim to introduce a new area of application for researchers in the field by exploring the relationship between almost symmetric Arf semigroups and rook polynomials through Young diagrams. The study addresses concepts such as numerical sets, numerical semigroups, partitions of positive integers, rook polynomials, and Young diagrams. It is well established in the literature that there are one-to-one correspondences among the sets of numerical sets, integer partitions, and Young diagrams. We utilized these correspondences to express the rook polynomials of the Young diagrams corresponding to almost symmetric Arf semigroups. It is known that various methods, such as the cell decomposition algorithm and the block decomposition algorithm, are used in the literature for the calculation of rook polynomials. In this study, we applied the block decomposition algorithm to calculate the rook polynomials of Young diagrams corresponding to almost symmetric Arf semigroups and presented the results we obtained. As a result, we determined the rook polynomials of the Young diagrams corresponding to almost symmetric Arf semigroups. We also obtained the rook polynomials of the complements of these Young diagrams.