$k$-Pairing: A Generalization of the Partition Pairing Theorems

Source abstract

We extend the partition pairing theory of Andrews and Dastidar by replacing pairs with groups of $k$ equal parts. Two weight-preserving bijections give combinatorial interpretations of the joint pairing index--width distribution and the negative-rank enumerations. The first combines conjugation with the Stockhofe--Keith correspondence and sends the $k$-pairing index and width to the number of parts and the largest part, respectively. Consequently, their joint distribution is independent of $k$ and is given by a Gaussian polynomial. We also obtain finite refinements that record the residual statistics. The second bijection sends simply $k$-paired partitions of negative $k$-pairing rank to $k$-regular partitions with marked internal gaps. In a fixed nonzero residue class, sign cancellation leaves rectangular partitions, while gap markings correspond to overlining choices with the smallest part not overlined. This gives direct combinatorial explanations of the divisor counts and the factor $1/2$ in the overpartition enumeration, extending the odd-divisor and odd-overpartition results of Andrews and Dastidar. Finally, motivated by the diagonal pairing of Andrews and Dastidar, we extend the construction to ordered tuples of $k$ Young diagrams. Applying the two-wing transfer operation to pairs of component diagrams defines an equivalence relation on these tuples. We show that two tuples are equivalent if and only if they have the same cell-multiplicity function, and that every equivalence class contains a unique representative whose component diagrams are nested. We also determine the cardinalities of the equivalence classes and identify the nested representatives with plane partitions of rectangular shape.