Computations On P-Partitions

In the early 1900s, Major Percy A. MacMahon developed Partition Analysis (i.e., the Omega operator) as a computational method for solving problems in connection with linear diophantine inequalities and equations. In 1972 Richard P. Stanley introduced P-partition as a common generalization of compositions and partitions. Partition Analysis has been given a new life by Gorge E. Andrews and showed the relevance in the current partition-theoretic research since 1997.Using the inclusion-exclusion principle, we derive a formula of generating functions for Stanley's P-partitions of the ordinal sum of posets. This formula simplifies the computations for many variations of plane partitions, such as plane partition polygons and plane partitions with diagonals or double diagonals. We illustrate the method by several examples, some of which are new variations of plane partitions.A k-gon partition is a non-decreasing sequence of k positive integers such that the last element is less than the sum of the others. By considering k-gon partitions, we derive the multivariable generating function for non-k-gon partitions, as given by Andrews, Paule and Riese. In addition, we provide an C++ package for Omega operator which is faster than E112 package.