Enriched 𝑃-Partitions

An (ordinary)PP-partition is an order-preserving map from a partially ordered set to a chain, with special rules specifying where equal values may occur. Examples include number-theoretic partitions (ordered and unordered, strict or unrestricted), plane partitions, and the semistandard tableaux associated with Schur’sSS-functions. In this paper, we introduce and develop a theory of enrichedPP-partitions; like ordinaryPP-partitions, these are order-preserving maps from posets to chains, but with different rules governing the occurrence of equal values. The principal examples of enrichedPP-partitions given here are the tableaux associated with Schur’sQQ-functions. In a sequel to this paper, further applications related to commutation monoids and reduced words in Coxeter groups will be presented.