Enriched 𝑃-Partitions

Abstract

Abstract. An (ordinary) P -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's S-functions. In this paper, we introduce and develop a theory of enriched P -partitions; like ordinary P -partitions, these are order-preserving maps from posets to chains, but with different rules governing the occurrence of equal values. The principal examples of enriched P -partitions given here are the tableaux associated with Schur's Q-functions. In a sequel to this paper, further applications related to commutation monoids and reduced words in Coxeter groups will be presented. IntroductionThis is the first of a series of related papers on the combinatorics of reduced expressions in Coxeter groups, pattern avoidance, commutation monoids, and Ppartitions. Our initial motivation for studying the title subject arose from investigations of symmetric functions associated with Coxeter groups, when it became apparent that this investigation involved fundamental combinatorial structures of independent interest.In retrospect, perhaps the best way to introduce this subject is as the completion of an analogy. One of the motivations guiding the development of Stanley's theory of (ordinary) P -partitions 1 has been the combinatorics of semistandard tableaux associated with Schur's S-functions. Indeed, an (ordinary) P -partition can be viewed as a type of generalized tableau-a mapping that assigns entries to the elements of a poset-with various rules for specifying when equality of adjacent entries is allowed. On the other hand, although it did not exist at the time of Stanley's monograph [St1], there is a combinatorial theory for Schur's Q-functions that parallels in many ways the corresponding theory for Schur's S-functions. In particular, Schur's Q-functions are generating functions for a type of tableau attached to shifted Young diagrams, the entries being subjected to a different set of rules than those used for Schur's S-functions. Enriched P -partitions are the generalized "tableaux" one obtains by keeping these new rules, but replacing the shifted Young diagrams with arbitrary partially ordered sets.Almost every aspect of the theory of ordinary P -partitions has an enriched counterpart. For example, in the ordinary theory, the descent set {i : w i > w i+1 } of

On the Distribution of Surface Extrema in Several One- and Two-dimensional Random Landscapes

“…Second, combinatorics of peaks of permutations has been recently reviewed by J. Stembridge in Ref. [19]. The peak algebra of Stembridge is a certain sub-algebra of the algebra of polynomials with infinitely many variables.…”

A lift of Schur's Q-functions to the peak algebra

“…Remark 4.13. In [20], the author also gave the following refinement of Schur's Q-functions in terms of peak functions:…”

Polytopes, Fibonacci numbers, Hopf algebras, and quasi-symmetric functions

“…, a k } we have M S ′ = M (a1+1,...,a k −a k−1 , n−a k ) . Stembridge [28] showed that the Abelian group Π ⊂ Qsym generated by the functions θ S is a subring, and rk Π 2n = c n−1 . According to [55], the ring Π is a Hopf subalgebra.…”

“…The algebra Π was introduced in [28] for studying so-called enriched P -partitions. One can show (see [56], and see Appendix E) that the algebra L * (D * ) ⊗ Q = R * (D * ) ⊗ Q ⊂ Qsym ⊗ Q defined by the conditions of Corollary 7.7 coincides with the Stembridge algebra Π ⊗ Q.…”

“…The proof of Proposition 1.83 can be obtained by following [17]. The difference in our proofs is that Bayer and Billera pass to a new matrix of face numbers of polytopes Q ∈ Ψ n , while we work directly with the flag numbers, using (28) and Lemma 1.84. In [17] the fact (important for us) that det K n = 1 was not pointed out.…”