Enriched P-partitions and peak algebras

By T. Kyle Petersen. 2005

104 citations recorded by the source corpus.

Abstract We develop a more general view of Stembridge's enriched P -partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group. Initially we focus on commutative peak algebras, spanned by sums of permutations with the same number of peaks, where we consider several variations on the definition of “peak.” Whereas Stembridge's enriched P -partitions are related to quasisymmetric functions (the dual coalgebra of Solomon's type A descent algebra), our generalized enriched P -partitions are related to type B quasisymmetric functions (the dual coalgebra of Solomon's type B descent algebra). Using these functions, we move on to explore (non-commutative) peak algebras spanned by sums of permutations with the same set of peaks. While some of these algebras have been studied before, our approach gives explicit structure constants with a combinatorial description.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5dcd6b8cd872b594adb78a2c03728b3f78c6699b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Recent Trends in Quasisymmetric Functions (2018, background): “A number of very interesting subtopics are therefore excluded from this article, including Stembridge’s subalgebra of peak quasisymmetric functions [122] and its associated structure ([16], [17], [19], [20], [34], [81]), Ehrenborg’s flag quasisymmetric function of a partially ordered set [34], color”
Permutations with given peak set (2012, background): “There has been a great deal of research into peaks of permutations, see [1, 2, 4, 6, 9, 10, 11, 12, 13, 15, 16, 17].”
Gamma-positivity in combinatorics and geometry (2017, background): “ng result combines [39, Theorem 4.7] with [94, Proposition 4.15] and provides a Bn-analog to Theorem 2.1. The permutation obtained from w∈ Bn by forgetting all signs is denoted by |w|. Theorem 2.10. ([39, 94]) For all n≥ 1, (19) Bn(x) = ⌊Xn/2⌋ i=0 γB n,i x i(1+x)n−2i, where γB n,i is equal to each o”