Trees, set compositions and the twisted descent algebra

2005/12/11 by Frédéric Patras, Patras, Frederic, Manfred Schocker +1 · 1 citation
Computer Science · Mathematics · #05A18 #05C05 #16W30 (Secondary) #17D99 (Primary) #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.math/0512227

openalex publication_date 2005/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We first show that increasing trees are in bijection with set compositions, extending simultaneously a recent result on trees due to Tonks and a classical result on increasing binary trees. We then consider algebraic structures on the linear span of set compositions (the twisted descent algebra). Among others, a number of enveloping algebra structures are introduced and studied in detail. For example, it is shown that the linear span of trees carries an enveloping algebra structure and embeds as such in an enveloping algebra of increasing trees. All our constructions arise naturally from the general theory of twisted Hopf algebras.

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• The noncommutative KP hierarchy and its solution via descent algebra

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