Lexical tableaux and quasisymmetric functions
John M. Campbell, Spencer Daugherty

TL;DR
This paper introduces lexical tableaux, a generalization of immaculate tableaux, establishing bijections with permutations and set-partitions, and constructs dual bases of Hopf algebras extsf{QSym} and extsf{NSym} based on these tableaux.
Contribution
It defines lexical tableaux, explores their combinatorial properties, and develops dual bases of Hopf algebras using these tableaux, extending the theory of immaculate tableaux.
Findings
Bijection between lexical tableaux and permutations with k disjoint cycles.
Introduction of dual bases of extsf{QSym} and extsf{NSym} using lexical tableaux.
Expansion formulas involving Kostka-like coefficients.
Abstract
There is a natural bijection between standard immaculate tableaux of composition shape and length and the set-partitions of into blocks, for the Stirling number of the second kind. We introduce a family of tableaux that we refer to as \emph{lexical tableaux} that generalize immaculate tableaux in such a way that there is a bijection between standard lexical tableaux of shape and length and the permutations on with disjoint cycles. In addition to the entries in the first column strictly increasing, the defining characteristic of lexical tableaux is that the word …
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algebraic structures and combinatorial models · Advanced Algebra and Geometry
