Lexical tableaux and quasisymmetric functions

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 $\alpha \vDash n$ and length $\ell(\alpha) = k$ and the $ \left\{ \begin{smallmatrix} n \\ k \end{smallmatrix} \right\} $ set-partitions of $\{ 1, 2, \ldots, n \}$ into $k$ blocks, for the Stirling number $ \left\{ \begin{smallmatrix} n \\ k \end{smallmatrix} \right\} $ 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 $\alpha \vDash n$ and length $\ell(\alpha) = k$ and the $ \left[ \begin{smallmatrix} n \\ k \end{smallmatrix} \right] $ permutations on $\{ 1, 2, \ldots, n \}$ with $k$ disjoint cycles. In addition to the entries in the first column strictly increasing, the defining characteristic of lexical tableaux is that the word $w$ formed by the consecutive labels in any row is the lexicographically smallest out of all cyclic permutations of $w$. This includes weakly increasing words, and thus, lexical tableaux provide a natural generalization of immaculate tableaux. Extending this generalization, we introduce a pair of dual bases of the Hopf algebras $\textsf{QSym}$ and $\textsf{NSym}$ defined in terms of lexical tableaux. We present two expansions of these bases, involving the monomial and fundamental bases (or, dually, the ribbon and complete homogeneous bases), using Kostka coefficient analogues and coefficients derived from standard lexical tableaux.