Lexical tableaux and quasisymmetric functions

TL;DR

This work introduces lexical tableaux, a natural generalization of immaculate tableaux, and proves a fundamental bijection between standard lexical tableaux of shape α ⊧ n with ℓ(α)=k and permutations on {1,...,n} with k disjoint cycles, via necklace-word row constraints. It develops dual bases of \\textsf{QSym} and \\textsf{NSym} built from lexical tableaux, providing positive expansions of monomial and fundamental bases (and dual ribbon/complete homogeneous bases) through Kostka-like coefficients and related counts. The authors establish that the set {\\mathfrak{L}^*_{\\alpha}} forms a basis for \\textsf{QSym}, and define a corresponding lexical basis for \\textsf{NSym}, with explicit connections between H, R, and lexical expansions. They also derive cancellation-free antipode formulas for two-row lexical basis elements and illustrate the approach with concrete examples, while outlining several open problems and directions for further study. Overall, the paper broadens the interplay between tableau combinatorics, cyclic invariance, and Hopf-algebraic bases, offering new tools and questions for the study of quasisymmetric and noncommutative symmetric functions.

Abstract

There is a natural bijection between standard immaculate tableaux of composition shape $α\vDash n$ and length $\ell(α) = k$ 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 $α\vDash n$ and length $\ell(α) = k$ 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.

Theorems & Definitions (42)

• Example 2.1
• Example 2.2
• Theorem 3.1
• proof
• Definition 3.2
• Theorem 3.3
• proof
• Example 3.4
• Example 3.5
• Theorem 3.6