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Linear function of a poset

Stefan Mitrovic

TL;DR

This work defines a new symmetric function $L_P$ for posets via a combinatorial Hopf algebra whose character counts linear extensions, linking it to Euler-type invariants of a generalized Rota framework. It analyzes expansions of $L_P$ in bases of $QSym$ and $Sym$, providing combinatorial interpretations for expansion coefficients through border-point structures and monotone colorings, and establishes a decomposition of posets into mountains via a multiplicative map $ abla$ that intertwines with Rev$(P)$. The results yield both positivity statements for special poset classes (e.g., $(2+2)$-free and $(2+1)$-free) and a framework of questions about border-point data, reversibility, and automorphism information, suggesting rich connections between poset combinatorics and symmetric-function theory. These contributions advance understanding of how poset structure governs symmetric-function expansions and offer a concrete decomposition mechanism with potential applications in algebraic combinatorics and related fields.

Abstract

Stanley and Grinberg introduced a symmetric function associated with digraphs and named it the Redei-Berge symmetric function. This function arises from a suitable combinatorial Hopf algebra on digraphs, which made it possible to assign the Redei-Berge function to posets. In this paper, we define a new combinatorial Hopf algebra of posets whose character is a close cousin of the Redei-Berge character for posets. Further, we investigate the properties of the symmetric function that arises from this algebra and explore its expansions in various natural bases of $QSym$ and $Sym$. Finally, we obtain an interesting method for decomposing a poset.

Linear function of a poset

TL;DR

This work defines a new symmetric function for posets via a combinatorial Hopf algebra whose character counts linear extensions, linking it to Euler-type invariants of a generalized Rota framework. It analyzes expansions of in bases of and , providing combinatorial interpretations for expansion coefficients through border-point structures and monotone colorings, and establishes a decomposition of posets into mountains via a multiplicative map that intertwines with Rev. The results yield both positivity statements for special poset classes (e.g., -free and -free) and a framework of questions about border-point data, reversibility, and automorphism information, suggesting rich connections between poset combinatorics and symmetric-function theory. These contributions advance understanding of how poset structure governs symmetric-function expansions and offer a concrete decomposition mechanism with potential applications in algebraic combinatorics and related fields.

Abstract

Stanley and Grinberg introduced a symmetric function associated with digraphs and named it the Redei-Berge symmetric function. This function arises from a suitable combinatorial Hopf algebra on digraphs, which made it possible to assign the Redei-Berge function to posets. In this paper, we define a new combinatorial Hopf algebra of posets whose character is a close cousin of the Redei-Berge character for posets. Further, we investigate the properties of the symmetric function that arises from this algebra and explore its expansions in various natural bases of and . Finally, we obtain an interesting method for decomposing a poset.
Paper Structure (10 sections, 25 theorems, 70 equations)

This paper contains 10 sections, 25 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalization of Rota's Hopf algebra
  4. Sets of border points
  5. Linear function of a poset
  6. Expansions of $L_P$
  7. Decomposition of a poset
  8. Further avenues
  9. Declarations and statements
  10. Data availability

Key Result

Lemma 3.3

Let $\mathcal{A}$ be a plucking with minimal sets $M_1, \ldots, M_k$ and $S\in\mathcal{A}$. If there exists $i\in S\setminus\bigcup_{i=1}^{k}M_i$, then $\chi_\mathcal{A}(S)=0$.

Theorems & Definitions (57)

  • Example 3.1
  • Example 3.2
  • Lemma 3.3
  • proof
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • Example 3.7
  • Lemma 3.8
  • Definition 4.1
  • ...and 47 more