[Paper Review] Counting faces of nestohedra

Vladimir Grujić, Tanja Stojadinović|arXiv (Cornell University)|Mar 26, 2017

Advanced Combinatorial Mathematics2 references3 citations

TL;DR

This paper presents a new algebraic formula for counting faces of nestohedra using a morphism from a combinatorial Hopf algebra of building sets to quasisymmetric functions. It introduces a $q$-analog $F_q(P_B)$, derives its recurrence relations, and shows that the $f$-polynomial of the nestohedron arises as the principal specialization of this quasisymmetric function, providing a deep algebraic characterization of face enumeration in nestohedra.

ABSTRACT

A new algebraic formula for the numbers of faces of nestohedra is obtained. The enumerator function $F(P_B)$ of positive lattice points in interiors of maximal cones of the normal fan of the nestohedron $P_B$ associated to a building set $B$ is described as a morphism from the certain combinatorial Hopf algebra of building sets to quasisymmetric functions. We define the $q$-analog $F_q(P_B)$ and derive its determining recurrence relations. The $f$-polynomial of the nestohedron $P_B$ appears as the principal specialization of the quasisymmetric function $F_q(P_B)$.

Motivation & Objective

• To develop an algebraic framework for enumerating faces of nestohedra using combinatorial Hopf algebras.
• To define a $q$-analog $F_q(P_B)$ of the face enumerator function for nestohedra.
• To derive recurrence relations that determine $F_q(P_B)$, enabling systematic computation of face counts.
• To establish a connection between the $f$-polynomial of a nestohedron and the principal specialization of a quasisymmetric function.

Proposed method

• The paper constructs a morphism from the combinatorial Hopf algebra of building sets to the algebra of quasisymmetric functions.
• It defines the enumerator function $F(P_B)$ as the generating function for positive lattice points in the interiors of maximal cones of the normal fan of $P_B$.
• The $q$-analog $F_q(P_B)$ is introduced to refine the face count, incorporating a parameter $q$ that tracks additional combinatorial structure.
• Recurrence relations are derived for $F_q(P_B)$ based on the structure of the building set $B$, enabling recursive computation.
• The $f$-polynomial of the nestohedron $P_B$ is identified as the principal specialization of $F_q(P_B)$, linking geometric and algebraic invariants.
• The framework leverages the algebraic properties of quasisymmetric functions to encode and compute face numbers systematically.

Experimental results

Research questions

• RQ1 How can the face enumeration of nestohedra be captured through an algebraic morphism from building set combinatorics to quasisymmetric functions?
• RQ2 What is the role of the $q$-analog $F_q(P_B)$ in refining the face count of nestohedra?
• RQ3 How do recurrence relations for $F_q(P_B)$ reflect the underlying combinatorial structure of the building set $B$?
• RQ4 In what way does the $f$-polynomial of $P_B$ emerge as a specialization of $F_q(P_B)$?
• RQ5 What is the significance of the normal fan's maximal cones in encoding face counts via lattice point enumeration?

Key findings

• The face enumerator $F(P_B)$ is realized as a morphism from the Hopf algebra of building sets to the algebra of quasisymmetric functions.
• The $q$-analog $F_q(P_B)$ is defined and shown to satisfy recurrence relations that depend on the structure of the building set $B$.
• The $f$-polynomial of the nestohedron $P_B$ is precisely the principal specialization of the quasisymmetric function $F_q(P_B)$, establishing a direct algebraic link to face enumeration.
• The framework provides a systematic method for computing face numbers of nestohedra using algebraic and combinatorial tools.
• The use of positive lattice points in the interiors of maximal cones of the normal fan enables a geometric interpretation of the face count via lattice point generating functions.
• The construction generalizes face enumeration in nestohedra by embedding it within the broader context of quasisymmetric functions and Hopf algebra structures.