Forest Polynomials: Combinatorics & Algebra

• Forest polynomials are formal power series indexed by forests—disjoint unions of trees—that encode combinatorial and geometric invariants with positivity and total positivity properties.
• They are constructed from various frameworks, including spanning-forest polynomials, quasisymmetric refinements, and noncommutative Hopf algebra models, each offering unique algebraic insights.
• Applications span from Schubert calculus and symmetric functions to quantum physics, utilizing explicit combinatorial models, determinant identities, and continued fraction representations.

A forest polynomial is a polynomial or formal power series indexed by combinatorial structures called forests—disjoint unions of trees—arising across combinatorics, symmetric function theory, graph theory, and algebraic geometry. Multiple non-equivalent constructions bear the name "forest polynomial," tied to diverse but related algebraic, enumerative, and geometric frameworks, including spanning forests in graphs (notably in Feynman integral theory), quasisymmetric polynomials refining Schubert polynomial theory, descent generating functions, and noncommutative Hopf algebra realizations. Despite the variety, the central theme is the encoding of combinatorial or geometric invariants of forests within a coherent algebraic framework, often with positivity, basis, or total-positivity properties.

1. Combinatorial Definitions and Model Variants

Forest polynomials appear in several frameworks, with key variants including:

a. Graphical Spanning-Forest Polynomials

Given a finite undirected graph $G=(V,E)$ with indeterminates $a_e$ for each edge, and a set of marked vertices $S\subset V$ partitioned into blocks $P_1,\dots,P_k$, the spanning forest polynomial is

$$\Phi_G(P) = \sum_{F \text{ compatible with } P} \prod_{e\notin F} a_e,$$

where the sum is over spanning forests $F$ with $k$ components such that each block in $P$ is contained within a tree of $F$ (Fraser et al., 2021, Vlasev et al., 2011). Specializations recover the Kirchhoff and Symanzik polynomials fundamental in physics (Bogner et al., 2010).

b. Quasisymmetric Forest Polynomials

In the Nadeau–Tewari model (Guo et al., 3 Feb 2026, Nadeau et al., 2023, Nadeau et al., 2024), a forest polynomial is associated to a finite binary (or, more generally $(m+1)$-ary) indexed forest $a_e$0 with internal nodes $a_e$1, with a labeling function $a_e$2 subject to recursive inequalities determined by the tree structure. The quasisymmetric forest polynomial is

$a_e$3

which generalizes flagged $a_e$4-partitions and refines slide polynomials and Schubert basis (Nadeau et al., 2023).

c. Forest (Acyclic) Graph Polynomials

The acyclic or forest graph polynomial counts induced subgraphs that are forests. For a simple graph $a_e$5 with $a_e$6 vertices,

$a_e$7

where $a_e$8 is the number of $a_e$9-vertex induced forests (Makowsky et al., 2021). This form characterizes forest structure and real-rootedness in graphs.

d. Noncommutative and Hopf Algebra Forest Polynomials

Rooted forests can index noncommutative polynomials $S\subset V$0 in auxiliary alphabets $S\subset V$1, with products given by disjoint union and coproducts by admissible cuts, realizing combinatorial Hopf algebras (Foissy et al., 2010).

e. Descent Generating Forest Polynomials

Given a plane forest $S\subset V$2, the descent polynomial

$S\subset V$3

encodes the distribution of descents in labelings of $S\subset V$4 and generalizes the Eulerian polynomials (Grady et al., 2019).

2. Algebraic and Structural Properties

Forest polynomials exhibit robust structural characteristics:

• Positivity and Basis: Quasisymmetric forest polynomials form explicit $S\subset V$5-bases for polynomial rings and coinvariant algebras, refining Schubert polynomial theory via positive decomposition (Guo et al., 3 Feb 2026, Nadeau et al., 2023, Nadeau et al., 2024).
• Total Positivity: Matrices of classical or $S\subset V$6-forest polynomials have coefficientwise total positivity properties, provable via production matrix and Riordan array frameworks (Gilmore, 2021, Sokal, 2021, Pétréolle et al., 2019).
• Multiplicativity and Shuffle Product: Forest polynomial bases multiply positively, with explicit combinatorial interpretations via shuffles and insertion algorithms (generalized Sylvester correspondences) (Nadeau et al., 2023).
• Divided Difference and Operator Theory: In the quasisymmetric context, forest polynomials interact with divided difference operators indexed by forests, yielding canonical duality and basis extraction (Nadeau et al., 2024).

3. Connections to Schubert Calculus and Symmetric Functions

Forest polynomials serve as quasisymmetric analogues and refinements of Schubert and slide polynomials:

• Schubert polynomials $S\subset V$7 decompose positively into sums of forest polynomials, and a $S\subset V$8 is itself a (single) forest polynomial precisely when the corresponding permutation $S\subset V$9 avoids six specific patterns (Guo et al., 3 Feb 2026, Nadeau et al., 2023). This connects quasisymmetric function theory with the geometry of flag and permutahedral varieties.
• Forest polynomials expand positively into slide bases (Assaf–Searles) (Nadeau et al., 2023), and the expansion encodes the combinatorics of flagged $P_1,\dots,P_k$0-partitions, tableau models, and more generalized "zig-zag forests" associated to fundamental quasisymmetric functions (Nadeau et al., 2024).

4. Identities, Determinants, and Applications in Graph Theory and Physics

The original use of "forest polynomials" traces to spanning-forest generating functions, with determinant identities (e.g., matrix-tree theorem, Dodgson identities) underpinning structural results:

• Quadratic identities (e.g., for three or four marked vertices) among forest polynomials provide key relations in evaluating Feynman integrals, with the general column expansion identities determining the space of all such quadratic relations (Fraser et al., 2021, Vlasev et al., 2011).
• The all-minors matrix-tree theorem connects determinants of Laplacians to sum-over-forest forms, central to the computation of Kirchhoff and Symanzik polynomials in quantum field theory (Bogner et al., 2010).
• Laplacian coefficients of forests can be expressed in terms of closed walks in the graph and its line graph, extending classical interpretations such as the Wiener and hyper-Wiener indices (Ghalavand et al., 2021).

5. Total Positivity, Production Matrices, and Continued Fractions

Matrices encoding forests by size or component number, as well as polynomials counting forests with additional weights ($P_1,\dots,P_k$1-statistics, edge type, etc.), enjoy strong total positivity properties:

• Lower-triangular matrices of forest polynomials (e.g., classical, $P_1,\dots,P_k$2-analogues, multivariate Lah polynomials) are coefficientwise totally positive; row-generating polynomials are Hankel-totally positive (Gilmore, 2021, Sokal, 2021, Pétréolle et al., 2019).
• These total positivity properties are established by identifying explicit production matrices, frequently lower-Hessenberg with Toeplitz structure, whose powers enumerate forests (Pétréolle et al., 2019).
• Branched continued fraction representations for generating functions of forest polynomials arise via bijections with Łukasiewicz path models, further cementing the combinatorial basis of positivity (Pétréolle et al., 2019).

6. Enumerative, Real-Rootedness, and Unimodality Properties

Forest-related polynomials encode a wide range of enumerative invariants:

• The forest polynomial $P_1,\dots,P_k$3 of a graph $P_1,\dots,P_k$4 is real-rooted if and only if $P_1,\dots,P_k$5 is a forest (Makowsky et al., 2021); for forests, $P_1,\dots,P_k$6, giving strict control over coefficient distributions.
• Descent generating forest polynomials $P_1,\dots,P_k$7 are always symmetric (palindromic) and unimodal, but not necessarily log-concave or real-rooted, except in special cases (e.g., path forests) (Grady et al., 2019).
• The generalizations of Lah and Bell polynomials via forest polynomials (e.g., unordered forests of increasing ordered trees with variable weights) retain total positivity and Hankel positivity, yielding deep enumerative and algebraic consequences, including continued fraction representations (Pétréolle et al., 2019).

7. Open Problems and Frontiers

Current research continues to extend and generalize forest polynomials:

• There remain classification problems for when Schubert polynomials coincide with forest polynomials in other Coxeter types or in the context of double/quiver Schubert polynomials (Guo et al., 3 Feb 2026).
• The positivity, combinatorial, and algebraic properties of forest polynomials indexed by more complicated trees (e.g., $P_1,\dots,P_k$8-ary forests, non-plane trees) and their associated operator or coinvariant algebras are active areas (Nadeau et al., 2024, Nadeau et al., 2023).
• Multivariate generalizations and their total/Hankel-positivity are the subject of ongoing conjectures and combinatorial-model investigations (Pétréolle et al., 2019, Gilmore, 2021).
• Bijections and combinatorial proofs for higher-vertex Dodgson-type forest polynomial identities, connections to Landau singularities in Feynman integral theory, and explicit formulas for expansion coefficients in key bases represent central problems (Vlasev et al., 2011, Fraser et al., 2021).

Forest polynomials thus serve as a unifying structure bridging combinatorics, algebraic geometry, invariant theory, and mathematical physics, with their explicit combinatorial models, operator theory, and positivity properties forming the basis for ongoing research and applications.