Polynomial Invariant on Rigid Boolean Functions

• The paper introduces a universal polynomial invariant derived from the double bialgebra structure on rigid boolean functions, generalizing chromatic polynomials to hypergraphs and matroids.
• It establishes a rigorous contraction–restriction coproduct and a convolutional recursion that underpin deletion–contraction expansions for computing these invariants.
• The approach has significant applications in graph theory, network reliability, and stochastic analysis, while revealing the intrinsic #P-hard complexity of evaluating the invariant.

A polynomial invariant on rigid boolean functions is a universal combinatorial polynomial associated to certain set functions defined on the power set of a finite set, known as rigid boolean functions. This invariant, arising from the structure of a double bialgebra, generalizes the chromatic polynomial of graphs to a much broader setting including hypergraphs and matroid rank functions. The construction is rooted in the theory of combinatorial species, bialgebras, and the interplay of two compatible coproducts, with significant connections to classical graph and matroid invariants (Foissy, 20 Jan 2026).

1. Boolean Functions and Twisted Bialgebra Structures

For any finite set $X$, a boolean function is a map $f:P(X)\to\mathbb{Z}$ satisfying $f(\emptyset)=0$. The collection $\mathrm{Bool}(X)$ forms a combinatorial species under relabeling by bijections. A two-parameter family of products $*_{q_1,q_2}$ is defined on $\mathrm{Bool}$: $$(f *_{q_1,q_2} g)(A) = q_1 f(A \cap X) + q_2 g(A \cap Y),$$ for $f\in \mathrm{Bool}(X)$, $g\in \mathrm{Bool}(Y)$, and $A\subseteq X\sqcup Y$ with $X\cap Y = \emptyset$. This product is associative and unital; it is commutative precisely when $q_1 = q_2$.

The primary coproduct is the restriction coproduct: $$\Delta(f) = \sum_{X = I \sqcup J} f|_I \otimes f|_J,$$ where $f|_I(A) = f(A \cap I)$. Combined, $(\mathrm{Bool}, *_{q_1, q_2}, \Delta)$ is a twisted bialgebra; passage to the bosonic Fock functor linearizes this into a genuine bialgebra for further study. The standard commutative case uses $q_1 = q_2 = 1$ and the product $*$ (Foissy, 20 Jan 2026).

2. Contraction–Restriction Coproduct and the Rigidity Obstruction

Attempting to define a second coproduct on boolean functions involves the operation of contraction with respect to an equivalence relation $\sim$ on $X$. Given $f\in\mathrm{Bool}(X)$, define: $$f/\!\sim (A) = f(\pi^{-1}(A)), \qquad f|_{\sim}(A) = \sum_{Y \in X/\!\sim} f(A \cap Y),$$ where $\pi: X \to X/\!\sim$ is the canonical quotient map. A natural candidate for a second coproduct is: $$\delta(f) = \sum_{\sim \in \mathcal{E}(f)} f/\!\sim \otimes f|_{\sim},$$ for some family of equivalences $\mathcal{E}(f)$. However, no choice of $\mathcal{E}(\cdot)$ on the full species $\mathrm{Bool}$ ensures the required bialgebra axioms: compatibility with $*$, $\Delta$, counit, and coassociativity. This necessitates restricting attention to a maximal subspecies $\mathrm{Bool}_{\max} \subset \mathrm{Bool}$ where these conditions hold.

3. Rigid Boolean Functions and the Double Bialgebra

A boolean function $f \in \mathrm{Bool}(X)$ is called indecomposable if it cannot be written nontrivially as $f = g * h$ for $X = Y \sqcup Z$. Its maximal factorization into indecomposables defines a canonical equivalence on $X$. The function $f$ is rigid if for all disjoint $A, B \subset X$,

$$f(A \cup B) = f(A) + f(B) \implies f|_{A\cup B} = f|_A * f|_B.$$

This property forces additive splittings of $f$ to correspond to true decompositions of $X$. Rigid boolean functions form the subspecies $\mathrm{Bool}_r$. On this subspecies, the families of weak and strong equivalences coincide, and a well-behaved contraction–restriction coproduct $\delta$ exists: $$\delta: \mathrm{Bool}_r \to \bigoplus_{X = I \sqcup J} \mathrm{Bool}_r(I) \otimes \mathrm{Bool}_r(J).$$ After Fock linearization, $(H_{\mathrm{Bool}_r}, *, \Delta, \delta)$ constitutes a connected double bialgebra (Foissy, 20 Jan 2026).

4. The Polynomial Invariant $\Phi_f(T)$ and Recursion

Within the structure of the connected double bialgebra, rigid boolean functions admit a unique double-bialgebra morphism into the classical binomial Hopf algebra $\mathbb{K}[T]$: $\Phi: H_{\mathrm{Bool}_r} \to \mathbb{K}[T]$ with

$\Phi(f)(1) = \varepsilon_\delta(f),$

where $\varepsilon_\delta$ is the counit associated to modular functions.

The polynomial $\Phi_f(T)$ of $f \in \mathrm{Bool}_r(X)$ satisfies a convolutional recursion: $$\Phi_{f}(T+S) = \sum_{\sim \in \mathcal{E}(f)} \Phi_{f/\!\sim}(T) \cdot \Phi_{f|_{\sim}}(S),$$ along with an expansion: $$\Phi_f(T) = \sum_{k \geq 1} \frac{1}{k!} \left((\Phi - \varepsilon_\delta)^{*k}(f)\right) T(T-1) \cdots (T-k+1).$$ Combinatorially,

$$\Phi_f(n) = \#\left\{ c:X \to \{1, \ldots, n\} \mid \forall i,\, f(c^{-1}(i)) \text{ is modular} \right\}$$

for $n \in \mathbb{N}_{>0}$, making $\Phi_f(T)$ a genuine polynomial of degree $|X|$ with integer coefficients (Foissy, 20 Jan 2026).

5. Relation to Chromatic and Tutte Polynomials

The invariant $\Phi_f(T)$ subsumes prominent combinatorial polynomials. For a hypergraph $H$ with indicator function $y(H)$,

$y(H)(A) = \#\{e \in E(H): e \subseteq A\},$

one has

$$\Phi_{y(H)}(n) = \text{chromatic polynomial of } H,$$

counting proper vertex colorings with $n$ colors such that no hyperedge of size at least $2$ is monochromatic.

For ordinary graphs, viewing $G$ as a $2$-uniform hypergraph recovers the classical chromatic polynomial of $G$. For matroids, if $f(A)$ is the matroid rank function, then

$$\Phi_f(n) = \#\{c:X \to [n]: \forall i,\, \{v_x: c(x)=i\} \text{ is independent}\},$$

which counts colorings with the independent set criterion for each fiber. The formalism also encompasses graphical matroids, yielding coloring counts respecting forest constraints on the edge set. Moreover, a bivariate refinement of $\Phi_f(T)$ enables recovery of the Tutte polynomial for matroids by encoding deletion and contraction through two variables (Foissy, 20 Jan 2026).

6. Computational Complexity and Applications

Evaluating $\Phi_f(n)$ is $\#P$–hard even in the case of graphs, reflecting the intrinsic complexity of underlying coloring and partition-counting problems. Nevertheless, the double bialgebraic approach yields deletion–contraction type recursions and expansions (notably the broken-circuit expansion) which are central to both theoretical understanding and algorithmic analysis. Applications of these invariants extend to graph and hypergraph coloring, network reliability assessment, moment–cumulant relationships in probability (via species duality), and the combinatorics of regularity structures for stochastic partial differential equations (Foissy, 20 Jan 2026).

7. Universality and Invariant Properties

The polynomial invariant $\Phi_f(T)$ is universal among invariants factoring through the double bialgebra $(H_{\mathrm{Bool}_r}, *, \Delta, \delta)$ into the binomial Hopf algebra. This universality ensures that $\Phi_f(T)$ captures the combinatorial essence of the class of rigid boolean functions, and classical polynomials such as the chromatic and Tutte polynomials are obtained as special cases. The construction provides a conceptual unification of these invariants within a double bialgebraic framework (Foissy, 20 Jan 2026).