Rank Generating Polynomials in Semimatroid Theory

• Rank generating polynomials of semimatroids are two-variable invariants that encode corank deficits and dependency excesses in central sets.
• They extend classical Tutte polynomial formulations by incorporating semimatroid axioms and affine hyperplane arrangements with recursive deletion–contraction rules.
• The active basis expansion and convolution identities provide combinatorial and topological insights, linking polynomial coefficients to Betti numbers and broken circuits.

Searching arXiv for recent and foundational papers on semimatroid rank generating polynomials, Tutte polynomials, and related topology. Rank generating polynomials of semimatroids are Tutte-type invariants that encode rank deficiency and dependency excess for central sets in a semimatroid, thereby extending the classical rank generating polynomials of matroids and graphs to the affine and semimatroidal setting. In the recent semimatroid literature, the central two-variable invariant is

$$R(\mathcal{C};\lambda,x):=\sum_{A\in\mathcal{C}}\lambda^{\,r(\mathcal{C})-r_{\mathcal{C}}(A)}x^{\,|A|-r_{\mathcal{C}}(A)},$$

while earlier work developed a topological rank-function generating polynomial whose coefficients are relative Betti numbers and whose evaluations are controlled by the semimatroid Tutte polynomial. Together these constructions show that semimatroid rank enumerators are simultaneously combinatorial, recursive, activity-theoretic, and topological objects (Fu, 1 Aug 2025, White, 2012, Houshan, 8 Jun 2025).

1. Semimatroids as the ambient structure

A semimatroid is presented in two equivalent notational forms in the cited literature: as a triple $\mathcal{C}=(S,\Delta,r)$, where $\Delta$ is a nonvoid simplicial complex on a finite set $S$, or as $(E,\mathcal{C},r_{\mathcal{C}})$, where $\mathcal{C}$ is a nonempty simplicial complex on a finite ground set $E$ and its members are called central sets. In both conventions, the rank function satisfies semimatroid axioms: $0\le r(X)\le |X|$, monotonicity under inclusion, submodularity when unions remain central, a closure-type condition when ranks agree on intersections, and an exchange-type condition when ranks increase. The maximal central sets all have the same rank, denoted $r(\mathcal{C})$ in the 2025 notation (Fu, 1 Aug 2025).

The standard semimatroid notions parallel matroid theory, but only inside the central-set complex. A set $X\in\mathcal{C}$ is independent if $\mathcal{C}=(S,\Delta,r)$0, maximal independent sets are bases, minimal dependent sets are circuits, loops are circuits of size $\mathcal{C}=(S,\Delta,r)$1, and bridges are elements contained in every basis. There is also a closure operator

$\mathcal{C}=(S,\Delta,r)$2

and the flats form a geometric semilattice $\mathcal{C}=(S,\Delta,r)$3 (Houshan, 8 Jun 2025).

The motivating geometric example is an affine hyperplane arrangement. If $\mathcal{C}=(S,\Delta,r)$4 is the family of intersecting subcollections and

$\mathcal{C}=(S,\Delta,r)$5

then $\mathcal{C}=(S,\Delta,r)$6 is a semimatroid. This places semimatroids between matroidal combinatorics and arrangement theory, with centrality replacing unrestricted subset summation (White, 2012).

2. Exact definition of the rank generating polynomial

The semimatroid rank generating polynomial is defined by

$\mathcal{C}=(S,\Delta,r)$7

It is obtained by summing only over central sets, with the exponent of $\mathcal{C}=(S,\Delta,r)$8 recording corank and the exponent of $\mathcal{C}=(S,\Delta,r)$9 recording nullity or dependency excess. In the terminology of the 2025 development, $\Delta$0 is the corank deficit and $\Delta$1 is the dependency excess (Fu, 1 Aug 2025).

This polynomial is not introduced in isolation. The same paper defines the size-corank polynomial

$\Delta$2

and then relates the two by

$\Delta$3

A recurrent point in the literature is therefore that $\Delta$4 is a compressed two-variable form of a broader semimatroid Tutte theory rather than an isolated invariant (Fu, 1 Aug 2025).

The usual semimatroid Tutte polynomial is

$\Delta$5

and the two invariants are related by the standard shift

$\Delta$6

Likewise, the characteristic polynomial

$\Delta$7

is recovered by the specialization

$\Delta$8

These identities place the rank generating polynomial exactly where it sits in matroid theory: as the shifted Tutte polynomial and a parent for characteristic-type specializations (Fu, 1 Aug 2025, Houshan, 8 Jun 2025).

3. Recursive structure and basis-activity expansion

The rank generating polynomial satisfies a deletion–contraction recurrence with the same case distinction as the semimatroid Tutte polynomial, after the appropriate variable shift: $\Delta$9 The presence of the final case is specific to semimatroids: the ground set may contain elements that are not central singletons, so centrality must be tracked explicitly (Fu, 1 Aug 2025).

Two special vanishing properties are particularly useful in later convolution identities: $S$0 and

$S$1

These conditions force restrictions to flats or cyclic flats after specialization, exactly as in matroid convolution formulas (Fu, 1 Aug 2025).

The activity expansion is the direct semimatroid analogue of Crapo’s basis expansion. Fix a linear order $S$2 on $S$3. For a basis $S$4, if $S$5 and $S$6, then the fundamental circuit $S$7 is the unique circuit contained in $S$8, and $S$9 is externally active if it is the minimum element of $(E,\mathcal{C},r_{\mathcal{C}})$0. For $(E,\mathcal{C},r_{\mathcal{C}})$1, the fundamental cocircuit $(E,\mathcal{C},r_{\mathcal{C}})$2 is the unique cocircuit contained in $(E,\mathcal{C},r_{\mathcal{C}})$3, and $(E,\mathcal{C},r_{\mathcal{C}})$4 is internally active if it is the minimum element of $(E,\mathcal{C},r_{\mathcal{C}})$5. Writing $(E,\mathcal{C},r_{\mathcal{C}})$6 and $(E,\mathcal{C},r_{\mathcal{C}})$7 for the externally and internally active sets, one has

$(E,\mathcal{C},r_{\mathcal{C}})$8

Accordingly, the coefficients are nonnegative integers when expanded in the shifted basis $(E,\mathcal{C},r_{\mathcal{C}})$9 and $\mathcal{C}$0, and they count bases according to internal and external activity statistics (Fu, 1 Aug 2025).

4. Convolution identities and polynomial interrelations

A central structural result is the semimatroid convolution identity

$\mathcal{C}$1

The same source also records the specialization

$\mathcal{C}$2

and the flat-restricted form

$\mathcal{C}$3

The restriction to flats is explained by the vanishing criterion: if $\mathcal{C}$4 is not a flat then $\mathcal{C}$5 has a loop, so the specialized factor at $\mathcal{C}$6 vanishes (Fu, 1 Aug 2025).

These formulas extend Kung’s convolution-multiplication identities from matroids to semimatroids. In the same 2025 program, they are situated alongside semimatroid versions of the Kook–Reiner–Stanton framework and alongside analogous convolution identities for the characteristic and Tutte polynomials. For example, one has

$\mathcal{C}$7

for loopless semimatroids, and

$\mathcal{C}$8

This suggests that semimatroid rank generating polynomials should be viewed as part of a closed convolutional calculus rather than as isolated enumerators (Houshan, 8 Jun 2025).

The same interrelations connect $\mathcal{C}$9 to the multivariate Tutte and dichromatic polynomials. The multivariate Tutte polynomial is

$E$0

the subset-corank polynomial is $E$1, and the dichromatic polynomial is the specialization

$E$2

In this hierarchy, $E$3 is the clean two-variable refinement obtained from size-corank by change of variables (Fu, 1 Aug 2025).

5. Topological rank-function generating polynomials

Earlier work on semimatroids studied a different but closely related generating object: the Poincaré polynomial of a monotone function

$E$4

where $E$5. For semimatroids, this construction is applied to the rank and nullity functions, so its coefficients are relative Betti numbers of the rank and nullity filtrations (White, 2012).

The decisive notion is homology decidability. A monotone function $E$6 is homology decidable if it has an optimal decision tree, meaning that the Morse inequalities become equalities: $E$7 This generalizes Jonsson’s semi-nonevasiveness. The paper proves a sufficient condition: if in a decision tree every evasive face $E$8 has dimension determined solely by the function value $E$9, then the tree is optimal and $0\le r(X)\le |X|$0 is homology decidable. That criterion is what allows the semimatroid rank and nullity functions to be handled inductively via deletion and contraction (White, 2012).

The resulting formulas give topological meanings to Tutte evaluations. For the rank function, the coefficients of $0\le r(X)\le |X|$1 are relative Betti numbers and are identified with coefficients of a specific evaluation of the semimatroid Tutte polynomial; for the nullity function the same is done at a different evaluation. The paper also shows that

$0\le r(X)\le |X|$2

where $0\le r(X)\le |X|$3 is the broken circuit complex. Thus the rank-function generating polynomial counts explicit no-broken-circuit faces produced by a collapse, rather than merely encoding an abstract specialization (White, 2012).

This produces a useful conceptual distinction. The 2025 polynomial $0\le r(X)\le |X|$4 is the direct semimatroid rank generating polynomial in the Tutte-theoretic sense, while the 2012 polynomial $0\le r(X)\le |X|$5 is a homological rank-function enumerator. The two are related because both are controlled by the same semimatroid Tutte data, but they package different information: $0\le r(X)\le |X|$6 is combinatorial and activity-theoretic, whereas $0\le r(X)\le |X|$7 records relative homology groups (Fu, 1 Aug 2025, White, 2012).

6. Broken circuits, collapses, and external connections

Broken-circuit theory is a common thread across the semimatroid polynomial literature. Fix a linear order on the ground set. A broken circuit is a set of the form $0\le r(X)\le |X|$8 for a circuit $0\le r(X)\le |X|$9, and the broken circuit complex $r(\mathcal{C})$0 is the subcomplex of faces containing no broken circuit. For semimatroids, $r(\mathcal{C})$1 is vertex-decomposable, its $r(\mathcal{C})$2-polynomial satisfies

$r(\mathcal{C})$3

and

$r(\mathcal{C})$4

Moreover,

$r(\mathcal{C})$5

These facts are important for rank generating questions because the evasive faces of the decision tree for the rank function are exactly the no-broken-circuit faces, so the topology, the decision tree, and the Tutte evaluations all align (White, 2012).

The broken-circuit theme also appears at the level of the characteristic polynomial. Writing

$r(\mathcal{C})$6

one has that $r(\mathcal{C})$7 equals the number of central subsets of size $r(\mathcal{C})$8 that contain no broken circuit. The coefficients are therefore order-independent, and for loopless semimatroids they alternate in sign. The same paper states that the unsigned coefficients are log-concave and unimodal, by identifying them with those of the corresponding pointed-matroid characteristic polynomial (Houshan, 8 Jun 2025).

Semimatroid rank generating polynomials are also tied to matroids, affine arrangements, and graph colorings. The rank extension

$r(\mathcal{C})$9

produces a matroid $X\in\mathcal{C}$0, and Ardila’s pointed-matroid viewpoint yields

$X\in\mathcal{C}$1

The assigning-matroid construction then recovers the semimatroid exactly from a compatible-circuit labeling on $X\in\mathcal{C}$2. For affine hyperplane arrangements, semimatroids provide the natural central-set model, and their characteristic and Tutte convolution identities specialize to the arrangement formulas. For assigning graphs and translated graphic arrangements, the same framework connects semimatroid-compatible polynomials to graph colorings over finite fields (Houshan, 8 Jun 2025).

A plausible implication is that the semimatroid rank generating polynomial occupies an intermediate position between ordinary matroid invariants and arrangement-specific invariants. It preserves the recursive, activity, and convolution structures familiar from matroid theory, but centrality restrictions and broken-circuit collapses supply additional geometric and topological content that is not visible in the unrestricted subset expansion alone.