Nested Set Complexes for Matroids

• Nested set complexes for matroids are combinatorial structures defined by building sets that stratify lattices and polyhedral subdivisions.
• They provide insights into Bergman fans and order complexes, ensuring properties like vertex decomposability, shellability, and Cohen–Macaulayness.
• Their applications span phylogenetics, network design, and moduli space compactifications, linking algebraic topology with combinatorics.

Nested set complexes for matroids provide a deep combinatorial and topological framework for understanding lattice-theoretic and polyhedral properties of matroidal structures. Developed initially in connection with building sets and lattice decompositions, these complexes unify and extend concepts such as order complexes, Bergman complexes, and polytopal subdivisions, yielding stratified refinements of matroid types and offering new enumerative and homological insights.

1. Lattice of Flats, Building Sets, and Nested Set Complexes

Given a loopless matroid $M$ on a finite ground set $E$, its lattice of flats $L(M)$ is a graded, geometric lattice of rank $r = \mathrm{rank}(M)$ with bottom element $\varnothing$ and top $E$ (Backman et al., 8 Jan 2026). A building set $B \subseteq L(M) \setminus \{\varnothing\}$ is defined so that for every $X \in L(M)$, the maximal elements of $B$ below $X$ yield a decomposition

$$\prod_{i=1}^k [\varnothing, F_i] \cong [\varnothing, X], \quad (Z_1, ..., Z_k) \mapsto Z_1 \vee \cdots \vee Z_k$$

where $F_1,...,F_k = \max \{ Y \in B : Y \leq X \}$ and $\vee$ is the join in the lattice. Equivalently, $B$ contains all connected flats and is closed under joins of overlapping elements (Dlugosch, 2011, Backman et al., 2024).

A nested set $N \subseteq B$ satisfies two conditions:

• $\max(B) \subseteq N$
• For every set $\{X_1,..., X_{\ell}\} \subseteq N$ of pairwise incomparable elements with $\ell \geq 2$, one has $X_1 \vee \cdots \vee X_{\ell} \notin B$.

The nested set complex $\Delta(L, B)$ is the simplicial complex whose vertices are $B \setminus \max(B)$, and whose faces are reduced nested sets $N \setminus \max(B)$. Facets of $\Delta(L,B)$ are inclusion-maximal nested sets, each of cardinality $r$ (Backman et al., 8 Jan 2026).

2. Polyhedral and Combinatorial Realizations

The nested set complex $\Delta(L,B)$ is strongly linked to the Bergman fan $\Sigma_{L,B}$, a complete simplicial fan whose cones are indexed by nested sets. For $X \in B$, define $e_X = \sum_{i \in X} e_i \in \mathbb{R}^E$. The Bergman fan consists of cones

$$\sigma_N = L_B + \mathrm{cone} \{ e_X : X \in N \setminus \max(B) \}$$

where $L_B = \mathrm{span} \{ e_F : F \in \max(B) \}$. Each cone corresponds to a face of $\Delta(L,B)$. By choosing a piecewise-linear function $\varphi$ with values $\varphi(e_X) = c_X$ for $X \in B$, one produces a polytopal normal complex $C_B$ homeomorphic to the link of zero in $\Sigma_{L,B}$ (Backman et al., 8 Jan 2026).

When the oriented matroid is realizable, iterated stellar subdivisions yield polytopal realizations: nested complexes are the face lattices of the iteratively subdivided positive tope, and the acyclonestohedron construction generalizes numerous known polytopes (e.g., nestohedra, graph associahedra, poset associahedra) (Mantovani et al., 19 Sep 2025).

3. Structural and Topological Properties

Nested set complexes for matroids are pure simplicial complexes whose dimension is $\mathrm{rank}(M) - |\max(B)| - 1$. Recent research establishes the following structural results:

• Vertex decomposability: For any matroid $M$ and any building set $G$ on $L(M)$, $\Delta(L,G)$ is vertex decomposable. This implies shellability and Cohen–Macaulayness (Coron et al., 8 Jan 2026).
• Convex ear decompositions: $\Delta(L,G)$ admits a convex ear decomposition. This is a sequence of subcomplexes where the first is the boundary of a polytope and each subsequent ear is a simplicial ball attached along its boundary (Coron et al., 8 Jan 2026).
• The $h$-vector of $\Delta(L,G)$ is strongly flawless and top-heavy, with coefficients satisfying majorization inequalities (Coron et al., 8 Jan 2026).

These properties extend and unify shelling results for the order complex and the Bergman complex, as well as their augmented versions.

4. Subdivisions and Refinement Hierarchies

The nested set complex forms a stratified hierarchy between the Bergman complex (coarse subdivision, faces correspond to flacets) and the order complex (fine subdivision, faces are chains of flats). For every building set $B$, $\Delta(L,B)$ is a subdivision of the Bergman complex, and hence its nested set fan refines the Bergman fan (Dlugosch, 2011).

Key decomposition formulas:

• Chain decomposition: Corresponds to chains of flats; the matroid type splits as a direct sum along the chain (Dlugosch, 2011).
• Nested set decomposition: For a face indexed by a nested set, the ground set is partitioned into blocks according to joins in the partition lattice; matroid types decompose into direct sums over interval matroids on those blocks (Dlugosch, 2011).
• Bergman decomposition: The finest, with each summand indivisible beyond connected components.

Refinements of building sets yield increasingly finer nested set decompositions of matroid types and further stratify matroidal polyhedral fans.

5. Enumeration: $f$-Vectors, $h$-Vectors, and Descent Statistics

Let $f_i$ denote the number of $i$-dimensional faces (cubes) in $C_B$, equivalently, the number of nested sets of cardinality $|\max(B)| + i$. The dimension $d = \mathrm{rank}(M) - |\max(B)| - 1$. The $h$-vector is defined by

$$h_0 + h_1 t + \cdots + h_d t^d = \sum_{k=0}^d f_{k-1} t^k (1 - t)^{d - k}$$

(Backman et al., 8 Jan 2026, Coron et al., 8 Jan 2026). For the order complex (maximal building set), the $h$-vector relates to the characteristic polynomial $\chi_M(\lambda)$; for other building sets, formulas involve Möbius functions of the corresponding intervals.

Maximal nested sets admit descent sets, allowing the $h$-polynomial to be written as a sum over maximal nested sets indexed by descent numbers. For special cases (e.g., the complex of trees), $h$-polynomials relate to second Eulerian polynomials enumerating Stirling permutations (Coron et al., 8 Jan 2026).

6. Embedding, Functoriality, and Convex Geometry of Building Sets

Building sets on the flat lattice of a matroid form a supersolvable convex geometry, closed under intersection and anti-exchange with respect to join operations (Backman et al., 2024). Every building set is functorial under matroid minors:

• Deletion and contraction induce poset embeddings preserving building sets and nested set complexes.
• The inclusion of face lattices into flat lattices produces embeddings of facial nested complexes inside Boolean nested complexes; positive Bergman complexes embed inside ordinary Bergman complexes (Mantovani et al., 19 Sep 2025).

These embedding results unify various families of nested complexes (Boolean, graph associahedra, hyperoctahedral, permutopermutohedra) and have applications in compactifications and cohomology rings.

7. Special Cases and Applications

Many combinatorially significant spaces arise as nested set complexes for particular building sets and lattices:

• For the partition lattice $\Pi_n$ with minimal building set, the nested set complex is the boundary complex of $\overline{\mathcal{M}}_{0,n+1}$, coinciding with the complex of phylogenetic trees $T_n$ (Coron et al., 8 Jan 2026).
• For Boolean and uniform matroids, nested set complexes yield standard simplices and spheres, respectively, with face counts and $h$-vectors determined by combinatorics of underlying sets (Backman et al., 2024).
• Applications extend to graphical matroids, relevant in network design and phylogenetics.

Remarks on generality: The line shelling order applies to nested set complexes for a wide range of building sets (graph associahedra, polymatroid fans, etc.), supporting shellability and vertex decomposability in each case (Backman et al., 8 Jan 2026, Coron et al., 8 Jan 2026).

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Nested set complexes for matroids synthesize geometric lattice theory, polyhedral combinatorics, and algebraic topology, yielding powerful tools for decomposing matroid types, enumerating faces, and analyzing topological and combinatorial properties of associated stratifications (Backman et al., 8 Jan 2026, Dlugosch, 2011, Coron et al., 8 Jan 2026, Mantovani et al., 19 Sep 2025, Backman et al., 2024).