Stirling Complexes in Graph Topology

• Stirling complexes are cellular models defined on trees for resource distribution, encoding combinatorial partitions via Stirling numbers.
• They exhibit a homotopy equivalence to wedges of spheres, where the sphere count and dimensions are determined by the parameters m and n.
• Generalizations like grouped Stirling complexes extend these ideas to multi-resource allocation with applications in robot motion planning and algebraic topology.

Stirling complexes are a family of cubical or cellular complexes naturally arising in the combinatorics and topology of resource distribution problems on graphs and trees, the study of configuration spaces, and in graph homology, where classical Stirling numbers of the first and second kind govern their algebraic and topological invariants. They generalize the discrete partitioning of labeled objects and encode deep connections between combinatorics, configuration space topology, graph theory, and operadic structures.

1. Definition and Basic Properties

Let $T$ be a tree on $m$ labeled vertices and $n \geq m$ the number of resources to be distributed. The Stirling complex $\operatorname{Str}(T, n)$ is the subcomplex of the $n$-fold Cartesian product $T^n$ (with its natural cubical structure) consisting of all $n$-tuples $c = (c_1, \dots, c_n)$ with $c_i \in V(T) \cup E(T)$ such that every vertex of $T$ appears at least once among the entries. The dimension of a cube is the number of coordinates assigned to edges; maximal cubes arise when $m$0 coordinates are edges, giving $m$1 (Kozlov, 2023).

Formally,

$m$2

where $m$3 is the set of vertices appearing as some $m$4.

The 0-cells of this complex have each coordinate in $m$5 and every vertex used at least once. The number of 0-cells is $m$6, with $m$7 the Stirling numbers of the second kind, counting surjections $m$8.

If $m$9 there are no valid configurations, so $n \geq m$0 is empty.

2. Homotopy Type and Sphere Decomposition

A fundamental result established by Kozlov (Kozlov, 2023) shows that $n \geq m$1 is, for any tree $n \geq m$2 with $n \geq m$3 vertices and $n \geq m$4, homotopy equivalent to a wedge of spheres of (exact) dimension $n \geq m$5:

$n \geq m$6

where $n \geq m$7, the number of spheres, depends only on $n \geq m$8 and $n \geq m$9, not the tree $\operatorname{Str}(T, n)$0. This is nontrivial, as the explicit cell structure does depend on the tree: for instance, the count of 0-cells and 1-cells for nonisomorphic trees with the same $\operatorname{Str}(T, n)$1 may differ, yet the homotopy types coincide.

The number $\operatorname{Str}(T, n)$2 satisfies the recurrence

$\operatorname{Str}(T, n)$3

with $\operatorname{Str}(T, n)$4 and $\operatorname{Str}(T, n)$5 for $\operatorname{Str}(T, n)$6, admitting closed-form expressions via inclusion–exclusion:

$\operatorname{Str}(T, n)$7

This homotopy-invariance is explained via an inductive construction where the complex is built as a homotopy colimit over vertex- and edge-labeled pieces, and the attaching maps become trivial due to the high connectivity of lower-dimensional Stirling complexes. Thus, the only invariants required are $\operatorname{Str}(T, n)$8 and $\operatorname{Str}(T, n)$9 (Kozlov, 2023).

3. Stirling Numbers and Combinatorial Structure

Stirling complexes encode the classical combinatorics of partitions and permutations:

• The Stirling numbers of the second kind $n$0 count the number of ways to partition $n$1 objects into $n$2 nonempty unlabeled parts. The 0-cells of $n$3 are in bijection with ordered surjections $n$4 and thus $n$5 in cardinality (Kozlov, 2023).
• In graph homology and the study of decorated tree complexes, the Stirling numbers of the first kind $n$6 appear as the ranks of certain homology groups in top degree (Ward, 2023).

The generating function for $n$7 is

$n$8

while $n$9 appears directly in the number of 0-cells and the enumeration of surjections.

4. Decorated-Tree Chain Complexes in Genus-1 Graph Homology

For commutative graph homology in genus $T^n$0 with $T^n$1 legs, the relevant Stirling complexes are chain complexes $T^n$2 where $T^n$3, defined as follows (Ward, 2023):

• The $T^n$4-chains are $T^n$5-vector spaces spanned by isomorphism classes $T^n$6, where $T^n$7 is a stable tree with $T^n$8 legs, $T^n$9 a distinguished vertex, $n$0 a $n$1-element subset of input flags at $n$2 (the "alternating flags"), $n$3 an ordering of internal edges (up to sign), and $n$4 an ordering of $n$5 (again up to sign). $n$6 is the number of internal edges.
• The differential $n$7 is given by explicit edge-contraction formulas, distinguishing whether the edge carries an alternating flag. The boundary map splits according to whether the contracted edge is alternating or not.

Each $n$8 is acyclic except in degree $n$9; the rank of $c = (c_1, \dots, c_n)$0 is $c = (c_1, \dots, c_n)$1.

The chain complexes $c = (c_1, \dots, c_n)$2 produce a direct sum decomposition of genus-1 commutative graph homology:

$c = (c_1, \dots, c_n)$3

Exemplary calculations for $c = (c_1, \dots, c_n)$4 show the matching of Betti numbers with entries of the Stirling triangle:

• For $c = (c_1, \dots, c_n)$5, $c = (c_1, \dots, c_n)$6, $c = (c_1, \dots, c_n)$7, $c = (c_1, \dots, c_n)$8.

5. Grouped Stirling Complexes and Generalizations

Grouped Stirling complexes $c = (c_1, \dots, c_n)$9, recently introduced, generalize the classic Stirling complexes by placing robots (resources) in $c_i \in V(T) \cup E(T)$0 colored groups over a graph $c_i \in V(T) \cup E(T)$1, with each group containing $c_i \in V(T) \cup E(T)$2 robots of color $c_i \in V(T) \cup E(T)$3 (Revelli et al., 10 Feb 2026). The defining constraints:

• Every vertex must host at least one robot.
• Robots of the same group must be separated by at least one full open edge.

The cellular structure consists of cells indexed by $c_i \in V(T) \cup E(T)$4-tuples $c_i \in V(T) \cup E(T)$5 with $c_i \in V(T) \cup E(T)$6, $c_i \in V(T) \cup E(T)$7, and separation constraints within $c_i \in V(T) \cup E(T)$8.

These complexes are always path-connected if there are at least three groups and the color vector is nontrivial ($c_i \in V(T) \cup E(T)$9, each $T$0). Dimensional formulas for cell counts exist in special cases, for example:

• For $T$1, there are $T$2 0-cells and $T$3 1-cells.
• For $T$4, the $T$5-cell count is

$T$6

When each $T$7, $T$8 recovers Kozlov's original Stirling complex. If some $T$9, the complex splits off a factor $m$00. The homology and cohomology structure remains mostly open outside trees and anchored cycles; connections to the Stirling numbers of the second kind may arise in certain combinatorial models.

6. Illustrative Examples and Applications

For $m$01 a tree with $m$02, $m$03 is homotopy equivalent to $m$04, with explicit geometric models:

• $m$05: $m$06 (two points connected)
• $m$07: $m$08 (hexagon)
• $m$09: $m$10 (rhombic dodecahedron)

For $m$11, $m$12, $m$13 is a wedge of $m$14 circles, independent of the tree's structure.

Grouped Stirling complexes for paths $m$15 with $m$16 are always wedges of circles, with the count provided by explicit formulas. For $m$17, $m$18, the complex is a hexagon (six 0-cells, six 1-cells).

In graph homology, the splitting via decorated-tree complexes provides computational tractability; for $m$19, the Betti numbers of the summands match $m$20.

7. Connections, Further Questions, and Open Problems

Stirling complexes and their grouped analogues are discrete configuration spaces, rich in both algebraic and topological structures:

• They serve as models for resource allocation and robot motion planning on graphs (Kozlov, 2023, Revelli et al., 10 Feb 2026).
• In genus-1 graph homology, constructed via Feynman transforms of commutative modular operads, Stirling complexes encode the direct sum decomposition of homology (Ward, 2023).
• Their homotopy invariance under tree isomorphism is a compelling categorical and topological property (Kozlov, 2023).

Active research investigates the computation of Betti numbers for grouped Stirling complexes in general graphs and color-vectors, the algebraic structure of their cohomology rings, topological complexity, Morse-theoretic cell filtrations, and the relations with classical combinatorial invariants such as the Stirling numbers of the second kind.

Open directions include cup-product structures, high-connectivity bounds, explicit Morse functions, and the implications for robot configuration spaces, as well as the deeper role of Stirling complex decompositions in graph-based algebraic topology (Revelli et al., 10 Feb 2026).