Grouped Stirling complexes

TL;DR

This work introduces grouped Stirling complexes $S_{\,\rvec}(G)$, a generalization of Stirling complexes that places $r$ colored robots on a graph with intra-color separation and universal vertex-occupancy constraints. The authors establish a closed cubical cell structure, derive non-emptiness conditions, and prove that $S_{\,\rvec}(G)$ is path-connected for connected $G$ whenever $\rvec$ is non-trivial with at least three colors. They provide exact cell counts for two natural color-vector families, showing that one family yields a wedge of circles and giving explicit formulas for the number of $0$- and $1$-cells, while the other family yields a general combinatorial count for $i$-cells. These results give detailed topological and combinatorial descriptions of grouped Stirling complexes, including homotopy types and connectivity properties that sharpen understanding of configuration-like spaces on graphs.

Abstract

Given a graph $G$, a configuration space of $G$ can be thought of as the set of all possible configurations of "robots" which can move throughout $G$, subject to some constraints. We introduce a type of configuration space which we call Grouped Stirling complexes, denoted by $S_{\vec r}(G)$, in which we place robots in groups subject to two constraints. First, there must be at least one robot on each vertex of $G$, and second, any two robots from the same group must be "separated by at least one full open edge" of $G$. The space $S_{\vec r}(G)$ has a closed cell structure, which means it can be built out of cells of various dimensions. Our main results show $S_{\vec r}(G)$ is path-connected, provided there are at least three groups, and determine the number of cells of $S_{\vec r}(G)$ in certain cases.

Figures (35)

• Figure 1: A graph $Y$ (left); configurations in $\mathrm{C}^2(Y)$ (middle) and $\mathrm{UC}^2(Y)$ (right)
• Figure 2: Configurations in $\mathrm{D}^2(Y)$ (left) and $\mathrm{UD}^2(Y)$ (right)
• Figure 3: The graph $Y$ (left); the configuration spaces $\mathrm{UD}^2(Y)$ (middle) and $\mathrm{D}^2(Y)$ (right)
• Figure 4: A graph $Y'$ (left); the configuration space $\mathrm{UD}^2(Y')$ (right)
• Figure 5: Configurations in $\mathrm{S}^5(Y)$ (left and middle) and in $S_{\vec{r}}(Y)$ with $\vec{r}=(3,2)$ (right)

Theorems & Definitions (37)

• Definition 2.1
• Example 2.2
• Remark
• Proposition 2.3
• proof
• Definition 2.4
• Definition 2.5
• Example 2.6
• Proposition 2.7
• proof