Numerical topology of the clique complex of the partition graph: Euler characteristic, clique counts, and sequence data

Abstract

We study the numerical topology of the clique complex $K_n=\mathrm{Cl}(G_n)$, where $G_n$ is the partition graph on the set of integer partitions of $n$. Building on the previously established homotopy equivalence $K_n \simeq \vee^{\,b_n} S^2$, we shift the focus from qualitative topology to its numerical content. Our main objects are the Euler characteristic $χ(K_n)$, the derived sequence $b_n=χ(K_n)-1$, the clique counts $c_r(n)$, and several related maximal-simplex counts. We develop two exact counting languages for the same invariant. The first is the direct clique-counting formula $χ(K_n)=\sum_{r\ge 1}(-1)^{r-1}c_r(n)$, which expresses Euler characteristic through clique counts in the partition graph. The second is a nerve-side formula arising from the canonical good cover by distinct full star- and full top-simplices, which yields $χ(K_n)=χ(N_n)$, where $N_n$ is the corresponding nerve. We further use the classification of maximal simplices into star-, top-, and edge-type pieces to formulate a local-to-global counting framework based on local admissibility data and global deduplication. The paper is primarily organizational and computational. It fixes a consistent counting dictionary, separates intrinsic global counts from auxiliary based counts, records exact data for the full main sequence package on $1\le n\le 25$, and extends the low-dimensional clique-count layer through $n=60$. We do not claim closed formulas for $χ(K_n)$ or for the full family of clique counts. Rather, the paper provides a framework in which such questions can be studied systematically.

Figures (2)

• Figure 1: Logarithmic plot of the low-dimensional clique counts $c_1(n),\dots,c_5(n)$ for $1\le n\le 60$. Each curve is drawn starting from its first positive value, and the first positive point is marked explicitly, so the logarithmic scale does not create artificial vertical segments from initial zeros.
• Figure 2: Logarithmic plot of $c_3(n)$ and $c_4(n)$ for $45\le n\le 60$, showing the first crossover at $n=60$.

Theorems & Definitions (14)

• Definition 2.1
• Proposition 2.1: Topology-to-counting dictionary
• proof
• Proposition 3.1
• proof
• Remark 3.1
• Proposition 3.2
• proof
• Proposition 3.3
• proof