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 $\chi(K_n)$, the derived sequence $b_n=\chi(K_n)-1$, 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 $\chi(K_n)=\sum_{r\ge 1}(-1)^{r-1}c_r(n)$, 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 $\chi(K_n)=\chi(N_n)$, 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 $\chi(K_n)$ or for the full family of clique counts. Rather, the paper provides a framework in which such questions can be studied systematically.