The homotopy type of the clique complex of the partition graph

Abstract

For each positive integer $n$, let $G_n$ be the graph whose vertices are the partitions of $n$, with edges corresponding to elementary transfers of one cell between two parts, followed by reordering. Let $K_n := \mathrm{Cl}(G_n)$ be the clique complex of $G_n$. We prove that $K_n$ is homotopy equivalent to a wedge of $2$-spheres. More precisely, $K_n$ is homotopy equivalent to a wedge of $b_n$ copies of $S^2$, where $b_n = χ(K_n) - 1$. Thus the homotopy type of $K_n$ is completely determined by its Euler characteristic. The proof has three main ingredients. First, we classify all cliques in $G_n$ via two canonical families of simplices, called star-simplices and top-simplices, and use them to build a canonical cover of $K_n$. Second, we pass to the corresponding nerve, construct a second natural cover, and show via the intersection poset of that cover that $K_n$ has the homotopy type of a CW-complex of dimension at most $2$. Third, using an explicit height function on partitions, we prove that $K_n$ is connected and simply connected. It follows that the reduced homology of $K_n$ is concentrated in degree $2$, where its rank is $χ(K_n) - 1$, and therefore $K_n$ has the homotopy type claimed above. We conclude with remarks on Euler characteristics, small examples, and the integer sequences arising from these complexes.