Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2603.18696 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908930013134848 |
|---|---|
| author | Lyudogovskiy, Fedor B. |
| author_facet | Lyudogovskiy, Fedor B. |
| contents | For a fixed integer $n$, let $G_n$ be the graph whose vertices are the partitions of $n$, with adjacency defined by a single elementary transfer of a cell in the Ferrers diagram. In a previous paper, the clique complex $K_n = \mathrm{Cl}(G_n)$ was studied from a global homotopy-theoretic point of view. This paper studies instead the local combinatorics of the graph $G_n$ itself. For a partition $λ=(s_1^{m_1},\dots,s_t^{m_t})$, where $s_1>\dots>s_t>0$, we describe the admissible transfers from $λ$ in terms of its block structure. This yields a bipartite graph $B(λ)$ obtained from $K_{t,t+1}$ by deleting two explicitly determined families of edges, corresponding to singleton support blocks and unit support gaps. We prove that the graph induced on the neighborhood of $λ$ in $G_n$ is isomorphic to the line graph $L(B(λ))$. As consequences, we obtain an explicit formula for the degree of $λ$, a classification of all cliques through $λ$, and a formula for the maximal dimension of a simplex of $K_n$ containing $λ$. These local invariants are shown to depend only on an ordered binary datum associated with the support of $λ$. The results provide a local structural description of the partition graph and a combinatorial language for the study of larger-scale features of $G_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18696 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local Morphology of the Partition Graph Lyudogovskiy, Fedor B. General Mathematics 05A17 (Primary), 05C75, 05E10 (Secondary) For a fixed integer $n$, let $G_n$ be the graph whose vertices are the partitions of $n$, with adjacency defined by a single elementary transfer of a cell in the Ferrers diagram. In a previous paper, the clique complex $K_n = \mathrm{Cl}(G_n)$ was studied from a global homotopy-theoretic point of view. This paper studies instead the local combinatorics of the graph $G_n$ itself. For a partition $λ=(s_1^{m_1},\dots,s_t^{m_t})$, where $s_1>\dots>s_t>0$, we describe the admissible transfers from $λ$ in terms of its block structure. This yields a bipartite graph $B(λ)$ obtained from $K_{t,t+1}$ by deleting two explicitly determined families of edges, corresponding to singleton support blocks and unit support gaps. We prove that the graph induced on the neighborhood of $λ$ in $G_n$ is isomorphic to the line graph $L(B(λ))$. As consequences, we obtain an explicit formula for the degree of $λ$, a classification of all cliques through $λ$, and a formula for the maximal dimension of a simplex of $K_n$ containing $λ$. These local invariants are shown to depend only on an ordered binary datum associated with the support of $λ$. The results provide a local structural description of the partition graph and a combinatorial language for the study of larger-scale features of $G_n$. |
| title | Local Morphology of the Partition Graph |
| topic | General Mathematics 05A17 (Primary), 05C75, 05E10 (Secondary) |
| url | https://arxiv.org/abs/2603.18696 |