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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.11837 |
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| _version_ | 1866908960455393280 |
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| author | Lyudogovskiy, Fedor B. |
| author_facet | Lyudogovskiy, Fedor B. |
| contents | Let $G_n$ be the partition graph whose vertices are the partitions of $n$, with adjacency given by elementary transfers of one cell between parts, followed by reordering. We study the support of a partition -- the set of distinct part sizes -- as a global vertex invariant of $G_n$. We show that support size $r$ occurs in $G_n$ if and only if $T_r=r(r+1)/2\le n$, so the maximal support size is $ρ(n)=\max\{r:T_r\le n\}$. We determine exactly how support changes along an edge: the support jump always lies in $\{-2,-1,0,1,2\}$, and we give an explicit birth-death formula in terms of the source and target part sizes. We also prove the degree bound $°(λ)\ge σ(λ)(σ(λ)-1)$ for every partition $λ$, with equality exactly for staircase partitions. In addition, support size is invariant under conjugation, the support-$1$ stratum consists exactly of rectangular partitions, and the coarse support-level graph always contains the chain $1-2-\cdots-ρ(n)$. We conclude with computational data for small $n$, including support-stratum counts, support-jump counts, and connectivity data for fixed-support subgraphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_11837 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Support and Support Jumps in the Partition Graph Lyudogovskiy, Fedor B. Combinatorics 05A17, 05C75, 05C90 Let $G_n$ be the partition graph whose vertices are the partitions of $n$, with adjacency given by elementary transfers of one cell between parts, followed by reordering. We study the support of a partition -- the set of distinct part sizes -- as a global vertex invariant of $G_n$. We show that support size $r$ occurs in $G_n$ if and only if $T_r=r(r+1)/2\le n$, so the maximal support size is $ρ(n)=\max\{r:T_r\le n\}$. We determine exactly how support changes along an edge: the support jump always lies in $\{-2,-1,0,1,2\}$, and we give an explicit birth-death formula in terms of the source and target part sizes. We also prove the degree bound $°(λ)\ge σ(λ)(σ(λ)-1)$ for every partition $λ$, with equality exactly for staircase partitions. In addition, support size is invariant under conjugation, the support-$1$ stratum consists exactly of rectangular partitions, and the coarse support-level graph always contains the chain $1-2-\cdots-ρ(n)$. We conclude with computational data for small $n$, including support-stratum counts, support-jump counts, and connectivity data for fixed-support subgraphs. |
| title | Support and Support Jumps in the Partition Graph |
| topic | Combinatorics 05A17, 05C75, 05C90 |
| url | https://arxiv.org/abs/2604.11837 |