Degree theory of the partition graph: exact maxima, profiles, and fibres
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918422855548928 |
|---|---|
| author | Lyudogovskiy, Fedor B. |
| author_facet | Lyudogovskiy, Fedor B. |
| contents | For the partition graph $G_n$, whose vertices are the partitions of $n$ and whose edges correspond to elementary unit transfers between parts, we develop a degree theory with three levels: exact value theory, exact profile theory, and fibre-level geometry. Writing $n=T_s+q$ with $T_s=s(s+1)/2$ and $0\le q\le s$, we prove that every degree-maximizing partition lies in the support-maximal stratum and obtain the exact formula \[ Δ_n=s(s-1)+\lfloor\sqrt{4q+1}\rfloor-1 \] for the maximal degree in $G_n$.
For a support-maximal partition $λ$, let $A(λ)$ and $B(λ)$ denote the numbers of active gap bonuses and multiplicity bonuses. We prove that the set of realized maximizing profiles is \[ Π_n=\{(a,b)\in\mathbb Z_{\ge0}^2:a+b=ρ(q),\ T_a+T_b\le q\}, \qquad ρ(q)=\lfloor\sqrt{4q+1}\rfloor-1. \] Thus the exact global theory stops at the profile level. For each realized profile we then study the corresponding fibre of maximizers: we prove nonemptiness, construct canonical representatives, obtain lower bounds for mixed fibres, and show that conjugation induces a bijection between the fibres for $(a,b)$ and $(b,a)$. We also classify exactly the first near-triangular fibre windows and formulate localization and stability questions for the remaining fixed-$q$ regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27248 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Degree theory of the partition graph: exact maxima, profiles, and fibres Lyudogovskiy, Fedor B. General Mathematics 05A17, 05C07, 05C35, 05C75 For the partition graph $G_n$, whose vertices are the partitions of $n$ and whose edges correspond to elementary unit transfers between parts, we develop a degree theory with three levels: exact value theory, exact profile theory, and fibre-level geometry. Writing $n=T_s+q$ with $T_s=s(s+1)/2$ and $0\le q\le s$, we prove that every degree-maximizing partition lies in the support-maximal stratum and obtain the exact formula \[ Δ_n=s(s-1)+\lfloor\sqrt{4q+1}\rfloor-1 \] for the maximal degree in $G_n$. For a support-maximal partition $λ$, let $A(λ)$ and $B(λ)$ denote the numbers of active gap bonuses and multiplicity bonuses. We prove that the set of realized maximizing profiles is \[ Π_n=\{(a,b)\in\mathbb Z_{\ge0}^2:a+b=ρ(q),\ T_a+T_b\le q\}, \qquad ρ(q)=\lfloor\sqrt{4q+1}\rfloor-1. \] Thus the exact global theory stops at the profile level. For each realized profile we then study the corresponding fibre of maximizers: we prove nonemptiness, construct canonical representatives, obtain lower bounds for mixed fibres, and show that conjugation induces a bijection between the fibres for $(a,b)$ and $(b,a)$. We also classify exactly the first near-triangular fibre windows and formulate localization and stability questions for the remaining fixed-$q$ regime. |
| title | Degree theory of the partition graph: exact maxima, profiles, and fibres |
| topic | General Mathematics 05A17, 05C07, 05C35, 05C75 |
| url | https://arxiv.org/abs/2603.27248 |