Degree theory of the partition graph: exact maxima, profiles, and fibres

TL;DR

This paper develops a comprehensive degree theory for the partition graph, providing exact formulas for maximum degrees, profiles, and fibres, and analyzing their geometric and combinatorial properties.

Contribution

It introduces a three-level degree theory for the partition graph, including exact maximum degree formulas, profile characterizations, and fibre structure analysis.

Findings

Derived an exact formula for the maximum degree in the partition graph.

Characterized the set of maximizing profiles and their properties.

Analyzed the structure and properties of fibres for each profile.

Abstract

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 \[ \Delta_n=s(s-1)+\lfloor\sqrt{4q+1}\rfloor-1 \] for the maximal degree in $G_n$. For a support-maximal partition $\lambda$, let $A(\lambda)$ and $B(\lambda)$ denote the numbers of active gap bonuses and multiplicity bonuses. We prove that the set of realized maximizing profiles is \[ \Pi_n=\{(a,b)\in\mathbb Z_{\ge0}^2:a+b=\rho(q),\ T_a+T_b\le q\}, \qquad \rho(q)=\lfloor\sqrt{4q+1}\rfloor-1. \] 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.