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

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.