Jump and Gradient Invariants in the Partition Graph

Abstract

We introduce edgewise jump invariants and gradient-type structures for the partition graph $G_n$, whose vertices are the partitions of $n$ and whose edges correspond to elementary transfers of one unit between parts. Previous work on $G_n$ has focused mainly on vertex-level invariants such as degree, local simplex dimension, and support size. Here we study how such invariants change along edges. For an oriented edge $e=(\hat{I}»,\hat{I}¼)$ and a vertex invariant $F$, we define the signed jump $\hat{I}{”}_{e}F=F(\hat{I}¼)-F(\hat{I}»)$ and focus on the basic jump signature \[ J(e)=(Δ_e d,Δ_eδ,Δ_eσ), \] where $d$ is degree, $\hat{I}´$ is local simplex dimension, and $\ddot{I}ƒ$ is support size. We prove that support jumps are universally bounded by $2$ and describe them in terms of local multiplicity data. We also develop a taxonomy of active, neutral, pure, and mixed transitions, relate nonzero jumps of integer-valued invariants to threshold-layer crossings, and discuss strict gradient orientations associated with real-valued vertex invariants. Finally, we formulate a reproducible protocol for a computational atlas of jump spectra, transition ranks, large-jump edges, and localization patterns. No large-scale computations are carried out here; the atlas is presented as a framework for subsequent work.

36 pages