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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2605.28981 |
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| _version_ | 1866913168701259776 |
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| author | Lyudogovskiy, Fedor B. |
| author_facet | Lyudogovskiy, Fedor B. |
| contents | 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 $e=(λ,μ)$ and a vertex invariant $F$, we define the signed jump $Δ_e F=F(μ)-F(λ)$ and focus on the basic jump signature \[ J(e)=(Δ_e d,Δ_eδ,Δ_eσ), \] where $d$ is degree, $δ$ is local simplex dimension, and $σ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28981 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Jump and Gradient Invariants in the Partition Graph Lyudogovskiy, Fedor B. Combinatorics 05A17, 05C75 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 $e=(λ,μ)$ and a vertex invariant $F$, we define the signed jump $Δ_e F=F(μ)-F(λ)$ and focus on the basic jump signature \[ J(e)=(Δ_e d,Δ_eδ,Δ_eσ), \] where $d$ is degree, $δ$ is local simplex dimension, and $σ$ 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. |
| title | Jump and Gradient Invariants in the Partition Graph |
| topic | Combinatorics 05A17, 05C75 |
| url | https://arxiv.org/abs/2605.28981 |