{"paper":{"title":"Enumeration of certain subsets of uprooted trees and spherical parking functions","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Chanchal Kumar, Gargi Lather, Nayana Shibu Deepthi","submitted_at":"2026-06-09T17:28:43Z","abstract_excerpt":"Spherical $G$-parking functions are a distinguished subset of standard monomials, arising from the skeleton ideals of the $G$-parking function ideal. Explicit spherical $G$-parking function enumeration formulas are known only in a few classes of graphs. In this paper, we consider a family of graphs $\\Gl$ ($1\\leq \\ell \\leq n-2$), obtained from the complete bipartite $K_{n+1}$ by deleting the $\\ell$ edges joining vertex $1$ to the vertices in $F_\\ell= \\{n-\\ell+1, \\ldots, n\\}$. The uprooted spanning trees of $\\Gl-\\{0\\}$ are counted by the set $\\UnFl$ of uprooted trees with the vertex set $[n]$ in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11137","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.11137/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}