{"paper":{"title":"$A_1^{(1)}$-Grounded partitions at levels $1$ and $2$, Part I: bijections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Benedek Dombos, Jihyeug Jang","submitted_at":"2025-08-04T17:52:36Z","abstract_excerpt":"Grounded partitions, introduced by Dousse and Konan, are coloured partitions satisfying difference conditions encoded by a matrix. For suitable choices of this matrix, their generating functions are known to coincide with characters of affine Lie algebras. In this paper, we study, from a combinatorial point of view, the grounded partitions introduced by Dousse, Hardiman and Konan and related to the Lie algebra $A_1^{(1)}$. Using the connection with characters, they showed that the generating function for these grounded partitions is an infinite product. We give direct combinatorial proofs of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.02664","kind":"arxiv","version":3},"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/2508.02664/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"}