{"paper":{"title":"Divisible Arm Lengths, Crystal Reflections, and Enumeration of Newly Found Decomposition Columns","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David J. Hemmer, Pavel Turek","submitted_at":"2026-06-26T17:49:30Z","abstract_excerpt":"In recent work the authors determine complete columns of symmetric-group decomposition matrices in odd prime characteristic $p$ labeled by $p$-regular partitions for which every hook of length divisible by $p$ has even arm length. In the present paper we enumerate these partitions and prove that each block of $p$-weight $w$ contains precisely \\[ \\binom{w+\\frac{p-3}{2}}{w} \\] such partitions.\n  More generally, for any integers $d,e>1$, we study and enumerate $d$-balanced $e$-regular partitions -- partitions for which every hook of length divisible by $e$ has arm length divisible by $d$. Our fir"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28305","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/2606.28305/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"}