{"paper":{"title":"k-Marked Dyson Symbols and Congruences for Moments of Cranks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Erin Y.Y. Shen, Kathy Q. Ji, William Y.C. Chen","submitted_at":"2013-12-07T10:10:49Z","abstract_excerpt":"By introducing $k$-marked Durfee symbols, Andrews found a combinatorial interpretation of $2k$-th symmetrized moment $\\eta_{2k}(n)$ of ranks of partitions of $n$. Recently, Garvan introduced the $2k$-th symmetrized moment $\\mu_{2k}(n)$ of cranks of partitions of $n$ in the study of the higher-order spt-function $spt_k(n)$. In this paper, we give a combinatorial interpretation of $\\mu_{2k}(n)$. We introduce $k$-marked Dyson symbols based on a representation of ordinary partitions given by Dyson, and we show that $\\mu_{2k}(n)$ equals the number of $(k+1)$-marked Dyson symbols of $n$. We then int"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.2080","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":""},"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"}