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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1312.2080","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-07T10:10:49Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"e0955ee380696cdbf63c26ded5c84e51b095cd243f7360edc75fe4b8f401d21d","abstract_canon_sha256":"421fcd22edc8926cf8993c464e79a754917bbca0f360a6543294eb6ed1ef8b69"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:17.380662Z","signature_b64":"WAM6cX+BjmldQNIrsy30a3BiuJ08GaRztPmXI4a5vzwfvpOWg4+cb1+GVZI1iICQ16iP8xJ8tp4vw2RGkneRDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5a2dd2f792e550eaf3cde3089f4644d64ebbdeee93e30714a86a520adba48324","last_reissued_at":"2026-05-18T03:05:17.380026Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:17.380026Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $n$. 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