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Let $\\mathcal{RSS}_k(n)$ be the family of {\\it reverse shuffle squares} in $[k]^{2n}$, words that can be partitioned into two disjoint subsequences which are reverses of each other. Henshall, Rampersad, and Shallit conjectured asymptotic formulas for the sizes of $\\mathcal{SS}_k(n)$ and $\\mathcal{RSS}_k(n)$ based on numerical evidence. We prove that\n  \\[\n  \\lvert \\mathcal{SS}_k(n) \\rvert=\\dfrac{1}{n+1}\\dbinom{2n}{n}k^n-\\dbinom{2n-1}{n+1}k^{n-1}+O"},"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":"2109.12455","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-09-25T22:50:03Z","cross_cats_sorted":[],"title_canon_sha256":"be646735e9dbea9b743100a19d13ffaac8a4835f8ac79848c1f6063dfd1a60e2","abstract_canon_sha256":"cddc57407fa3547a3578fbd5a9f220f1e28e2710f50e0389b0bd05e8fa162a5c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:14:04.177095Z","signature_b64":"t0E3XksHWlf11HQ002cIFYrqSJAOcNKTFWj/f0vr250/7qiKypyx5P4lH7W8VrDQyKavGxyYX1pTwK6aT6fcDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"752218e23eaf9e4034bf5b0b70bc5c4b48fa02965e20c866273d069bd4fc5838","last_reissued_at":"2026-07-05T07:14:04.176636Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:14:04.176636Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Shuffle Squares and Reverse Shuffle Squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Emily Huang, Ihyun Nam, Rishubh Thaper, Xiaoyu He","submitted_at":"2021-09-25T22:50:03Z","abstract_excerpt":"Let $\\mathcal{SS}_k(n)$ be the family of {\\it shuffle squares} in $[k]^{2n}$, words that can be partitioned into two disjoint identical subsequences. Let $\\mathcal{RSS}_k(n)$ be the family of {\\it reverse shuffle squares} in $[k]^{2n}$, words that can be partitioned into two disjoint subsequences which are reverses of each other. Henshall, Rampersad, and Shallit conjectured asymptotic formulas for the sizes of $\\mathcal{SS}_k(n)$ and $\\mathcal{RSS}_k(n)$ based on numerical evidence. We prove that\n  \\[\n  \\lvert \\mathcal{SS}_k(n) \\rvert=\\dfrac{1}{n+1}\\dbinom{2n}{n}k^n-\\dbinom{2n-1}{n+1}k^{n-1}+O"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.12455","kind":"arxiv","version":2},"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/2109.12455/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2109.12455","created_at":"2026-07-05T07:14:04.176693+00:00"},{"alias_kind":"arxiv_version","alias_value":"2109.12455v2","created_at":"2026-07-05T07:14:04.176693+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.12455","created_at":"2026-07-05T07:14:04.176693+00:00"},{"alias_kind":"pith_short_12","alias_value":"OURBRYR6V6PE","created_at":"2026-07-05T07:14:04.176693+00:00"},{"alias_kind":"pith_short_16","alias_value":"OURBRYR6V6PEANF7","created_at":"2026-07-05T07:14:04.176693+00:00"},{"alias_kind":"pith_short_8","alias_value":"OURBRYR6","created_at":"2026-07-05T07:14:04.176693+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OURBRYR6V6PEANF7LMFXBPC4JN","json":"https://pith.science/pith/OURBRYR6V6PEANF7LMFXBPC4JN.json","graph_json":"https://pith.science/api/pith-number/OURBRYR6V6PEANF7LMFXBPC4JN/graph.json","events_json":"https://pith.science/api/pith-number/OURBRYR6V6PEANF7LMFXBPC4JN/events.json","paper":"https://pith.science/paper/OURBRYR6"},"agent_actions":{"view_html":"https://pith.science/pith/OURBRYR6V6PEANF7LMFXBPC4JN","download_json":"https://pith.science/pith/OURBRYR6V6PEANF7LMFXBPC4JN.json","view_paper":"https://pith.science/paper/OURBRYR6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2109.12455&json=true","fetch_graph":"https://pith.science/api/pith-number/OURBRYR6V6PEANF7LMFXBPC4JN/graph.json","fetch_events":"https://pith.science/api/pith-number/OURBRYR6V6PEANF7LMFXBPC4JN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OURBRYR6V6PEANF7LMFXBPC4JN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OURBRYR6V6PEANF7LMFXBPC4JN/action/storage_attestation","attest_author":"https://pith.science/pith/OURBRYR6V6PEANF7LMFXBPC4JN/action/author_attestation","sign_citation":"https://pith.science/pith/OURBRYR6V6PEANF7LMFXBPC4JN/action/citation_signature","submit_replication":"https://pith.science/pith/OURBRYR6V6PEANF7LMFXBPC4JN/action/replication_record"}},"created_at":"2026-07-05T07:14:04.176693+00:00","updated_at":"2026-07-05T07:14:04.176693+00:00"}