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Callan conjectured that $b(n)=p(n)$ for all $n$, which was proved by Bernardi, Duplantier, and Nadeau.\n  We propose a refinement of Callan's original conjecture. Let $b(n,d)$ denote the number of ballot permutations with $d$ descents. Let $p(n,d)$ denote the number of odd order permutations with $M(\\pi)=d$, "},"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":"1810.00993","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-10-01T22:02:18Z","cross_cats_sorted":[],"title_canon_sha256":"22d3c302809eab8c5c42fe46494b216fba4180e104589af5bad097da35ee0016","abstract_canon_sha256":"775c9dde12436166b220ce43318e2114996d55eb1e88234ae1fda95759f9b067"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:19.481819Z","signature_b64":"MMbJ0LYyz/9pYEQ9u9S+BI+clR/RXPGizyFe77LlYUKJuEpeDsaG17VoRb19kDyTH3VsberCj1xoK/UUmYOdDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8a1a6796fdd94f3b0c92bff58a2100b3d0b54fa17ead1ddcd2016625528a698","last_reissued_at":"2026-05-17T23:51:19.481128Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:19.481128Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ballot Permutations and Odd Order Permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sam Spiro","submitted_at":"2018-10-01T22:02:18Z","abstract_excerpt":"A permutation $\\pi$ is ballot if, for all $k$, the word $\\pi_1\\cdots \\pi_k$ has at least as many ascents as it has descents. Let $b(n)$ denote the number of ballot permutations of order $n$, and let $p(n)$ denote the number of permutations which have odd order in the symmetric group $S_n$. Callan conjectured that $b(n)=p(n)$ for all $n$, which was proved by Bernardi, Duplantier, and Nadeau.\n  We propose a refinement of Callan's original conjecture. Let $b(n,d)$ denote the number of ballot permutations with $d$ descents. 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