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These \"R-permutations\" depict the min length coset reps for the quotient of S_n by the parabolic subgroup W_J, with J the complement of R. We refer to those that blockwise avoid the pattern 312 as \"312-avoiding R-permutations\" and define the \"parabolic R-Catalan number\" to be the number of them. Let lambda be a partition of N with at most n parts whose set of shape column lengths less than n is R. We show that the number"},"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":"1612.06323","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-19T19:46:46Z","cross_cats_sorted":[],"title_canon_sha256":"115f94c913b89949100a9ed839afcb33143ddb1fa2549f699ff8f7f49964adf3","abstract_canon_sha256":"a65043fdd1d45f1f6da6b1290f5f203badc787a16f4855e9716d54877b19cda2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:57.790631Z","signature_b64":"2JnoakfVsktX13tonvdbozUrhXnnHrWi2A8r2a3dJAxnq20Da8hnxQooRy3eAVRpke3vyVnGmRUlgXCp8TxVCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e497963876b7a1d2750d2c05e9834829dc8b308ac655dac383f558f24f968eb","last_reissued_at":"2026-05-18T00:49:57.789978Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:57.789978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Parabolic Catalan numbers count flagged Schur functions; Convexity of tableau sets for Demazure characters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Matthew J. Willis, Robert A. Proctor","submitted_at":"2016-12-19T19:46:46Z","abstract_excerpt":"Shuffles are n-multipermutations with suit multiplicities given by a subset R of {1,..,n-1}. Their inverses are ordered partitions of {1,..,n} whose block sizes derive from R. These \"R-permutations\" depict the min length coset reps for the quotient of S_n by the parabolic subgroup W_J, with J the complement of R. We refer to those that blockwise avoid the pattern 312 as \"312-avoiding R-permutations\" and define the \"parabolic R-Catalan number\" to be the number of them. Let lambda be a partition of N with at most n parts whose set of shape column lengths less than n is R. 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