{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:45:19Z","timestamp":1787323519608,"version":"build-2736575974"},"reference-count":26,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1601961"],"award-info":[{"award-number":["DMS-1601961"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>We discuss two surprising properties of a family of polynomials that generalize the Mahonian $q$-Catalan polynomials and, more generally, the $q$-Schr\u00f6der polynomials. By interpreting them as $\\mathfrak{sl}_2$-characters, we show that the rational $q$-Schr\u00f6der polynomials are parity-unimodal, which means that the even- and odd-degree coefficients are separately unimodal. Second, we show that they exhibit a $q=-1$ phenomenon. This is a special case of a more general cyclic sieving phenomenon for certain transitive $S_n$-actions, deduced from Molien's formula.<\/jats:p>","DOI":"10.1137\/20m1370112","type":"journal-article","created":{"date-parts":[[2021,9,13]],"date-time":"2021-09-13T12:37:38Z","timestamp":1631536658000},"page":"2049-2069","source":"Crossref","is-referenced-by-count":3,"title":["Parity-Unimodality and a Cyclic Sieving Phenomenon for Necklaces"],"prefix":"10.1137","volume":"35","author":[{"given":"Eric N.","family":"Stucky","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,9,13]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s00026-015-0293-6"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-03-11824-4"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1155\/S1073792803210205"},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"S. C. Billey, M. Konvalinka, and J. P. Swanson,\n                      On the Distribution of the Major Index on Standard Young Tableaux\n                      , preprint,https:\/\/arxiv.org\/abs\/2005.10341(2020).","DOI":"10.1016\/j.aam.2019.101972"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.37236\/5681"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/pdq027"},{"key":"atypb7","unstructured":"D. Callan and L. Smiley,\n                      Noncrossing Partitions under Rotation and Reflection\n                      , preprint,https:\/\/arxiv.org\/abs\/math\/0510447(2005)."},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1090\/S1088-4165-03-00214-0"},{"key":"atypb9","unstructured":"Z. Ding,\n                      Dihedral Symmetries of Non-crossing Partition Lattices\n                      , Ph.D. thesis, University of Miami, Coral Gables, FL, 2016."},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/BF01367762"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1155\/IMRN.2005.2409"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1112\/S0024611502013825"},{"key":"atypb13","unstructured":"P. Etingof and X. Ma,\n                      Lecture Notes on Cherednik Algebras\n                      , preprint,https:\/\/arxiv.org\/abs\/1001.0432(2010)."},{"key":"atypb14","unstructured":"P. Galashin and T. Lam,\n                      Positroids, Knots, and $q,t$-Catalan Numbers\n                      , preprint,https:\/\/arxiv.org\/abs\/2012.09745(2020)."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1142\/S1793042119501069"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022450120589"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/BF01077957"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/BF01075283"},{"key":"atypb19","first-page":"1152","volume":"52","author":"Molien V. F.","year":"1897","journal-title":"Sitzunger Preuss. Akad. Wiss."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/s00026-018-0408-y"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2004.04.009"},{"key":"atypb22","first-page":"067","volume":"16","author":"Shen L.","year":"2020","journal-title":"Symmetry Integrability Geom. Methods Appl."},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1111\/j.1749-6632.1989.tb16434.x"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(94)90112-0"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2016.09.006"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.37236\/8870"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1370112","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:07:14Z","timestamp":1787321234000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1370112"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":26,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/20M1370112"],"URL":"https:\/\/doi.org\/10.1137\/20m1370112","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}