{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:19:25Z","timestamp":1759335565106,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p> A $321$-$k$-gon-avoiding permutation $\\pi$ avoids $321$ and the following four patterns: $$k(k+2)(k+3)\\cdots(2k-1)1(2k)23\\cdots(k-1)(k+1),$$ $$k(k+2)(k+3)\\cdots(2k-1)(2k)12\\cdots(k-1)(k+1),$$ $$(k+1)(k+2)(k+3)\\cdots(2k-1)1(2k)23\\cdots k,$$ $$(k+1)(k+2)(k+3)\\cdots(2k-1)(2k)123\\cdots k.$$ The $321$-$4$-gon-avoiding permutations were introduced and studied by Billey and Warrington [BW] as a class of elements of the symmetric group whose Kazhdan-Lusztig, Poincar\u00e9 polynomials, and the singular loci of whose Schubert varieties have fairly simple formulas and descriptions. Stankova and West [SW] gave an exact enumeration in terms of linear recurrences with constant coefficients for the cases $k=2,3,4$. In this paper, we extend these results by finding an explicit expression for the generating function for the number of $321$-$k$-gon-avoiding permutations on $n$ letters. The generating function is expressed via Chebyshev polynomials of the second kind.<\/jats:p>","DOI":"10.37236\/1677","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:23:39Z","timestamp":1578709419000},"source":"Crossref","is-referenced-by-count":4,"title":["$321$-Polygon-Avoiding Permutations and Chebyshev Polynomials"],"prefix":"10.37236","volume":"9","author":[{"given":"Toufik","family":"Mansour","sequence":"first","affiliation":[]},{"given":"Zvezdelina","family":"Stankova","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2003,1,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v9i2r5\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v9i2r5\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:10:16Z","timestamp":1579324216000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v9i2r5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,1,22]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2002,10,31]]}},"URL":"https:\/\/doi.org\/10.37236\/1677","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2003,1,22]]},"article-number":"R5"}}