{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:11Z","timestamp":1753893851583,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The research on pattern-avoidance has yielded so far limited knowledge on Wilf-ordering of permutations. The Stanley-Wilf limits $\\lim_{n\\rightarrow \\infty} \\sqrt[n]{|S_n(\\tau)|}$ and further works suggest asymptotic ordering of layered versus monotone patterns. Yet, B\u00f3na has provided essentially the only known up to now result of its type on complete ordering of $S_k$ for $k=4$: $|S_n(1342)| &lt; |S_n(1234)| &lt; |S_n(1324)|$ for $n\\geq 7$, along with some other sporadic examples in Wilf-ordering.  We give a different proof of this result by ordering $S_3$ up to the stronger shape-Wilf-order: $|S_Y(213)|\\leq |S_Y(123)|\\leq |S_Y(312)|$ for any Young diagram $Y$, derive as a consequence that $|S_Y(k+2,k+1,k+3,\\tau)|\\leq |S_Y(k+1,k+2,k+3,\\tau)|\\leq |S_Y(k+3,k+1,k+2,\\tau)|$ for any $\\tau\\in S_k$, and find out when equalities are obtained. (In particular, for specific $Y$'s we find out that $|S_Y(123)|=|S_Y(312)|$ coincide with every other Fibonacci term.) This strengthens and generalizes B\u00f3na's result to arbitrary length permutations. While all length-3 permutations have been shown in numerous ways to be Wilf-equivalent, the current paper distinguishes between and orders these permutations by employing all Young diagrams. This opens up the question of whether shape-Wilf-ordering of permutations, or some generalization of it, is not the \"true\" way of approaching pattern-avoidance ordering.<\/jats:p>","DOI":"10.37236\/974","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:48:59Z","timestamp":1578718139000},"source":"Crossref","is-referenced-by-count":0,"title":["Shape-Wilf-Ordering on Permutations of Length 3"],"prefix":"10.37236","volume":"14","author":[{"given":"Zvezdelina","family":"Stankova","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2007,8,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1r56\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1r56\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:01:47Z","timestamp":1579320107000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v14i1r56"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,8,20]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2007,1,3]]}},"URL":"https:\/\/doi.org\/10.37236\/974","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2007,8,20]]},"article-number":"R56"}}