{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T17:15:28Z","timestamp":1765041328301,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We study relationships between permutation statistics and pattern-functions, counting the number of times particular patterns occur in a permutation. This allows us to write several familiar statistics as linear combinations of pattern counts, both in terms of a permutation and in terms of its image under the fundamental bijection. We use these enumerations to resolve the question of characterizing so-called \"shallow\" permutations, whose depth (equivalently, disarray\/displacement) is minimal with respect to length and reflection length. We present this characterization in several ways, including vincular patterns, mesh patterns, and a new object that we call \"arrow patterns.\" Furthermore, we specialize to characterizing and enumerating shallow involutions and shallow cycles, encountering the Motzkin and large Schr\u00f6der numbers, respectively.<\/jats:p>","DOI":"10.37236\/10858","type":"journal-article","created":{"date-parts":[[2022,12,16]],"date-time":"2022-12-16T10:05:37Z","timestamp":1671185137000},"source":"Crossref","is-referenced-by-count":2,"title":["Pattern-Functions, Statistics, and Shallow Permutations"],"prefix":"10.37236","volume":"29","author":[{"given":"Yosef","family":"Berman","sequence":"first","affiliation":[]},{"given":"Bridget Eileen","family":"Tenner","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2022,12,16]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i4p43\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i4p43\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,16]],"date-time":"2022-12-16T10:05:38Z","timestamp":1671185138000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v29i4p43"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,16]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,10,6]]}},"URL":"https:\/\/doi.org\/10.37236\/10858","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2022,12,16]]},"article-number":"P4.43"}}