{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T15:06:59Z","timestamp":1776870419011,"version":"3.51.2"},"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>We introduce and study new refinements of inversion statistics for permutations, such as $k$-step inversions, (the number of inversions with fixed position differences) and non-inversion sums (the sum of the differences of positions of the non-inversions of a permutation). We also provide a distribution function for non-inversion sums, a distribution function for $k$-step inversions that relates to the Eulerian polynomials, and special cases of distribution functions for other statistics we introduce, such as $(\\le\\!\\!k)$-step inversions and $(k_1,k_2)$-step inversions (that fix the value separation as well as the position).\u00a0 We connect our refinements to other work, such as inversion tops that are $0$ modulo a fixed integer $d$, left boundary sums of paths, and marked meshed patterns.\u00a0\u00a0 Finally, we use non-inversion sums to show that for every number $n&gt;34$, there is a permutation such that the dot product of that permutation and the identity permutation (of the same length) is $n$.<\/jats:p>","DOI":"10.37236\/1993","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:32:37Z","timestamp":1578713557000},"source":"Crossref","is-referenced-by-count":3,"title":["Refined Inversion Statistics on Permutations"],"prefix":"10.37236","volume":"19","author":[{"given":"Joshua","family":"Sack","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Henning","family":"\u00dalfarsson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2012,1,27]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p29\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p29\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T22:42:31Z","timestamp":1579300951000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i1p29"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,1,27]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2012,2,15]]}},"URL":"https:\/\/doi.org\/10.37236\/1993","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,1,27]]},"article-number":"P29"}}