{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:30Z","timestamp":1753893810572,"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>A bicoloured Dyck path is a Dyck path in which each up-step is assigned one of two colours, say, red and green. We say that a permutation $\\pi$ is $\\sigma$-segmented if every occurrence $o$ of $\\sigma$ in $\\pi$ is a segment-occurrence (i.e., $o$ is a contiguous subword in $\\pi$). We show combinatorially the following two results:  The $132$-segmented permutations of length $n$ with $k$ occurrences of $132$ are in one-to-one correspondence with bicoloured Dyck paths of length $2n-4k$ with $k$ red up-steps. Similarly, the $123$-segmented permutations of length $n$ with $k$ occurrences of $123$ are in one-to-one correspondence with bicoloured Dyck paths of length $2n-4k$ with $k$ red up-steps, each of height less than $2$. We enumerate the permutations above by enumerating the corresponding bicoloured Dyck paths. More generally, we present a bivariate generating function for the number of bicoloured Dyck paths of length $2n$ with $k$ red up-steps, each of height less than $h$. This generating function is expressed in terms of Chebyshev polynomials of the second kind.<\/jats:p>","DOI":"10.37236\/1936","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:10:10Z","timestamp":1578719410000},"source":"Crossref","is-referenced-by-count":0,"title":["Counting Segmented Permutations Using Bicoloured Dyck Paths"],"prefix":"10.37236","volume":"12","author":[{"given":"Anders","family":"Claesson","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2005,8,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r39\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r39\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:47:58Z","timestamp":1579322878000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v12i1r39"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,8,17]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2005,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1936","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2005,8,17]]},"article-number":"R39"}}