{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:22Z","timestamp":1753893862171,"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>Inspired by a new mathematical model for bobbin lace, this paper considers finite lattice paths formed from the set of step vectors $\\mathfrak{A}=$$\\{\\rightarrow,$ $\\nearrow,$ $\\searrow,$ $\\uparrow,$ $\\downarrow\\}$ with the restriction that vertical steps $(\\uparrow, \\downarrow)$ cannot be consecutive. The set $\\mathfrak{A}$ is the union of the well known Motzkin step vectors $\\mathfrak{M}=$$\\{\\rightarrow,$ $\\nearrow,$ $\\searrow\\}$ with the vertical steps $\\{\\uparrow, \\downarrow\\}$. An explicit bijection $\\phi$ between the exhaustive set of\u00a0vertically constrained\u00a0paths formed from $\\mathfrak{A}$ and a bisection of the paths generated by $\\mathfrak{M}S$ is presented. In a similar manner, paths with the step vectors $\\mathfrak{B}=$$\\{\\nearrow,$ $\\searrow,$ $\\uparrow,$ $\\downarrow\\}$, the union of Dyck step vectors and constrained vertical steps, are examined.\u00a0 We show, using the same $\\phi$ mapping, that there is a bijection between\u00a0vertically constrained\u00a0$\\mathfrak{B}$ paths and the subset of Motzkin paths avoiding horizontal steps at even indices.\u00a0 Generating functions are derived to enumerate these\u00a0vertically constrained, partially directed paths when restricted to the half and quarter-plane.\u00a0 Finally, we extend Schr\u00f6der and Delannoy step sets in a similar manner and find a bijection between these paths and a subset of Schr\u00f6der paths that are smooth (do not change direction) at a regular horizontal interval.<\/jats:p>","DOI":"10.37236\/7799","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T07:08:04Z","timestamp":1578640084000},"source":"Crossref","is-referenced-by-count":5,"title":["Vertically Constrained Motzkin-Like Paths Inspired by Bobbin Lace"],"prefix":"10.37236","volume":"26","author":[{"given":"Veronika","family":"Irvine","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stephen","family":"Melczer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Frank","family":"Ruskey","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2019,5,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i2p16\/7829","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i2p16\/7829","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:14:46Z","timestamp":1579234486000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i2p16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,3]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,4,5]]}},"URL":"https:\/\/doi.org\/10.37236\/7799","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2019,5,3]]},"article-number":"P2.16"}}