{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T19:27:29Z","timestamp":1768678049115,"version":"3.49.0"},"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 consider domino tilings of $3$-dimensional cubiculated regions. A three-dimensional domino is a $2\\times 1\\times\u00a0 1$ rectangular cuboid. We are particularly interested in regions of the form $\\mathcal{R}_N = \\mathcal{D} \\times [0,N]$ where $\\mathcal{D} \\subset \\mathbb{R}^2$ is a fixed quadriculated disk. In dimension $3$, the twist associates to each tiling $\\mathbf{t}$ an integer $\\operatorname{Tw}(\\mathbf{t})$. We prove that, when $N$ goes to infinity, the twist follows a normal distribution.\r\nA flip is a local move: two neighboring parallel dominoes are removed and placed back in a different position. The twist is invariant under flips. A quadriculated disk $\\mathcal{D}$ is regular if, whenever two tilings $\\mathbf{t}_0$ and $\\mathbf{t}_1$ of $\\mathcal{R}_N$ satisfy $\\operatorname{Tw}(\\mathbf{t}_0) = \\operatorname{Tw}(\\mathbf{t}_1)$, $\\mathbf{t}_0$ and $\\mathbf{t}_1$ can be joined by a sequence of flips provided some extra vertical space is allowed.\r\nMany large disks are regular, including rectangles $\\mathcal{D} = [0,L] \\times [0,M]$ with $LM$ even and $\\min\\{L,M\\} \\ge 3$. For regular disks, we describe the larger connected components under flips of the set of tilings of the region $\\mathcal{R}_N = \\mathcal{D} \\times [0,N]$. As a corollary, let $p_N$ be the probability that two random tilings $\\mathbf{T}_0$ and $\\mathbf{T}_1$ of $\\mathcal{D} \\times [0,N]$ can be joined by a sequence of flips conditional to their twists being equal. Then $p_N$ tends to $1$ if and only if $\\mathcal{D}$ is regular.\r\nUnder a suitable equivalence relation, the set of tilings has a group structure, the {\\em domino group} $G_{\\mathcal{D}}$. These results illustrate the fact that the domino group dictates many properties of the space of tilings of the cylinder $\\mathcal{R}_N = \\mathcal{D} \\times [0,N]$, particularly for large $N$.<\/jats:p>","DOI":"10.37236\/9779","type":"journal-article","created":{"date-parts":[[2021,2,11]],"date-time":"2021-02-11T10:11:14Z","timestamp":1613038274000},"source":"Crossref","is-referenced-by-count":5,"title":["Domino Tilings of Cylinders: Connected Components under Flips and Normal Distribution of the Twist"],"prefix":"10.37236","volume":"28","author":[{"given":"Nicolau C.","family":"Saldanha","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2021,2,12]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i1p28\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i1p28\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,11]],"date-time":"2021-02-11T10:11:15Z","timestamp":1613038275000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i1p28"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,12]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1,14]]}},"URL":"https:\/\/doi.org\/10.37236\/9779","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,2,12]]},"article-number":"P1.28"}}