{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:02Z","timestamp":1753893782239,"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>Let $s_d(n)$ be the number of distinct decompositions of the $d$-dimensional hypercube with $n$ rectangular regions that can be obtained via a sequence of splitting operations. We prove that the generating series $y = \\sum_{n \\geq 1} s_d(n)x^n$ satisfies the functional equation $x = \\sum_{n\\geq 1} \\mu_d(n)y^n$, where $\\mu_d(n)$ is the $d$-fold Dirichlet convolution of the M\u00f6bius function. This generalizes a recent result by Goulden et al., and shows that $s_1(n)$ also gives the number of natural exact covering systems of $\\mathbb{Z}$ with $n$ residual classes. We also prove an asymptotic formula for $s_d(n)$ and describe a bijection between $1$-dimensional decompositions and natural exact covering systems.<\/jats:p>","DOI":"10.37236\/11234","type":"journal-article","created":{"date-parts":[[2024,5,3]],"date-time":"2024-05-03T16:14:45Z","timestamp":1714752885000},"source":"Crossref","is-referenced-by-count":0,"title":["Decompositions of Unit Hypercubes and the Reversion of a Generalized M\u00f6bius Series"],"prefix":"10.37236","volume":"31","author":[{"given":"Yu Hin (Gary)","family":"Au","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2024,5,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i2p26\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i2p26\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,3]],"date-time":"2024-05-03T16:14:45Z","timestamp":1714752885000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v31i2p26"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,3]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,4,5]]}},"URL":"https:\/\/doi.org\/10.37236\/11234","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2024,5,3]]},"article-number":"P2.26"}}