{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T14:59:41Z","timestamp":1773845981078,"version":"3.50.1"},"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>Let $S_{m,n}$ be the graph on the vertex set ${\\Bbb Z}_m \\times {\\Bbb Z}_n$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\\pm 1)$ or $(a,b) = (c \\pm 1,d)$ modulo $(m,n)$. We present a formula for the Euler characteristic of the simplicial complex $\\Sigma_{m,n}$ of independent sets in $S_{m,n}$. In particular, we show that the unreduced Euler characteristic of $\\Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. For general $m$ and $n$, we relate the Euler characteristic of $\\Sigma_{m,n}$ to certain periodic rhombus tilings of the plane. Using this correspondence, we settle another conjecture due to Fendley et al., which states that all roots of $\\det (xI-T_m)$ are roots of unity, where $T_m$ is a certain transfer matrix associated to $\\{\\Sigma_{m,n} : n \\ge 1\\}$. In the language of statistical mechanics, the reduced Euler characteristic of $\\Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.<\/jats:p>","DOI":"10.37236\/1093","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:59:13Z","timestamp":1578718753000},"source":"Crossref","is-referenced-by-count":21,"title":["Hard Squares with Negative Activity  and Rhombus Tilings of the Plane"],"prefix":"10.37236","volume":"13","author":[{"given":"Jakob","family":"Jonsson","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2006,8,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v13i1r67\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v13i1r67\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:10:02Z","timestamp":1579320602000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v13i1r67"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,8,7]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2006,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1093","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,8,7]]},"article-number":"R67"}}