{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T15:36:13Z","timestamp":1759073773995,"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 $C_{m,n}$ be the graph on the vertex set $\\{1, \\ldots, m\\} \\times \\{0, \\ldots, n-1\\}$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\\pm 1)$ or $(a,b) = (c \\pm 1,d)$, where the second index is computed modulo $n$.  One may view $C_{m,n}$ as a unit square grid on a cylinder with circumference $n$ units.  For odd $n$, we prove that the Euler characteristic of the simplicial complex $\\Sigma_{m,n}$ of independent sets in $C_{m,n}$ is either $2$ or $-1$, depending on whether or not $\\gcd(m-1,n)$ is divisble by $3$. The proof relies heavily on previous work due to Thapper, who reduced the problem of computing the Euler characteristic of $\\Sigma_{m,n}$ to that of analyzing a certain subfamily of sets with attractive properties. The situation for even $n$ remains unclear.  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\/71","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:22:18Z","timestamp":1578716538000},"source":"Crossref","is-referenced-by-count":8,"title":["Hard Squares with Negative Activity on Cylinders with Odd Circumference"],"prefix":"10.37236","volume":"16","author":[{"given":"Jakob","family":"Jonsson","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,3,23]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i2r5\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i2r5\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T03:05:13Z","timestamp":1579316713000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i2r5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,3,23]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2009,2,11]]}},"URL":"https:\/\/doi.org\/10.37236\/71","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,3,23]]},"article-number":"R5"}}