{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T22:44:43Z","timestamp":1775774683856,"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 $a_1,\\ldots,a_n$ be distinct, positive integers with $(a_1,\\ldots,a_n)=1$, and let k be an arbitrary field.  Let $H(a_1,\\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\\ldots,t^{a_n}]$.  We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms.  By way of contrast, it is known that no such expression exists for any $n\\geq4$.<\/jats:p>","DOI":"10.37236\/1729","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T07:03:15Z","timestamp":1578639795000},"source":"Crossref","is-referenced-by-count":27,"title":["Short Generating Functions for some Semigroup Algebras"],"prefix":"10.37236","volume":"10","author":[{"given":"Graham","family":"Denham","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2003,9,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v10i1r36\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v10i1r36\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:06:51Z","timestamp":1579324011000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v10i1r36"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,9,17]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2003,1,6]]}},"URL":"https:\/\/doi.org\/10.37236\/1729","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,9,17]]},"article-number":"R36"}}