{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:51Z","timestamp":1753893831533,"version":"3.41.2"},"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 $I$ be the ideal generated by alternating polynomials in two sets of $n$ variables. Haiman proved that the $q,t$-Catalan number is the Hilbert series of the bi-graded vector space $M(=\\bigoplus_{d_1,d_2}M_{d_1,d_2})$ spanned by a minimal set of generators for $I$. In this paper we give simple upper bounds on $\\text{dim }M_{d_1, d_2}$ in terms of number of partitions, and find all bi-degrees $(d_1,d_2)$ such that $\\dim M_{d_1, d_2}$ achieve the upper bounds. For such bi-degrees, we also find explicit bases for $M_{d_1, d_2}$.<\/jats:p>","DOI":"10.37236\/645","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:43:26Z","timestamp":1578714206000},"source":"Crossref","is-referenced-by-count":0,"title":["$q,t$-Catalan Numbers and Generators for the Radical Ideal defining the Diagonal Locus of $({\\mathbb C}^2)^n$"],"prefix":"10.37236","volume":"18","author":[{"given":"Kyungyong","family":"Lee","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Li","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,8,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p158\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p158\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:07:27Z","timestamp":1579302447000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p158"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,8,5]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/645","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,8,5]]},"article-number":"P158"}}