{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,21]],"date-time":"2025-09-21T17:28:00Z","timestamp":1758475680714,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In this note, we provide a new proof for the number of partitions of $n$ having subpartitions of length $\\ell$ with gap $d$. Moreover, by generalizing the definition of a subpartition, we show what is counted by $q$-expansion\\[\\prod_{n=1}^{\\infty} \\frac{1}{1-q^n} \\sum_{n=0}^{\\infty} (-1)^n q^{(an^2 + bn)\/2}\\]and how fast it grows. Moreover, we prove there is a special sign pattern for the coefficients of $q$-expansion\\[\\prod_{n=1}^{\\infty} \\frac{1}{1-q^n} \\left( 1 - 2 \\sum_{n=0}^{\\infty} (-1)^n q^{(an^2 + bn)\/2} \\right).\\]<\/jats:p>","DOI":"10.37236\/3526","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:53:45Z","timestamp":1578704025000},"source":"Crossref","is-referenced-by-count":3,"title":["On the Subpartitions of the Ordinary Partitions, II"],"prefix":"10.37236","volume":"21","author":[{"given":"Byungchan","family":"Kim","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eunmi","family":"Kim","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2014,10,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i4p21\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i4p21\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:43:13Z","timestamp":1579257793000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v21i4p21"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,10,30]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2014,10,2]]}},"URL":"https:\/\/doi.org\/10.37236\/3526","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2014,10,30]]},"article-number":"P4.21"}}