{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:25:28Z","timestamp":1759335928873,"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>In 2012 Andrews and Merca gave a new expansion for partial sums of Euler's pentagonal number series and expressed \\[\\sum_{j=0}^{k-1}(-1)^j(p(n-j(3j+1)\/2)-p(n-j(3j+5)\/2-1))=(-1)^{k-1}M_k(n)\\] where $M_k(n)$ is the number of partitions of $n$ where $k$ is the least integer that does not occur as a part and there are more parts greater than $k$ than there are less than $k$. We will show that $M_k(n)=C_k(n)$ where $C_k(n)$ is the number of partition pairs $(S, U)$ where $S$ is a partition with parts greater than $k$, $U$ is a partition with $k-1$ distinct parts all of which are greater than the smallest part in $S$, and the sum of the parts in $S \\cup U$ is $n$. We use partition pairs to determine what is counted by three similar expressions involving linear combinations of pentagonal numbers.\u00a0Most of the results will be presented analytically and combinatorially.<\/jats:p>","DOI":"10.37236\/4917","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T18:39:22Z","timestamp":1578681562000},"source":"Crossref","is-referenced-by-count":8,"title":["Interpreting the Truncated Pentagonal Number Theorem using Partition Pairs"],"prefix":"10.37236","volume":"22","author":[{"given":"Louis W.","family":"Kolitsch","sequence":"first","affiliation":[]},{"given":"Michael","family":"Burnette","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2015,6,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i2p55\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i2p55\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:15:29Z","timestamp":1579238129000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i2p55"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,22]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2015,4,14]]}},"URL":"https:\/\/doi.org\/10.37236\/4917","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2015,6,22]]},"article-number":"P2.55"}}