{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:32Z","timestamp":1753893812451,"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 $\\alpha_k(\\lambda)$ denote the number of $k$-hooks in a partition $\\lambda$ and let\u00a0$b(n,k)$ be the maximum value of $\\alpha_k(\\lambda)$ among partitions of $n$.\u00a0Amdeberhan\u00a0posed a conjecture on the generating function\u00a0of $b(n,1)$. We give a proof of this conjecture. In general,\u00a0we obtain a formula that can be used to determine $b(n,k)$. This leads to a generating function formula for $b(n,k)$.\u00a0We introduce the notion of\u00a0nearly $k$-triangular partitions. We show that for any $n$,\u00a0there is a nearly $k$-triangular partition which can be transformed into\u00a0a partition of $n$ that attains the maximum number\u00a0of $k$-hooks. The operations for the transformation enable us to compute the\u00a0number $b(n,k)$.<\/jats:p>","DOI":"10.37236\/2991","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:31:53Z","timestamp":1578706313000},"source":"Crossref","is-referenced-by-count":0,"title":["On the Maximum Number of $k$-Hooks of Partitions of $n$"],"prefix":"10.37236","volume":"20","author":[{"given":"Anna R.B.","family":"Fan","sequence":"first","affiliation":[]},{"given":"Harold R.L.","family":"Yang","sequence":"additional","affiliation":[]},{"given":"Rebecca T.","family":"Yu","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,6,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p49\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p49\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:18:49Z","timestamp":1579259929000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i2p49"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,6,7]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2013,4,9]]}},"URL":"https:\/\/doi.org\/10.37236\/2991","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,6,7]]},"article-number":"P49"}}