{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:27Z","timestamp":1753893867758,"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>Given a word, we are interested in the structure of its contiguous subwords split into $k$ blocks of equal length, especially in the homogeneous and anti-homogeneous cases.\u00a0 We introduce the notion of $(\\mu_1,\\dots,\\mu_k)$-block-patterns, words of the form $w = w_1\\cdots w_k$ where, when $\\{w_1,\\dots,w_k\\}$ is partitioned via equality, there are $\\mu_s$ sets of size $s$ for each $s \\in \\{1,\\dots,k\\}$.\u00a0 This is a generalization of the well-studied $k$-powers and the $k$-anti-powers\u00a0 recently introduced by Fici, Restivo, Silva, and Zamboni, as well as a refinement of the\u00a0 $(k,\\lambda)$-anti-powers introduced by Defant. We generalize the anti-Ramsey-type results of Fici et al. to $(\\mu_1,\\dots,\\mu_k)$-block-patterns and improve their bounds on $N_\\alpha(k,k)$, the minimum length such that every word of length $N_\\alpha(k,k)$ on an alphabet of size $\\alpha$ contains a $k$-power or $k$-anti-power.\u00a0 We also generalize their results on infinite words avoiding $k$-anti-powers to the case of $(k,\\lambda)$-anti-powers.\u00a0 We provide a few results on the relation between $\\alpha$ and\u00a0 $N_\\alpha(k,k)$ and find the expected number of $(\\mu_1,\\dots,\\mu_k)$-block-patterns in a word of length $n$.<\/jats:p>","DOI":"10.37236\/8032","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T14:55:44Z","timestamp":1578668144000},"source":"Crossref","is-referenced-by-count":2,"title":["$(k,\\lambda)$-Anti-Powers and Other Patterns in Words"],"prefix":"10.37236","volume":"25","author":[{"given":"Amanda","family":"Burcroff","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2018,11,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i4p41\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i4p41\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:21:03Z","timestamp":1579234863000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i4p41"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,30]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,10,5]]}},"URL":"https:\/\/doi.org\/10.37236\/8032","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2018,11,30]]},"article-number":"P4.41"}}