{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:29:31Z","timestamp":1759638571231,"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>We prove that the property of being closed (resp., palindromic, rich, privileged trapezoidal, balanced) is expressible in first-order logic for automatic (and some related) sequences. It therefore follows that the characteristic function of those $n$ for which an automatic sequence $\\bf x$ has a closed (resp., palindromic, privileged, rich, trapezoidal, balanced) factor of length\u00a0 $n$ is itself automatic. For privileged words this requires a new characterization of the privileged property. We compute the corresponding characteristic functions for various famous sequences, such as the Thue-Morse sequence, the Rudin-Shapiro sequence, the ordinary\u00a0 paperfolding sequence, the period-doubling sequence, and the Fibonacci sequence. Finally, we also show that the function counting the total number of palindromic factors in the prefix of length $n$ of a $k$-automatic sequence is not $k$-synchronized.<\/jats:p>","DOI":"10.37236\/5752","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T21:56:32Z","timestamp":1578693392000},"source":"Crossref","is-referenced-by-count":10,"title":["Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences"],"prefix":"10.37236","volume":"23","author":[{"given":"Luke","family":"Schaeffer","sequence":"first","affiliation":[]},{"given":"Jeffrey","family":"Shallit","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,2,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i1p25\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i1p25\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:36:14Z","timestamp":1579239374000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i1p25"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,2,5]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,1,11]]}},"URL":"https:\/\/doi.org\/10.37236\/5752","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,2,5]]},"article-number":"P1.25"}}