{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T20:10:50Z","timestamp":1774555850629,"version":"3.50.1"},"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>For a family \\(\\mathcal{F}\\) of words of length \\(n\\) drawn from an alphabet \\(A=[r]=\\{1,\\dots,r\\}\\), Danh and Daykin defined the deletion shadow \\(\\Delta \\mathcal{F}\\) as the family containing all words that can be made by deleting one letter of a word of \\(\\mathcal{F}\\). They asked, given the size of such a family, how small its deletion shadow can be, and answered this with a Kruskal-Katona type result when the alphabet has size \\(2\\). However, Leck showed that no ordering can give such a result for larger alphabets. The minimal shadow has been known for families of size \\(s^n\\), where the optimal family has form \\([s]^n\\). We give the minimal shadow for many intermediate sizes between these levels, showing that families of the form \"all words in \\([s]^n\\) in which the symbol \\(s\\) appears at most \\(k\\) times\" are optimal. This proves a conjecture of Bollob\u00e1s and Leader. Our proof uses some fractional techniques that may be of independent interest.<\/jats:p>","DOI":"10.37236\/14141","type":"journal-article","created":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T19:48:29Z","timestamp":1774554509000},"source":"Crossref","is-referenced-by-count":0,"title":["New Optima for the Deletion Shadow"],"prefix":"10.37236","volume":"33","author":[{"given":"Benedict","family":"Randall Shaw","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2026,3,27]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p58\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p58\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T19:48:29Z","timestamp":1774554509000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i1p58"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,27]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/14141","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,27]]},"article-number":"P1.58"}}