{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:16Z","timestamp":1753893796996,"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>A positive list (list of positive integers) is protographic if its merger with all but finitely many positive graphic lists is graphic.  Define the family ${\\cal P}_s$ of $s$-protogaphic lists by letting ${\\cal P}_0$ be the family of positive graphic lists and letting ${\\cal P}_s$ for $s&gt;0$ be the family of positive lists whose merger with all but finitely many lists in ${\\cal P}_{s-1}$ is in ${\\cal P}_{s-1}$. The main result is that $X\\in{\\cal P}_s$ if and only if $t(X)\\in {\\cal P}_{s-1}$, where $t(X)$ is the list obtained from $X$ by subtracting one from each term of $X$ (deleting those that become $0$) and appending a 1 for each term of $X$.  A corollary is that the maximum number of iterations to reach a graphic list from an $n$-term even list with sum $2k$ is $k-n+1$ (when $k\\ge n$), achieved by the unique such list having one term larger than 1.<\/jats:p>","DOI":"10.37236\/1757","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:23:55Z","timestamp":1578720235000},"source":"Crossref","is-referenced-by-count":0,"title":["Graphic and Protographic Lists of Integers"],"prefix":"10.37236","volume":"11","author":[{"given":"Dmitry","family":"Fon-Der-Flaass","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Douglas B.","family":"West","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2004,1,2]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r4\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r4\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:05:12Z","timestamp":1579323912000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i1r4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,1,2]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2004,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/1757","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2004,1,2]]},"article-number":"R4"}}