{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:22:26Z","timestamp":1759335746326,"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 present uniformly available simple enumerative formulae for unrooted planar $n$-edge maps (counted up to orientation-preserving isomorphism) of numerous classes including arbitrary, loopless, non-separable, eulerian maps and plane trees.  All the formulae conform to a certain pattern with respect to the terms of the sum over $t\\mid n,\\,t\\! &lt; \\!n.$ Namely, these terms, which correspond to non-trivial automorphisms of the maps, prove to be of the form $\\phi\\left({n\\over t}\\right)\\alpha\\,r^t {k\\,t\\choose t}$, where $\\phi(m)$ is the Euler function, $k$ and $r$ are integer constants and $\\alpha$ is a constant or takes only two rational values.  On the contrary, the main, \"rooted\" summand corresponding to $t=n$ contains an additional factor which is a rational function of $n$.  Two simple new enumerative results are deduced for bicolored eulerian maps. A collateral aim is to briefly survey recent and old results of unrooted planar map enumeration.<\/jats:p>","DOI":"10.37236\/1841","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T21:30:59Z","timestamp":1578691859000},"source":"Crossref","is-referenced-by-count":6,"title":["Enumerative Formulae for Unrooted Planar Maps: a Pattern"],"prefix":"10.37236","volume":"11","author":[{"given":"Valery A.","family":"Liskovets","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2004,12,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r88\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r88\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:52:26Z","timestamp":1579305146000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i1r88"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,12,7]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2004,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/1841","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2004,12,7]]},"article-number":"R88"}}