{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:37Z","timestamp":1753893817777,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $\\alpha$  be a formal variable and $F_w$ be a weighted species of structures (class of structures closed under weight-preserving isomorphisms) of the form ${F}_{w} = E({F}_{w}^{c})$, where $E$ and $F_w^c$ respectively denote the   species of sets and of connected $F_w$-structures. Multiplying by $\\alpha$ the weight of each $F_w^c$-structure yields the species ${F}_{{w}^{( \\alpha   )}} = E({F}_{ \\alpha w}^{c})$. We introduce a \"universal\" virtual weighted species, $\\Lambda ^{(\\alpha)}$, such that $F_{w^{(\\alpha)}} = \\Lambda^{( \\alpha)}\\, \\circ \\, F_w^+$, where   $F_w^+$ denotes the species of non-empty $F_w$-structures. Using general properties of $\\Lambda^{( \\alpha)}$ , we compute the various enumerative power series $G(x)$, $\\widetilde{G}(x)$, $\\overline{G}(x)$, $G(x;q)$, $G\\langle{x;q}\\rangle$, ${Z}_{G}(x_1,x_2,x_3,\\ldots)$, ${\\Gamma }_{G}(x_1,x_2,x_3,\\ldots)$, for $G = F_{w^{(\\alpha)}}$, in terms of $F_w$. Special instances of our formulas include the exponential formula, ${F}_{{w}^{(\\alpha )}}(x)=\\exp(\\alpha F_{w}(x))=({F}_{w}(x){)}^{\\alpha }$,  cyclotomic identities, and their $q$-analogues. The virtual weighted species, $\\Lambda ^{(\\alpha)}$, is, in fact, a new combinatorial lifting of the function ${(1+x)}^{\\alpha }$.<\/jats:p>","DOI":"10.37236\/1270","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:51:57Z","timestamp":1578689517000},"source":"Crossref","is-referenced-by-count":1,"title":["An Extension of the Exponential Formula in  Enumerative Combinatorics"],"prefix":"10.37236","volume":"3","author":[{"given":"Gilbert","family":"Labelle","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pierre","family":"Leroux","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[1995,7,15]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v3i2r12\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v3i2r12\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:31:31Z","timestamp":1579321891000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v3i2r12"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,7,15]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[1996,1,24]]}},"URL":"https:\/\/doi.org\/10.37236\/1270","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[1995,7,15]]},"article-number":"R12"}}