{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T04:22:55Z","timestamp":1768969375126,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $\\psi(\\boldsymbol{x}, \\boldsymbol{y},\\boldsymbol{z}; 1, 1+\\beta)$ with an additional parameter $\\beta$ that might be interpreted as a continuous deformation of the rooted bipartite maps generating series. Indeed, it has a property that for $\\beta \\in \\{0,1\\}$, it specializes to the rooted, orientable (general, i.e. orientable or not, respectively) bipartite maps generating series. They made the following conjecture: coefficients of $\\psi$ are polynomials in $\\beta$ with positive integer coefficients that can be written as a multivariate generating series of rooted, general bipartite maps, where the exponent of $\\beta$ is an integer-valued statistics that in some sense \"measures the non-orientability\" of the corresponding bipartite map.We show that except two special values of $\\beta = 0,1$ for which the combinatorial interpretation of the coefficients of $\\psi$ is known, there exists a third special value $\\beta = -1$ for which the coefficients of $\\psi$ indexed by two partitions $\\mu,\\nu$, and one partition with only one part are given by rooted, orientable bipartite maps with arbitrary face degrees and black\/white vertex degrees given by $\\mu$\/$\\nu$, respectively. We show that this evaluation corresponds, up to a sign, to a top-degree part of the coefficients of $\\psi$. As a consequence, we introduce a collection of integer-valued statistics of maps $(\\eta)$ such that the top-degree of the multivariate generating series of rooted, bipartite maps with only one face (called unicellular) with respect to $\\eta$ gives the top degree of the appropriate coefficients of $\\psi$. Finally, we show that $b$ conjecture holds true for all rooted, unicellular bipartite maps of genus at most $2$.<\/jats:p>","DOI":"10.37236\/6130","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:50:28Z","timestamp":1578671428000},"source":"Crossref","is-referenced-by-count":6,"title":["Top Degree Part in $b$-Conjecture for Unicellular Bipartite Maps"],"prefix":"10.37236","volume":"24","author":[{"given":"Maciej","family":"Do\u0142\u0119ga","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,8,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p24\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i3p24\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:50:26Z","timestamp":1579236626000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i3p24"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,8,11]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,7,14]]}},"URL":"https:\/\/doi.org\/10.37236\/6130","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,8,11]]},"article-number":"P3.24"}}