{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T04:01:05Z","timestamp":1769054465458,"version":"3.49.0"},"reference-count":31,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2006,10,6]],"date-time":"2006-10-06T00:00:00Z","timestamp":1160092800000},"content-version":"vor","delay-in-days":4115,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1995,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Mallows and Riordan \u201cThe Inversion Enumerator for Labeled Trees,\u201d <jats:italic>Bulletin of the American Mathematics Society<\/jats:italic>, vol. 74 [1968] pp. 92\u201094) first defined the inversion polynomial, <jats:italic>J<jats:sub>n<\/jats:sub>(q)<\/jats:italic> for trees with <jats:italic>n<\/jats:italic> vertices and found its generating function. In the present work, we define inversion polynomials for ordered, plane, and cyclic trees, and find their values at <jats:italic>q<\/jats:italic> = 0, \u00b1 1. Our techniques involve the use of generating functions (including Lagrange inversion), hypergeometric series, and binomial coefficient identities, induction, and bijections. We also derive asymptotic formulae for those results for which we do not have a closed form. \u00a9 1995 John Wiley &amp; Sons, Inc.<\/jats:p>","DOI":"10.1002\/jgt.3190190402","type":"journal-article","created":{"date-parts":[[2007,5,26]],"date-time":"2007-05-26T13:52:35Z","timestamp":1180187555000},"page":"435-459","source":"Crossref","is-referenced-by-count":13,"title":["Enumeration of trees by inversions"],"prefix":"10.1002","volume":"19","author":[{"given":"Ira M.","family":"Gessel","sequence":"first","affiliation":[]},{"given":"Bruce E.","family":"Sagan","sequence":"additional","affiliation":[]},{"given":"Yeong\u2010Nan","family":"Yeh","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,6]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"Generalized Hypergeometric Series","author":"Bailey W. N.","year":"1935"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1137\/1016082"},{"key":"e_1_2_1_4_2","volume-title":"Foundations of Applied Combinatorics","author":"Bender E. A.","year":"1991"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-55251-0_2"},{"key":"e_1_2_1_6_2","first-page":"508","article-title":"Note sur une \u00c9quation aux diff\u00e9rences finies","volume":"3","author":"Catalan E.","year":"1838","journal-title":"J. Math."},{"key":"e_1_2_1_7_2","article-title":"Some explanations of Dobinski's formula","author":"Chen B.","journal-title":"Stud. Appl. Math."},{"key":"e_1_2_1_8_2","unstructured":"W. Y. C.Chen Ph.D. thesis M.I.T. Cambridge MA (1991)."},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.87.24.9635"},{"key":"e_1_2_1_10_2","volume-title":"Asymptotic Methods in Analysis","author":"DeBruijn N. G.","year":"1981"},{"key":"e_1_2_1_11_2","volume-title":"100 Great Problems of Elementary Mathematics","author":"D\u00f6rrie H.","year":"1965"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(86)90004-X"},{"key":"e_1_2_1_13_2","unstructured":"D.Foata La S\u00e9rie G\u00e9n\u00e9ratrice Exponetielle dans les Probl\u00e9mes d'Enum\u00e9ration S\u00e9minaire de Math\u00e9matique Sup\u00e9rieurs No. 54. Presses de l'Universit\u00e9 de Montr\u00e9al Montr\u00e9al (1974)."},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(79)90108-0"},{"key":"e_1_2_1_15_2","volume-title":"Combinatorial Identities","author":"Gould H. W.","year":"1972"},{"key":"e_1_2_1_16_2","volume-title":"Concrete Mathematics: A Foundation for Computer Science","author":"Graham R. L.","year":"1989"},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(81)90052-9"},{"key":"e_1_2_1_18_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02107572"},{"key":"e_1_2_1_19_2","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(81)90041-4"},{"key":"e_1_2_1_20_2","doi-asserted-by":"publisher","DOI":"10.1016\/0747-7171(92)90037-5"},{"key":"e_1_2_1_21_2","series-title":"Combinatoire Enum\u00e9rative, Montr\u00e9al, 1985, Proceedings, Lecture Notes in Mathematics","first-page":"210","volume-title":"Combinatorial resolution of systems of differential equations, I. Ordinary differential equations","author":"Leroux P.","year":"1986"},{"key":"e_1_2_1_22_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1968-11888-9"},{"key":"e_1_2_1_23_2","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1978-085-0"},{"key":"e_1_2_1_24_2","series-title":"Canadian Mathematics Monographs","volume-title":"Counting Labeled Trees","author":"Moon J. W.","year":"1970"},{"key":"e_1_2_1_25_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(82)80037-1"},{"key":"e_1_2_1_26_2","article-title":"Hypergeometric solutions of linear recurrences with polynomial coefficients","author":"Petkov\u0161ek M.","journal-title":"J. Symbolic Comput."},{"key":"e_1_2_1_27_2","volume-title":"New Directions in the Theory of Graphs","author":"Robinson R. W.","year":"1973"},{"key":"e_1_2_1_28_2","first-page":"549","article-title":"Sur le nombre de mani\u00e8res d'effectuer un produit de n facteurs","volume":"3","author":"Rodrigues M. O.","year":"1838","journal-title":"J. Math."},{"key":"e_1_2_1_29_2","volume-title":"Generalized Hypergeometric Functions","author":"Slater L. J.","year":"1966"},{"key":"e_1_2_1_30_2","volume-title":"Generatingfunctionology","author":"Wilf H. S.","year":"1990"},{"key":"e_1_2_1_31_2","series-title":"Proceedings, Lecture Notes in Math.","first-page":"210","volume-title":"Combinatoire Enum\u00e9rative, Montr\u00e9al, 1985","author":"Yeh Y.\u2010N.","year":"1986"},{"key":"e_1_2_1_32_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0747-7171(08)80044-2"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190190402","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190190402","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T19:36:58Z","timestamp":1698262618000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190190402"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,7]]},"references-count":31,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1995,7]]}},"alternative-id":["10.1002\/jgt.3190190402"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190190402","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,7]]}}}