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2014 | 12 | 12 | 1852-1870
Tytuł artykułu

Secant tree calculus

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A true Tree Calculus is being developed to make a joint study of the two statistics “eoc” (end of minimal chain) and “pom” (parent of maximum leaf) on the set of secant trees. Their joint distribution restricted to the set {eoc-pom ≤ 1} is shown to satisfy two partial difference equation systems, to be symmetric and to be expressed in the form of an explicit three-variable generating function.
Wydawca
Czasopismo
Rocznik
Tom
12
Numer
12
Strony
1852-1870
Opis fizyczny
Daty
wydano
2014-12-01
online
2014-07-20
Twórcy
autor
Bibliografia
  • [1] Désiré André, Développement de sec x et tan x, C. R. Math. Acad. Sci. Paris, 88 (1879), 965–979.
  • [2] Désiré André, Sur les permutations alternées, J. Math. Pures et Appl., 7 (1881), 167–184.
  • [3] Louis Comtet, Advanced Combinatorics, D. Reidel/Dordrecht-Holland, Boston, 1974. http://dx.doi.org/10.1007/978-94-010-2196-8
  • [4] R.C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Arch. Wisk., 14 (1966), 241–246.
  • [5] Dominique Foata; Guo-Niu Han, The doubloon polynomial triangle, Ramanujan J., 23(2010), 107–126 (The Andrews Festschrift). http://dx.doi.org/10.1007/s11139-009-9194-9
  • [6] Dominique Foata; Guo-Niu Han, Doubloons and q-secant numbers, Münster J. of Math., 3(2010), 129–150.
  • [7] Dominique Foata; Guo-Niu Han, Doubloons and new q-tangent numbers, Quarterly J. Math., 62(2011), 417–432. http://dx.doi.org/10.1093/qmath/hap043
  • [8] Dominique Foata; Guo-Niu Han, Finite difference calculus for alternating permutations, J. Difference Equations and Appl., 19(2013), 1952–1966. http://dx.doi.org/10.1080/10236198.2013.794794
  • [9] Dominique Foata; Guo-Niu Han, Tree Calculus for Bivariate Difference Equations, J. Difference Equations and Appl., 2014, to appear, 36 pages.
  • [10] Markus Fulmek, A continued fraction expansion for a q-tangent function, Sém. Lothar. Combin., B45b(2000), 3pp.
  • [11] Yoann Gelineau; Heesung Shin; Jiang Zeng, Bijections for Entringer families, Europ. J. Combin., 32(2011), 100–115. http://dx.doi.org/10.1016/j.ejc.2010.07.004
  • [12] Guo-Niu Han; Arthur Randrianarivony; Jiang Zeng, Un autre q-analogue des nombres d’Euler, The Andrews Festschrift. Seventeen Papers on Classical Number Theory and Combinatorics, D. Foata, G.-N. Han eds., Springer-Verlag, Berlin Heidelberg, 2001, pp. 139–158. Sém. Lothar. Combin., Art. B42e, 22 pp.
  • [13] Charles Jordan, Calculus of Finite Differences, Röttig and Romwalter, Budapest, 1939.
  • [14] M. Josuat-Vergès, A q-enumeration of alternating permutations, Europ. J. Combin., 31(2010), 1892–1906. http://dx.doi.org/10.1016/j.ejc.2010.01.008
  • [15] Sergey Kitaev; Jeffrey Remmel, Quadrant Marked Mesh Patterns in Alternating Permutations, Sém. Lothar. Combin., B68a (2012), 20pp.
  • [16] A. G. Kuznetsov; I. M. Pak; A. E. Postnikov, Increasing trees and alternating permutations, Uspekhi Mat. Nauk, 49(1994), 79–110.
  • [17] Niels Nielsen, Traité élémentaire des nombres de Bernoulli, Paris, Gauthier-Villars, 1923.
  • [18] Christiane Poupard, De nouvelles significations énumératives des nombres d’Entringer, Discrete Math., 38(1982), 265–271. http://dx.doi.org/10.1016/0012-365X(82)90293-X
  • [19] Christiane Poupard, Deux propriétés des arbres binaires ordonnés stricts, Europ. J. Combin., 10(1989), 369–374. http://dx.doi.org/10.1016/S0195-6698(89)80009-5
  • [20] Christiane Poupard, Two other interpretations of the Entringer numbers, Europ. J. Combin., 18(1997), 939–943. http://dx.doi.org/10.1006/eujc.1997.0147
  • [21] Helmut Prodinger, Combinatorics of geometrically distributed random variables: new q-tangent and q-secant numbers, Int. J. Math. Math. Sci., 24(2000), 825–838. http://dx.doi.org/10.1155/S0161171200004439
  • [22] Helmut Prodinger, A Continued Fraction Expansion for a q-Tangent Function: an Elementary Proof, Sém. Lothar. Combin., B60b (2008), 3 pp.
  • [23] Richard P. Stanley, A Survey of Alternating Permutations, in Combinatorics and graphs, 165–196, Contemp. Math., 531, Amer. Math. Soc. Providence, RI, 2010. http://dx.doi.org/10.1090/conm/531/10466
  • [24] Heesung Shin; Jiang Zeng, The q-tangent and q-secant numbers via continued fractions, Europ. J. Combin., 31(2010), 1689–1705. http://dx.doi.org/10.1016/j.ejc.2010.04.003
  • [25] Xavier G. Viennot, Séries génératrices énumératives, chap. 3, Lecture Notes, 160 p., 1988, notes de cours donnés à l’École Normale Supérieure Ulm (Paris), UQAM (Montréal, Québec) et Université de Wuhan (Chine) http://web.mac.com/xgviennot/Xavier_Viennot/cours.html.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-014-0429-7
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