×

Finite difference calculus for alternating permutations. (English) Zbl 1284.39001

The article deals with the following difference equation \[ \Delta^2f_n(k) + 4f_{n-1}(k) = 0 \;\;(n \geq 2, \;1 \leq k \leq 2n - 3),\tag{1} \] considered by Christiane Poupard; the unknown function \(f = (f_n(k))\) (\(n \geq 1\), \(1 \leq k \leq 2n-1\)) is usually written in the following form: \[ \begin{matrix} &&&& f_1(1) &&&& \\ &&& f_2(1) & f_2(2) & f_2(3) &&& \\ && f_3(1) & f_3(2) & f_3(3) & f_3(4) & f_3(5) && \\ & f_4(1) & f_4(2) & f_4(3) & f_4(4) & f_4(5) & f_4(6) & f_4(7) & \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \end{matrix}\tag{2} \] The main results of the article are the exact formulas for solutions of (1) under the following initial conditions: \[ f_1(1) = 1; \;f_n(1) = 0 \;\;\text{and} \;\;f_n(2) = 2 \sum_k f_{n-1}(k) \;\;\text{for} \;\;n \geq 2;\tag{3} \]
\[ f_1(1) = 1; \;f_n(1) = f_n(2n - 1) = 0 \;\;\text{for} \;\;n \geq 2;\tag{4} \]
\[ f_1(1) = 1; \;f_n(1) = \sum_k f_{n-1}(k) \;\;\text{and} \;\;f_n(2) = 3 \sum_k f_{n-1}(k) \;\;\text{for} \;\;n \geq 2;\tag{5} \]
\[ f_1(1) = 1; \;f_n(1) = f_n(2n - 1) = \sum_k f_{n-1}(k) \;\;\text{for} \;\;n \geq 2.\tag{6} \] In particular, it is shown that the solution of (1) satisfying (3) or (4) are the same; the similar statement holds for the solutions satisfying (5) and (6). The corresponding solutions (2) generate so-called tangent and secant numbers. The end of the article presents some properties of these numbers and describes some curious connections between these numbers and the corresponding binary trees.

MSC:

39A05 General theory of difference equations

Software:

OEIS
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] André D., C.R. Math. Acad. Sci. Paris 88 pp 965– (1879)
[2] André D., J. Math. Pures Appl. 7 pp 167– (1881)
[3] DOI: 10.1016/S0195-6698(80)80027-8 · Zbl 0455.10006 · doi:10.1016/S0195-6698(80)80027-8
[4] Andrews G., Proc. Am. Math. Soc. 68 pp 380– (1978)
[5] DOI: 10.1007/978-94-010-2196-8 · doi:10.1007/978-94-010-2196-8
[6] Entringer R.C., Nieuw Arch. Wisk. 14 pp 241– (1966)
[7] Foata D., Proc. Am. Math. Soc. 81 pp 143– (1981)
[8] DOI: 10.1007/s11139-009-9194-9 · Zbl 1218.05013 · doi:10.1007/s11139-009-9194-9
[9] Foata D., Münster J. Math. 3 pp 129– (2010)
[10] DOI: 10.1090/S0002-9939-09-10144-2 · Zbl 1226.05023 · doi:10.1090/S0002-9939-09-10144-2
[11] DOI: 10.1093/qmath/hap043 · Zbl 1225.05031 · doi:10.1093/qmath/hap043
[12] Foata D., Manuscript (unabridged version) 71 pages (1971)
[13] D.Foata and M.P.Schützenberger, Nombres d’Euler et permutations alternantes, in A Survey of Combinatorial Theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Co., 1971), North-Holland, Amsterdam, 1973, pp. 173–187.
[14] Fulmek M., Sém. Lothar. Combin. 45 (2000)
[15] DOI: 10.1016/j.ejc.2010.07.004 · Zbl 1201.05004 · doi:10.1016/j.ejc.2010.07.004
[16] G.N.Han, The Poupard Statistics on Tangent and Secant Trees, Strasbourg, preprint (2013), 12 p.
[17] G.N.Han, A.Randrianarivony, and J.Zeng, Un autre q-analogue des nombres d’Euler, in The Andrews Festschrift. Seventeen Papers on Classical Number Theory and Combinatorics, D.Foata and G.N.Han, eds. Springer-Verlag, Berlin Heidelberg. Sém. Lothar. Combin., Art. B42e, 22 pp, 2001, pp. 139–158.
[18] C.Jordan, Calculus of Finite Differences, Röttig and Romwalter, Budapest, 1939.
[19] DOI: 10.1016/j.ejc.2010.01.008 · Zbl 1207.05007 · doi:10.1016/j.ejc.2010.01.008
[20] Kitaev S., Sém. Lothar. Combin. 68 (2012)
[21] Kuznetsov A.G., Uspekhi Mat. Nauk 49 pp 79– (1994)
[22] N.Nielsen, Traité élémentaire des nombres de Bernoulli, Gauthier-Villars, Paris, 1923.
[23] DOI: 10.1016/0012-365X(82)90293-X · Zbl 0482.05006 · doi:10.1016/0012-365X(82)90293-X
[24] DOI: 10.1016/S0195-6698(89)80009-5 · Zbl 0692.05022 · doi:10.1016/S0195-6698(89)80009-5
[25] DOI: 10.1006/eujc.1997.0147 · Zbl 0886.05011 · doi:10.1006/eujc.1997.0147
[26] DOI: 10.1155/S0161171200004439 · Zbl 0965.05012 · doi:10.1155/S0161171200004439
[27] Prodinger H., Sém. Lothar. Combin. 60 (2008)
[28] DOI: 10.1016/j.ejc.2010.04.003 · Zbl 1258.05003 · doi:10.1016/j.ejc.2010.04.003
[29] N.J.A. Sloane, The On-line Encyclopedia of Integer Sequences, http://www.otis.org. · Zbl 1274.11001
[30] DOI: 10.1017/CBO9780511609589 · doi:10.1017/CBO9780511609589
[31] DOI: 10.1090/conm/531/10466 · doi:10.1090/conm/531/10466
[32] X.G. Viennot, Séries génératrices énumératives, chap. 3, Lecture Notes, 160 pp., 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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.