Voir la notice de l'article provenant de la source Cambridge University Press
Abramson, Morton. Permutations Related to Secant, Tangent and Eulerian Numbers. Canadian mathematical bulletin, Tome 22 (1979) no. 3, pp. 281-291. doi: 10.4153/CMB-1979-034-9
@article{10_4153_CMB_1979_034_9,
author = {Abramson, Morton},
title = {Permutations {Related} to {Secant,} {Tangent} and {Eulerian} {Numbers}},
journal = {Canadian mathematical bulletin},
pages = {281--291},
year = {1979},
volume = {22},
number = {3},
doi = {10.4153/CMB-1979-034-9},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1979-034-9/}
}
TY - JOUR AU - Abramson, Morton TI - Permutations Related to Secant, Tangent and Eulerian Numbers JO - Canadian mathematical bulletin PY - 1979 SP - 281 EP - 291 VL - 22 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1979-034-9/ DO - 10.4153/CMB-1979-034-9 ID - 10_4153_CMB_1979_034_9 ER -
[1] 1. Abramson, Morton, Some generalized Eulerian numbers, Proc. Fifth Southeastern Conference on Combinatorics, graph theory and computing, Florida Atlantic University, Boca Raton, Florida, 1974, pp. 159-172. Google Scholar
[2] 2. Abramson, Morton, A simple solution of Simon Newcomb's problem, J. of Comb. Theory, “A”, 18 (1975), pp. 223-225. Google Scholar
[3] 3. Abramson, Morton, A note on permutations with fixed pattern, J. of Comb. Theory, “A”, 19 (1975), pp. 237-239. Google Scholar
[4] 4. Abramson, Morton, Enumeration of sequences by levels and rises, Discrete Math., 12 (1975), pp. 101-112. Google Scholar
[5] 5. Abramson, Morton, Sequences by number of w-rises, Canad. Math. Bull., 18 (1975), pp. 317-319. Google Scholar
[6] 6. Abramson, Morton and Promislow, David, Enumeration of arrays by column rises, J. of Comb. Theory, “A”, 24 (1978), pp. 247-250. Google Scholar
[7] 7. André, D., Développements de see x et de tan x, C. R. Acad. Sci. Paris, 88 (1879), pp. 965-967. Google Scholar
[8] 8. André, D., Sur les permutations alternées, J. Math. Pures Appl., 7 (1881), p. 167. Google Scholar
[9] 9. Blundon, W. J., Solution to “A permutation problem”, 4755 [1957, 596], Amer. Math. Monthly, 65 (1958), p. 533. Google Scholar
[10] 10. Carlitz, L., Some arithmetic properties of the Oliver functions, Math. Ann., 128 (1955), pp. 412-419. Google Scholar
[11] 11. Carlitz, L., Eulerian numbers and polynomials, Math. Magazine, 32 (1959), pp. 247-260. Google Scholar
[12] 12. Carlitz, L., Eulerian numbers and polynomials of higher order, Duke Math. J., 27 (1960), pp. 401-423. Google Scholar
[13] 13. Carlitz, L., A note on Eulerian numbers, Arch. Math., 14 (1963), pp. 383-390. Google Scholar
[14] 14. Carlitz, L., Extended Bernoulli and Eulerian numbers, Duke Math. J., 31 (1964), pp. 667-689. Google Scholar
[15] 15. Carlitz, L., Enumeration of sequences by rises and falls: a refinement of the Simon Newcomb problem, Duke Math. J., 39 (1972), pp. 267-280. Google Scholar
[16] 16. Carlitz, L., Eulerian numbers and operators, The Theory of Arithmetic Functions, pp. 65-70, Lecture Notes in Math., Vol. 251, Springer, Berlin, 1972. Google Scholar
[17] 17. Carlitz, Leonard, Enumeration of a special class of permutations by rises, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 412-460 (1973), pp. 189-196. Google Scholar
[18] 18. Carlitz, L., Enumeration of permutations by rises and cycle structure, J. Reine Angew. Math., 262/263 (1973), pp. 220-233. Google Scholar
[19] 19. Carlitz, L., Enumeration of up-down permutations by number of rises, Pacifie J. Math., 45 (1973), pp. 49-58. Google Scholar
[20] 20. Carlitz, L., Enumeration of up-down sequences, discrete Math., 4 (1973), pp. 273-286. Google Scholar
[21] 21. Carlitz, L., Eulerian numbers and operators, collect. Math., 24 (1973) pp. 175-200. Google Scholar
[22] 22. Carlitz, L., Permutations with prescribed pattern, Math. Nachr., 58 (1973), 31-53. Google Scholar
[23] 23. Carlitz, L., Permutations and sequences, Advances in Math., 14 (1974), pp. 92-120. Google Scholar
[24] 24. Carlitz, L., A combinatorial property of q-Eulerian numbers, Amer. Math. Monthly, 82 (1975), pp. 51-54. Google Scholar
[25] 25. Carlitz, L., Generating functions for a special class of permutations, Proc. Amer. Math. Soc, 47 (1975), pp. 251-256. Google Scholar
[26] 26. Carlitz, L., Permutations, sequences, and special functions, SLAM Rev., 17 (1975), pp. 298-321. Google Scholar
[27] 27. Carlitz, L., Combinatorial property of a special polynomial sequence, Canad. Math. Bull., 20 (1977), pp. 183-188. Google Scholar
[28] 28. Carlitz, L., Kurtz, D. C., Scoville, R., and Stackelberg, O. P., Asymptotic properties of Eulerian numbers, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 23 (1972), pp. 47-54. Google Scholar
[29] 29. Carlitz, L. and Riordan, J., Congruences for Eulerian Numbers, Duke Math. J., 20 (1953), pp. 339-343. Google Scholar
[30] 30. Carlitz, L., Roselle, D. P. and Scoville, R. A., Permutations and sequences with repetitions by number of increases, J. Comb. Theory, 1 (1966), pp. 350-374. Google Scholar
[31] 31. Carlitz, L. and Scoville, Richard, Tangent numbers and operators, Duke Math. J., 39 (1972), pp. 413-429. Google Scholar
[32] 32. Carlitz, L. and Scoville, Richard, Up-down sequences, Duke Math. J., 39 (1972), pp. 583-598. Google Scholar
[33] 33. Carlitz, L. and Scoville, Richard, Enumeration of rises and falls by position, Discrete Math., 5, (1973), pp. 45-59. Google Scholar
[34] 34. Carlitz, L. and Scoville, Richard, Enumeration of permutations by rises, falls, rising maxima, and falling maxima, Acta Math. Acad. Sci. Hungar., 25 (1974), pp. 269-277. Google Scholar
[35] 35. Carlitz, L. and Scoville, Richard, Generalized Eulerian numbers: combinatorial applications, J. Reine Angew. Math., 265 (1974), pp. 110-137. Google Scholar
[36] 36. Carlitz, L. and Scoville, Richard, Enumeration of up-down permutations by upper records, Monatshefte fur Mathematik, 79 (1975), pp. 3-12. Google Scholar
[37] 37. Carlitz, L. and Scoville, Richard, Generating functions for certain types of permutations, J. Combinatorial Theory “A”, 18 (1975), pp. 262-275. Google Scholar
[38] 38. Carlitz, L. and Scoville, Richard, Eulerian numbers and operators, Fibonacci Quart., 13 (1975), pp. 71-83. Google Scholar
[39] 39. Carlitz, L. and Scoville, Richard, Some permutation problems, J. Comb. Theory “A, 22 (1977), pp. 129-145. Google Scholar
[40] 40. Carlitz, L., Scoville, Richard and Vaughan, Theresa, Enumeration of permutations and sequences with restrictions, Duke Math. J., 40 (1973), pp. 723-741. Google Scholar
[41] 41. Carlitz, L., Scoville, R. and Vaughan, T., Enumeration of pairs of permutations and sequences, Bull. Amer. Soc, 80 (1974), pp. 881-884. Google Scholar
[42] 42. Carlitz, L., Scoville, R., and Vaughan, T., Enumeration of pairs of permutations, Discrete Math., 14 (1976), pp. 215-239. Google Scholar
[43] 43. Carlitz, L., Scoville, R. and Vaughan, T., Enumeration of pairs of sequences by rises, falls, and levels, Manuscripta Math., 19 (1976), pp. 211-243. Google Scholar
[44] 44. Carlitz, L. and Vaughan, T., Enumeration of sequences of given specification according to rises, falls, and maxima, Discrete Math., 8 (1974), pp. 147-167. Google Scholar
[45] 45. Comtet, L., Analyse Combinatoire, vol. 1, pp. 63-64, vol. 2, pp. 82-86, Paris, 1970. Google Scholar
[46] 46. Debruijn, N. E., Permutations with given ups and downs, Nieuw Archief voor Wiskunde, 18 (1970), pp. 61-65. Google Scholar
[47] 47. Dillon, J. F. and Roselle, D. P., Eulerian numbers of higher order, Duke Math. J., 35 (1968), pp. 247-256. Google Scholar
[48] 48. Dillon, J. F. and Roselle, D. P., Simon Newcomb's problem, SIAM J. Appl. Math., 17 (1969), pp. 1086-1097. Google Scholar
[49] 49. Donaghey, Robert, Alternating permutations and binary increasing trees, J. Comb. Theory, “A”, 18 (1975), pp. 141-148. Google Scholar
[50] 50. Drane, F. B. and Roselle, D. P., A sequence of polynomials related to the Eulerian polynomials, Utilitas Math., 9 (1976), pp. 33-37. Google Scholar
[51] 51. Dumont, Dominique, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), pp. 305-318. Google Scholar
[52] 52. Entringer, R. C., A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Arch. V. Wiskunde, 14 (1966), pp. 241-246. Google Scholar
[53] 53. Etringer, R. C., A note on enumeration of permutations of ( 1, …, n) by number of maxima, Gac. Mat. (Madrid), 23 (1971), pp. 67-69. Google Scholar
[54] 54. Euler, L., Institutiones Calculi Differentiate, St. Petersburg, 1755. Google Scholar
[55] 55. Foata, Dominique, Groupes de réarrangements et nombres d' Euler, C. R. Acad. Sci. Paris Sér. A-B, 275 (1972), A 1147-A 1150. Google Scholar
[56] 56. Foata, Dominque and Schùtzenberger, Marcel-P., Théorie géométrique des polynômes eulériens, in “Lecture Notes in Mathematics,” No. 138, Springer-Verlag, New York, 1970. Google Scholar
[57] 57. Foata, D. and Schùtzenberger, M. P., Nombres d' Euler et permutations alternantes, A Survey of Combinatorial Theory, pp. 173-187, North-Holland, Amsterdam, 1973. Google Scholar
[58] 58. Foata, Dominque and Strehl, Volker, Euler numbers and variations of permutations, Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973), Tomo I, pp. 119-131. Atti dei Convegni Lincei, No. 17, Accad. Naz. Lincei, Roma, 1976. Google Scholar
[59] 59. Foulkes, H. O., Enumeration of permutations with prescribed up-down and inversion sequences, Discrete Math., 15 (1976), pp. 235-252. Google Scholar
[60] 60. Foulkes, H. O., A nonrecursive combinatorial rule for Eulerian numbers, J. Comb. Theory, “A”, 22 (1977), pp. 246-248. Google Scholar
[61] 61. Gessel, Ira Martin, Generating Functions and Enumeration of sequences, Ph.d. Thesis, M.I.T. 1977. Google Scholar
[62] 62. Gessel, Ira and Stanley, Richard P., Stirling Polynomials, J. of Comb. Theory, 24 (1978), pp. 24-33. Google Scholar
[63] 63. Gould, H. W., Comment on Problem 67-5, “up-down” permutations, SIAM Review, 10 (1968), pp. 225-226. Google Scholar
[64] 64. Gould, H. W., Explicit formulas for Bernoulli numbers, Amer. Math. Monthly, 79 (1972), pp. 44-51. Google Scholar
[65] 65. Johnson, Allan Wm. Jr. Solution to, “ A difference equation in two variables,” Problem E 2609 [1976, 567], Amer. Math. Monthly, 84 (1977), pp. 826-827. Google Scholar
[66] 66. Knop, Robert E., A note on hypercube partitions, J. Comb. Theory, “A”, 15 (1973), pp. 338-342. Google Scholar
[67] 67. Kreweras, Germain, Sur une extension du problème dit “de Simon Newcomb”, C.R. Acad. Sci. Paris Sér. A-B, 263 (1966), pp. A43-A45. Google Scholar
[68] 68. Kurtz, David C., A note on concavity properties of triangular arrays of numbers, J. of Comb. Theory “A”, 13 (1972), pp. 135-139. Google Scholar
[69] 69. Netto, E., Lehrbuch der Combinatorik, Berlin, 1927. Google Scholar
[70] 70. Olivier, L., Bemerkungen uber eine Art von Funktionen, welche Eigenschaften haben, wie die Cosinus und Sinus, J. Reine Angew. Math., 2 (1827), pp. 243-251. Google Scholar
[71] 71. Riordan, J., Triangular permutation numbers, Proc. Amer. Math. Soc, 2 (1951), pp. 429- 432. Google Scholar
[72] 72. Riordan, J., An Introduction to Combinatorial Analysis, J. Wiley, New York, 1958. Google Scholar
[73] 73. Roselle, D. P., Permutations by number of rises and successions, Proc. Amer. Math. Soc, 19 (1968), pp. 8-16. Google Scholar
[74] 74. Rosen, J., The number of product-weighted lead codes for ballots and its relation to the Ursell-functions of the Linear Ising Model, J. Comb. Theory, “A”, 20 (1976), p. 377. Google Scholar
[75] 75. Sorin, B., La distribution combinatoire du nombre de différences premières positives, Rev. Inst. Internat. Statist., 39 (1971), pp. 9-20. Google Scholar
[76] 76. Stanley, Richard P., Ordered Structures and Partitions, Memoirs of the Amer. Math. Soc, No. 119, Amer. Math. Soc, Providence, R. L, 1972. Google Scholar
[77] 77. Stanley, Richard P., Binomial posets, Môbius inversion, and permutation enumeration, J. Combinatorial Theory “A”, 20 (1976), pp. 336-356. Google Scholar
[78] 78. Strehl, Volker, Enumeration of alternating permutations according to peak sets, J. of Comb. Theory, “A”, 24 (1978), pp. 238-240. Google Scholar
[79] 79. Tanny, S., A probabilistic interpretation of Eulerian numbers, Duke Math. J., 40 (1973), pp. 717-722. Google Scholar
[80] 80. West, Don, Solution to “The Smith college diploma problem” Problem E 2404 [1973, 316], Amer. Math. Monthly, 81 (1974), pp. 286-289. Google Scholar
[81] 81. Worpitzky, J., Studien über die Bernoullischen und Eulerschen Zahlen, J. fur die reine und angewandte Math., 94 (1883), pp. 203-232. Google Scholar
Cité par Sources :