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 A186358 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k up-down cycles (0<=k<=n). A cycle (b(1), b(2), ...) is said to be up-down if, when written with its smallest element in the first position, it satisfies b(1)b(3)<... . 4
 1, 0, 1, 0, 1, 1, 1, 1, 3, 1, 4, 6, 7, 6, 1, 19, 35, 30, 25, 10, 1, 114, 210, 190, 125, 65, 15, 1, 799, 1468, 1351, 840, 420, 140, 21, 1, 6392, 11760, 10820, 6692, 3185, 1176, 266, 28, 1, 57527, 105905, 97458, 60058, 28098, 10479, 2856, 462, 36, 1, 575270, 1059306, 975140, 599640, 278500, 103593, 30345, 6210, 750, 45, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS Sum of entries in row n is n!. T(n,0) = A186359(n). Sum_{k=0..n} k * T(n,k) = A186360(n). LINKS Table of n, a(n) for n=0..65. Emeric Deutsch and Sergi Elizalde, Cycle up-down permutations, arXiv:0909.5199 [math.CO], 2009; and also, Australas. J. Combin. 50 (2011), 187-199. FORMULA E.g.f.: (1-sin z)^(1-t)/(1-z). The trivariate e.g.f. H(t,s,z) of the permutations of {1,2,...,n} with respect to size (marked by z), number of up-down cycles (marked by t), and number of cycles that are not up-down (marked by s) is given by H(t,s,z) = (1-sin z)^(s-t)/(1-z)^s. EXAMPLE T(3,0)=1 because we have (123). T(4,2)=7 because we have (1)(243), (142)(3), (132)(4), (13)(24), (12)(34), (143)(2), and (14)(23). Triangle starts: 1; 0,1; 0,1,1; 1,1,3,1; 4,6,7,6,1; 19,35,30,25,10,1; MAPLE G := (1-sin(z))^(1-t)/(1-z): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 10 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: for n from 0 to 10 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form MATHEMATICA m = maxExponent = 11; (CoefficientList[# + O[t]^m, t] Range[0, m-1]!&) /@ CoefficientList[(1 - Sin[z])^{1-t}/(1-z) + O[z]^m, z] // Flatten (* Jean-François Alcover, Aug 07 2018 *) CROSSREFS Cf. A186359, A168360. Sequence in context: A071315 A072267 A024934 * A170839 A049918 A357182 Adjacent sequences: A186355 A186356 A186357 * A186359 A186360 A186361 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Feb 20 2011 STATUS approved

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Last modified April 12 10:51 EDT 2024. Contains 371632 sequences. (Running on oeis4.)