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(2)>b(3)<... .

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

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, A186360.

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