A153743 Number of elements in wreath product C_4 wr S_n that alternate up/not-up with respect to a weak product ordering.
4, 10, 100, 565, 9356, 79584, 1844492, 20922625, 623457040, 8840131486, 321957866768, 5478133336309, 235789017471008, 4680625831294820, 232457094647793632, 5273696164520751265, 296832635265929103616
Offset: 1
Keywords
Examples
Viewing elements in one-line notation as a list of ordered pairs with first entries in [4] and second entries forming a permutation in S_n, two of the 100 up/not-up elements for n=3 are (1,2) (4,3) (3,1) and (1,1) (1,3) (4,2). Note that the first element goes up/down and the second goes up/not-up with respect to the weak product ordering on ordered pairs.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..400
- A. Niedermaier and J. Remmel, Analogues of Up-down Permutations for Colored Permutations, J. Int. Seq. 13 (2010), 10.5.6.
Programs
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Mathematica
Rest[CoefficientList[Series[(6 + 6*Sin[x] + 18*x*Cos[x] - 9 x^2*Sin[x] - x^3*Cos[x])/(6*Cos[x] - 18*x*Sin[x] - 9 x^2*Cos[x] + x^3*Sin[x]), {x, 0, 40}], x]*Range[0, 40]!] (* G. C. Greubel, Aug 27 2016 *)
Formula
E.g.f.: (6 + 6*sin(x) + 18*x*cos(x) - 9*x^2*sin(x) - x^3*cos(x)) / (6*cos(x) - 18*x*sin(x) - 9*x^2*cos(x) + x^3*sin(x)).