A153743 - OEIS

%I #24 Jul 30 2023 18:24:31

%S 4,10,100,565,9356,79584,1844492,20922625,623457040,8840131486,

%T 321957866768,5478133336309,235789017471008,4680625831294820,

%U 232457094647793632,5273696164520751265,296832635265929103616

%N Number of elements in wreath product C_4 wr S_n that alternate up/not-up with respect to a weak product ordering.

%H G. C. Greubel, <a href="/A153743/b153743.txt">Table of n, a(n) for n = 1..400</a>

%H A. Niedermaier and J. Remmel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Remmel/remmel.html">Analogues of Up-down Permutations for Colored Permutations</a>, J. Int. Seq. 13 (2010), 10.5.6.

%F 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)).

%e 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.

%t 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 *)

%K nonn

%O 1,1

%A _Andrew Niedermaier_, Dec 31 2008