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Number of permutations of [n] having exactly two alternating descents.
2

%I #15 Apr 30 2018 04:22:08

%S 2,7,36,182,1056,6669,46348,350240,2866632,25260211,238550196,

%T 2403846954,25749417440,292191829273,3501634042140,44194287012404,

%U 585947863072440,8142408896706495,118341851934179140,1795485894869716670,28386889319060192592

%N Number of permutations of [n] having exactly two alternating descents.

%C Index i is an alternating descent of permutation p if either i is odd and p(i) > p(i+1), or i is even and p(i) < p(i+1).

%H Alois P. Heinz, <a href="/A302895/b302895.txt">Table of n, a(n) for n = 3..483</a>

%H D. Chebikin, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r132">Variations on descents and inversions in permutations</a>, The Electronic J. of Combinatorics, 15 (2008), #R132.

%F a(n) ~ (4 - Pi)^2 * 2^(n + 3/2) * n^(n + 5/2) / (exp(n) * Pi^(n + 5/2)). - _Vaclav Kotesovec_, Apr 29 2018

%F E.g.f.: ((4 + (-2 + x)*x - 4*cos(x))*(1 + cos(x)) + (4 + (-6 + x)*x)*sin(x))/(3 + 2*cos(x) - cos(2*x) - 2*(2 + cos(x))*sin(x)). - _Vaclav Kotesovec_, Apr 30 2018

%e a(3) = 2: 213, 312.

%e a(4) = 7: 1243, 1342, 2134, 2341, 3124, 4123, 4321.

%p b:= proc(u, o) option remember; series(`if`(u+o=0, 1,

%p add(b(o+j-1, u-j)*x, j=1..u)+

%p add(b(o-j, u-1+j), j=1..o)), x, 4)

%p end:

%p a:= n-> coeff(b(n, 0), x, 3):

%p seq(a(n), n=3..30);

%t nmax = 30; Drop[CoefficientList[Series[((4 + (-2 + x)*x - 4*Cos[x])*(1 + Cos[x]) + (4 + (-6 + x)*x)*Sin[x])/(3 + 2*Cos[x] - Cos[2*x] - 2*(2 + Cos[x])*Sin[x]), {x, 0, nmax}], x] * Range[0, nmax]!, 3] (* _Vaclav Kotesovec_, Apr 30 2018 *)

%Y Column k=3 of A145876.

%K nonn

%O 3,1

%A _Alois P. Heinz_, Apr 15 2018