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

%I #16 Apr 30 2018 04:51:15

%S 61,594,6669,67054,704834,7570716,84889638,992272308,12127342203,

%T 154898419006,2066994606155,28788990664242,418074366639272,

%U 6322807317984024,99466230062507580,1625658804523009416,27571506609797250441,484700416772477950602,8822485993502063393465

%N Number of permutations of [n] having exactly five 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="/A302898/b302898.txt">Table of n, a(n) for n = 6..400</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)^5 * 2^(n + 5/2) * n^(n + 11/2) / (5! * Pi^(n + 11/2) * exp(n)). - _Vaclav Kotesovec_, Apr 29 2018

%F E.g.f.: (120*cos(x)^3 + (x^5 - 10*x^4 + 60*x^3 - 240*x^2 + 600*x - 120*sin(x) - 360)*cos(x)^2 + ((- 5*x^4 + 40*x^3 - 180*x^2 + 480*x - 240)*sin(x) - 25*x^4 + 200*x^3 - 540*x^2 + 480*x - 240)*cos(x) + (- 13*x^5 + 130*x^4 - 540*x^3 + 1200*x^2 - 1200*x + 480)*sin(x) - 17*x^5 + 170*x^4 - 660*x^3 + 1200*x^2 - 1200*x + 480)/((120*sin(x) - 360)*cos(x)^2 - 480*sin(x) + 480). - _Vaclav Kotesovec_, Apr 30 2018

%e a(6) = 61: 214365, 215364, 215463, 216354, ..., 635142, 635241, 645132, 645231.

%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, 7)

%p end:

%p a:= n-> coeff(b(n, 0), x, 6):

%p seq(a(n), n=6..30);

%t nmax = 30; Drop[CoefficientList[Series[(120*Cos[x]^3 + (x^5 - 10*x^4 + 60*x^3 - 240*x^2 + 600*x - 120*Sin[x] - 360)*Cos[x]^2 + ((- 5*x^4 + 40*x^3 - 180*x^2 + 480*x - 240)*Sin[x] - 25*x^4 + 200*x^3 - 540*x^2 + 480*x - 240)*Cos[x] + (- 13*x^5 + 130*x^4 - 540*x^3 + 1200*x^2 - 1200*x + 480)*Sin[x] - 17*x^5 + 170*x^4 - 660*x^3 + 1200*x^2 - 1200*x + 480)/((120*Sin[x] - 360)*Cos[x]^2 - 480*Sin[x] + 480), {x, 0, nmax}], x] * Range[0, nmax]!, 6] (* _Vaclav Kotesovec_, Apr 30 2018 *)

%Y Column k=6 of A145876.

%K nonn

%O 6,1

%A _Alois P. Heinz_, Apr 15 2018