A302898 Number of permutations of [n] having exactly five alternating descents.

61, 594, 6669, 67054, 704834, 7570716, 84889638, 992272308, 12127342203, 154898419006, 2066994606155, 28788990664242, 418074366639272, 6322807317984024, 99466230062507580, 1625658804523009416, 27571506609797250441, 484700416772477950602, 8822485993502063393465

Offset

6,1

Comments

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).

Links

Alois P. Heinz, Table of n, a(n) for n = 6..400

D. Chebikin, Variations on descents and inversions in permutations, The Electronic J. of Combinatorics, 15 (2008), #R132.

Formula

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

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

Example

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

Maple

b:= proc(u, o) option remember; series(`if`(u+o=0, 1,
       add(b(o+j-1, u-j)*x, j=1..u)+
       add(b(o-j, u-1+j),   j=1..o)), x, 7)
    end:
a:= n-> coeff(b(n, 0), x, 6):
seq(a(n), n=6..30);

Mathematica

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 *)

Crossrefs

Column k=6 of A145876.

Sequence in context: A061002 A303412 A209548 * A234926 A355602 A069595

Adjacent sequences: A302895 A302896 A302897 * A302899 A302900 A302901

Keyword

nonn

Author

Alois P. Heinz, Apr 15 2018

Status

approved