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

%I #13 Apr 30 2018 04:53:26

%S 272,3407,46348,556952,6847224,84889638,1085246904,14322115212,

%T 195951082944,2781436057021,40985527637668,626827892111140,

%U 9945795998932920,163614736611741324,2788498384849238640,49195762917367001256,897689701635240236352,16927557342294928274187

%N Number of permutations of [n] having exactly six 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="/A302899/b302899.txt">Table of n, a(n) for n = 7..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)^6 * 2^(n + 5/2) * n^(n + 13/2) / (6! * Pi^(n + 13/2) * exp(n)). - _Vaclav Kotesovec_, Apr 29 2018

%F E.g.f.: (1440*cos(x)^4 + (- x^6 + 6*x^5 - 30*x^4 + 120*x^3 - 360*x^2 + 720*x)*cos(x)^3 + ((- x^6 + 18*x^5 - 150*x^4 + 840*x^3 - 3240*x^2 + 7920*x - 4320)*sin(x) - 29*x^6 + 420*x^5 - 2610*x^4 + 9120*x^3 - 18360*x^2 + 16560*x - 7200)*cos(x)^2 + 28*x*((x^5 - (129/14)*x^4 + (255/7)*x^3 - 90*x^2 + (900/7)*x - 360/7)*sin(x) + 31*x^5*(1/14) - 321*x^4*(1/14) + 645*x^3*(1/7) - 1170*x^2*(1/7) + 900*x*(1/7) - 360/7)*cos(x) + (90*x^6 - 1104*x^5 + 5640*x^4 - 15120*x^3 + 21600*x^2 - 17280*x + 5760)*sin(x) + 90*x^6 - 1056*x^5 + 5160*x^4 - 13680*x^3 + 21600*x^2 - 17280*x + 5760)/(720*cos(x)^4 + (720*sin(x) - 2160)*cos(x)^3 + (2880*sin(x) - 5760)*cos(x)^2 + (- 2880*sin(x) + 2880)*cos(x) - 5760*sin(x) + 5760). - _Vaclav Kotesovec_, Apr 30 2018

%e a(7) = 272: 2143657, 2143756, 2153647, 2153746, ..., 7561423, 7562314, 7562413, 7563412.

%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, 8)

%p end:

%p a:= n-> coeff(b(n, 0), x, 7):

%p seq(a(n), n=7..30);

%t nmax = 30; Drop[CoefficientList[Series[(1440*Cos[x]^4 + (- x^6 + 6*x^5 - 30*x^4 + 120*x^3 - 360*x^2 + 720*x)*Cos[x]^3 + ((- x^6 + 18*x^5 - 150*x^4 + 840*x^3 - 3240*x^2 + 7920*x - 4320)*Sin[x] - 29*x^6 + 420*x^5 - 2610*x^4 + 9120*x^3 - 18360*x^2 + 16560*x - 7200)*Cos[x]^2 + 28*x*((x^5 - (129/14)*x^4 + (255/7)*x^3 - 90*x^2 + (900/7)*x - 360/7)*Sin[x] + 31*x^5*(1/14) - 321*x^4*(1/14) + 645*x^3*(1/7) - 1170*x^2*(1/7) + 900*x*(1/7) - 360/7)*Cos[x] + (90*x^6 - 1104*x^5 + 5640*x^4 - 15120*x^3 + 21600*x^2 - 17280*x + 5760)*Sin[x] + 90*x^6 - 1056*x^5 + 5160*x^4 - 13680*x^3 + 21600*x^2 - 17280*x + 5760)/(720*Cos[x]^4 + (720*Sin[x] - 2160)*Cos[x]^3 + (2880*Sin[x] - 5760)*Cos[x]^2 + (- 2880*Sin[x] + 2880)*Cos[x] - 5760*Sin[x] + 5760), {x, 0, nmax}], x] * Range[0, nmax]!, 7] (* _Vaclav Kotesovec_, Apr 30 2018 *)

%Y Column k=7 of A145876.

%K nonn

%O 7,1

%A _Alois P. Heinz_, Apr 15 2018