A302902 - OEIS

%I #19 Apr 30 2018 05:00:19

%S 50521,1160026,25260211,473324450,8650769675,154898419006,

%T 2781436057021,50412205403030,928559550102410,17440458896525180,

%U 334876925319944690,6583281405926363500,132633340608724861210,2740015852551381054980,58057801350608276240150

%N Number of permutations of [n] having exactly nine 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).

%C From _Vaclav Kotesovec_, Apr 29 2018: (Start)

%C In general, number of permutations of [n] having exactly k alternating descents (column k+1 of A145876) is asymptotic to a(n,k) ~ (4 - Pi)^k * 2^(n + 2) * n^k * n! / (k! * Pi^(n + k + 1)).

%C Equivalently, a(n,k) ~ (4 - Pi)^k * 2^(n + 5/2) * n^(n + k + 1/2) / (k! * Pi^(n + k + 1/2) * exp(n)).

%C (End)

%H Alois P. Heinz, <a href="/A302902/b302902.txt">Table of n, a(n) for n = 10..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)^9 * 2^(n + 5/2) * n^(n + 19/2) / (9! * Pi^(n + 19/2) * exp(n)). - _Vaclav Kotesovec_, Apr 29 2018

%F E.g.f.: (362880*cos(x)^5 + (x^9 - 18*x^8 + 216*x^7 - 2016*x^6 + 15120*x^5 - 90720*x^4 + 423360*x^3 - 1451520*x^2 + 3265920*x - 362880*sin(x) - 1814400)*cos(x)^4 + ((- 9*x^8 + 144*x^7 - 1512*x^6 + 12096*x^5 - 75600*x^4 + 362880*x^3 - 1270080*x^2 + 2903040*x - 1451520)*sin(x) - 1107*x^8 + 17712*x^7 - 137592*x^6 + 713664*x^5 - 2646000*x^4 + 6894720*x^3 - 11430720*x^2 + 8709120*x - 2903040)*cos(x)^3 + ((- 251*x^9 + 4518*x^8 - 40392*x^7 + 247968*x^6 - 1134000*x^5 + 3900960*x^4 - 9737280*x^3 + 15966720*x^2 - 13063680*x + 4354560)*sin(x) - 3653*x^9 + 65754*x^8 - 543240*x^7 + 2776032*x^6 - 9752400*x^5 + 24040800*x^4 - 40219200*x^3 + 42094080*x^2 - 26127360*x + 7257600)*cos(x)^2 + ((9108*x^8 - 145728*x^7 + 1037232*x^6 - 4354560*x^5 + 11793600*x^4 - 20321280*x^3 + 20321280*x^2 - 11612160*x + 2903040)*sin(x) + 13572*x^8 - 217152*x^7 + 1502928*x^6 - 5806080*x^5 + 13608000*x^4 - 20321280*x^3 + 20321280*x^2 - 11612160*x + 2903040)*cos(x) + (11092*x^9 - 199656*x^8 + 1599696*x^7 - 7499520*x^6 + 22680000*x^5 - 45722880*x^4 + 60963840*x^3 - 52254720*x^2 + 26127360*x - 5806080)*sin(x) + 11588*x^9 - 208584*x^8 + 1666224*x^7 - 7741440*x^6 + 23042880*x^5 - 45722880*x^4 + 60963840*x^3 - 52254720*x^2 + 26127360*x - 5806080)/((362880*sin(x) - 1814400)*cos(x)^4 + (- 4354560*sin(x) + 7257600)*cos(x)^2 + 5806080*sin(x) - 5806080). - _Vaclav Kotesovec_, Apr 30 2018

%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, 11)

%p     end:

%p a:= n-> coeff(b(n, 0), x, 10):

%p seq(a(n), n=10..30);

%t nmax = 30; Drop[CoefficientList[Series[(362880*Cos[x]^5 + (x^9 - 18*x^8 + 216*x^7 - 2016*x^6 + 15120*x^5 - 90720*x^4 + 423360*x^3 - 1451520*x^2 + 3265920*x - 362880*Sin[x] - 1814400)*Cos[x]^4 + ((- 9*x^8 + 144*x^7 - 1512*x^6 + 12096*x^5 - 75600*x^4 + 362880*x^3 - 1270080*x^2 + 2903040*x - 1451520)*Sin[x] - 1107*x^8 + 17712*x^7 - 137592*x^6 + 713664*x^5 - 2646000*x^4 + 6894720*x^3 - 11430720*x^2 + 8709120*x - 2903040)*Cos[x]^3 + ((- 251*x^9 + 4518*x^8 - 40392*x^7 + 247968*x^6 - 1134000*x^5 + 3900960*x^4 - 9737280*x^3 + 15966720*x^2 - 13063680*x + 4354560)*Sin[x] - 3653*x^9 + 65754*x^8 - 543240*x^7 + 2776032*x^6 - 9752400*x^5 + 24040800*x^4 - 40219200*x^3 + 42094080*x^2 - 26127360*x + 7257600)*Cos[x]^2 + ((9108*x^8 - 145728*x^7 + 1037232*x^6 - 4354560*x^5 + 11793600*x^4 - 20321280*x^3 + 20321280*x^2 - 11612160*x + 2903040)*Sin[x] + 13572*x^8 - 217152*x^7 + 1502928*x^6 - 5806080*x^5 + 13608000*x^4 - 20321280*x^3 + 20321280*x^2 - 11612160*x + 2903040)*Cos[x] + (11092*x^9 - 199656*x^8 + 1599696*x^7 - 7499520*x^6 + 22680000*x^5 - 45722880*x^4 + 60963840*x^3 - 52254720*x^2 + 26127360*x - 5806080)*Sin[x] + 11588*x^9 - 208584*x^8 + 1666224*x^7 - 7741440*x^6 + 23042880*x^5 - 45722880*x^4 + 60963840*x^3 - 52254720*x^2 + 26127360*x - 5806080)/((362880*Sin[x] - 1814400)*Cos[x]^4 + (- 4354560*Sin[x] + 7257600)*Cos[x]^2 + 5806080*Sin[x] - 5806080), {x, 0, nmax}], x] * Range[0, nmax]!, 10] (* _Vaclav Kotesovec_, Apr 30 2018 *)

%Y Column k=10 of A145876.

%K nonn

%O 10,1

%A _Alois P. Heinz_, Apr 15 2018