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

%I #13 Apr 30 2018 04:55:50

%S 1385,21682,350240,4945368,70145634,992272308,14322115212,

%T 211595659320,3216832016019,50412205403030,815486339550108,

%U 13622914005990480,235041722344009380,4187522527966916520,77010173788311008040,1461190162226869057872,28588437379997078589117

%N Number of permutations of [n] having exactly seven 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="/A302900/b302900.txt">Table of n, a(n) for n = 8..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)^7 * 2^(n + 5/2) * n^(n + 15/2) / (7! * Pi^(n + 15/2) * exp(n)). - _Vaclav Kotesovec_, Apr 29 2018

%F E.g.f.: (5040*cos(x)^4 + (7*x^6 - 84*x^5 + 630*x^4 - 3360*x^3 + 12600*x^2 - 30240*x + 5040*sin(x) + 15120)*cos(x)^3 + ((x^7 - 14*x^6 + 126*x^5 - 840*x^4 + 4200*x^3 - 15120*x^2 + 35280*x - 20160)*sin(x) + 60*x^7 - 840*x^6 + 5544*x^5 - 23520*x^4 + 67200*x^3 - 120960*x^2 + 105840*x - 40320)*cos(x)^2 + ((- 196*x^6 + 2352*x^5 - 12600*x^4 + 40320*x^3 - 75600*x^2 + 60480*x - 20160)*sin(x) - 434*x^6 + 5208*x^5 - 25200*x^4 + 60480*x^3 - 75600*x^2 + 60480*x - 20160)*cos(x) + (- 298*x^7 + 4172*x^6 - 25200*x^5 + 85680*x^4 - 176400*x^3 + 211680*x^2 - 141120*x + 40320)*sin(x) - 332*x^7 + 4648*x^6 - 27720*x^5 + 90720*x^4 - 176400*x^3 + 211680*x^2 - 141120*x + 40320)/(5040*cos(x)^4 + (20160*sin(x) - 40320)*cos(x)^2 - 40320*sin(x) + 40320). - _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, 9)

%p end:

%p a:= n-> coeff(b(n, 0), x, 8):

%p seq(a(n), n=8..30);

%t nmax = 30; Drop[CoefficientList[Series[(5040*Cos[x]^4 + (7*x^6 - 84*x^5 + 630*x^4 - 3360*x^3 + 12600*x^2 - 30240*x + 5040*Sin[x] + 15120)*Cos[x]^3 + ((x^7 - 14*x^6 + 126*x^5 - 840*x^4 + 4200*x^3 - 15120*x^2 + 35280*x - 20160)*Sin[x] + 60*x^7 - 840*x^6 + 5544*x^5 - 23520*x^4 + 67200*x^3 - 120960*x^2 + 105840*x - 40320)*Cos[x]^2 + ((- 196*x^6 + 2352*x^5 - 12600*x^4 + 40320*x^3 - 75600*x^2 + 60480*x - 20160)*Sin[x] - 434*x^6 + 5208*x^5 - 25200*x^4 + 60480*x^3 - 75600*x^2 + 60480*x - 20160)*Cos[x] + (- 298*x^7 + 4172*x^6 - 25200*x^5 + 85680*x^4 - 176400*x^3 + 211680*x^2 - 141120*x + 40320)*Sin[x] - 332*x^7 + 4648*x^6 - 27720*x^5 + 90720*x^4 - 176400*x^3 + 211680*x^2 - 141120*x + 40320)/(5040*Cos[x]^4 + (20160*Sin[x] - 40320)*Cos[x]^2 - 40320*Sin[x] + 40320), {x, 0, nmax}], x] * Range[0, nmax]!, 8] (* _Vaclav Kotesovec_, Apr 30 2018 *)

%Y Column k=8 of A145876.

%K nonn

%O 8,1

%A _Alois P. Heinz_, Apr 15 2018