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

%I #17 Apr 30 2018 04:22:01

%S 5,26,182,1196,8699,67054,556952,4945368,46901985,473324450,

%T 5068610666,57423496708,686431008183,8636011443670,114084907758220,

%U 1579087434474288,22855092931547133,345270138380243850,5434925832561946750,89001957427837012060

%N Number of permutations of [n] having exactly three 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="/A302896/b302896.txt">Table of n, a(n) for n = 4..483</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)^3 * 2^(n + 5/2) * n^(n + 7/2) / (3! * Pi^(n + 7/2) * exp(n)). - _Vaclav Kotesovec_, Apr 29 2018

%F E.g.f.: (6*cos(x)^2 + (3*x^2 - 12*x + 6*sin(x) + 6)*cos(x) + (x^3 - 6*x^2 + 18*x - 12)*sin(x) + 2*x^3 - 12*x^2 + 18*x - 12)/(6*cos(x)^2 + 12*sin(x) - 12). - _Vaclav Kotesovec_, Apr 30 2018

%e a(4) = 5: 2143, 3142, 3241, 4132, 4231.

%e a(5) = 26: 12435, 12534, 13425, 13524, 14523, 21345, 21543, 23415, 23514, 24513, 31245, 31542, 32541, 34512, 41235, 41532, 42531, 43215, 43521, 51234, 51432, 52431, 53214, 53421, 54213, 54312.

%p b:= proc(u, o) option remember; series(`if`(u+o=0, 1,

%p add(b(o-j, u-1+j), j=1..o)), x, 5)

%p end:

%p a:= n-> coeff(b(n, 0), x, 4):

%p seq(a(n), n=4..30);

%t nmax = 25; Drop[CoefficientList[Series[(6*Cos[x]^2 + (3*x^2 - 12*x + 6*Sin[x] + 6)*Cos[x] + (x^3 - 6*x^2 + 18*x - 12)*Sin[x] + 2*x^3 - 12*x^2 + 18*x - 12)/(6*Cos[x]^2 + 12*Sin[x] - 12), {x, 0, nmax}], x] * Range[0, nmax]!, 4] (* _Vaclav Kotesovec_, Apr 30 2018 *)

%Y Column k=4 of A145876.

%K nonn

%O 4,1

%A _Alois P. Heinz_, Apr 15 2018

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Last modified September 20 06:28 EDT 2024. Contains 376052 sequences. (Running on oeis4.)