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Number of permutations of [2n+1] having exactly n alternating descents.
3

%I #16 Dec 21 2020 07:17:06

%S 1,2,36,1196,76840,7570716,1085246904,211595659320,53984412657360,

%T 17440458896525180,6960292943873805976,3362366089440205308072,

%U 1933633403768889597292336,1305355624659052356741634136,1022196734801743485304805455920,919035074839303194470918726496240

%N Number of permutations of [2n+1] having exactly n 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="/A302903/b302903.txt">Table of n, a(n) for n = 0..100</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) = A145876(2n+1,n+1) = A302905(2n+1).

%F a(n) ~ sqrt(3) * 2^(2*n + 2) * n^(2*n + 1) / (sqrt(5) * exp(2*n)). - _Vaclav Kotesovec_, Apr 29 2018

%e a(1) = 2: 123, 321.

%e a(2) = 36: 12345, 12543, 13542, 14325, 14532, 15324, 15423, 21354, 21453, 23541, 24315, 24531, 25314, 25413, 31254, 31452, 32145, 32451, 34215, 34521, 35214, 35412, 41253, 41352, 42135, 42351, 43125, 45213, 45312, 51243, 51342, 52134, 52341, 53124, 54123, 54321.

%p b:= proc(u, o) option remember; expand(`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)))

%p end:

%p a:= n-> coeff(b(2*n+1, 0), x, n+1):

%p seq(a(n), n=0..20);

%t b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1,

%t Sum[b[o + j - 1, u - j]*x, {j, 1, u}] +

%t Sum[b[o - j, u - 1 + j], {j, 1, o}]]];

%t a[n_] := Coefficient[b[2n + 1, 0], x, n + 1];

%t a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ *)

%Y Bisection (odd part) of A302905.

%Y Cf. A145876.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Apr 15 2018