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A302905
Number of permutations of [n] having exactly ceiling(n/2)-1 alternating descents.
4
1, 1, 1, 2, 7, 36, 182, 1196, 8699, 76840, 704834, 7570716, 84889638, 1085246904, 14322115212, 211595659320, 3216832016019, 53984412657360, 928559550102410, 17440458896525180, 334876925319944690, 6960292943873805976, 147563833511292247796, 3362366089440205308072
OFFSET
0,4
COMMENTS
a(0) = 1 by convention.
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).
LINKS
D. Chebikin, Variations on descents and inversions in permutations, The Electronic J. of Combinatorics, 15 (2008), #R132.
FORMULA
a(n) = A145876(n,ceiling(n/2)) for n > 0.
EXAMPLE
a(2) = 1: 12.
a(3) = 2: 123, 321.
a(4) = 7: 1234, 1432, 2431, 3214, 3421, 4213, 4312.
MAPLE
b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
add(b(o+j-1, u-j)*x, j=1..u)+
add(b(o-j, u-1+j), j=1..o)))
end:
a:= n-> coeff(b(n, 0), x, ceil(n/2)):
seq(a(n), n=0..25);
MATHEMATICA
b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1,
Sum[b[o + j - 1, u - j]*x, {j, u}] +
Sum[b[o - j, u - 1 + j], {j, o}]]];
a[n_] := Coefficient[b[n, 0], x, Ceiling[n/2]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 31 2021, after Alois P. Heinz *)
CROSSREFS
Bisections give: A302904 (even part), A302903 (odd part).
Cf. A145876.
Sequence in context: A060814 A192816 A302895 * A378787 A249691 A129261
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Apr 15 2018
STATUS
approved