A263789 Triangle read by rows: T(n,k) (n>=0, 0<=k<=floor(n/2)) is the number of permutations of n and k valleys (considered cyclically).

1, 1, 0, 2, 0, 6, 0, 16, 8, 0, 40, 80, 0, 96, 528, 96, 0, 224, 2912, 1904, 0, 512, 14592, 23040, 2176, 0, 1152, 69120, 221184, 71424, 0, 2560, 316160, 1858560, 1372160, 79360, 0, 5632, 1413632, 14353152, 20252672, 3891712, 0, 12288, 6223872, 104742912

Offset

0,4

Comments

Conjecture: column k > 0 is asymptotic to n * 2^(n-2*k) * k^(n-1). - Vaclav Kotesovec, Oct 26 2015

Links

Alois P. Heinz, Rows n = 0..200, flattened

FindStat - Combinatorial Statistic Finder, The number of cyclic valleys and cyclic peaks of a permutation.

Formula

T(n,k) = n*A008303(n-1, k-1) for n > 1. - Andrew Howroyd, May 13 2020

Example

Triangle begins:
  1;
  1;
  0,  2;
  0,  6;
  0, 16,   8;
  0, 40,  80;
  0, 96, 528, 96;
  ...

Maple

b:= proc(u, o, t) option remember; expand(`if`(u+o=0, x,
      add(b(u-j, o+j-1, 0), j=1..u)*`if`(min(t, n)>0, x, 1)+
      add(b(u+j-1, o-j, 1), j=1..o)))
    end:
T:= n-> `if`(n<2, 1, (p-> seq(n*coeff(p, x, i)
        , i=0..degree(p)))(b(n-1, 0$2))):
seq(T(n), n=0..14);  # Alois P. Heinz, Oct 28 2015

Mathematica

b[u_, o_, t_] := b[u, o, t] = Expand[If[u+o == 0, x, Sum[b[u-j, o+j-1, 0], {j, 1, u}]*If[Min[t, n] > 0, x, 1] + Sum[b[u+j-1, o-j, 1], {j, 1, o}]]]; T[n_] := If[n<2, 1, Function[p, Table[n*Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n-1, 0, 0]]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 24 2017, after Alois P. Heinz *)

Crossrefs

Columns k=1-6 give: A057711 (for n>1), A159710, A159711, A159712, A159713, A159714.

Row sums give A000142.

Cf. A000142, A008303.

Sequence in context: A285479 A327369 A296620 * A081153 A369278 A126869

Adjacent sequences: A263786 A263787 A263788 * A263790 A263791 A263792

Keyword

nonn,tabf

Author

Christian Stump, Oct 26 2015

Extensions

More terms from Alois P. Heinz, Oct 26 2015

Status

approved