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
Examples
Triangle begins: 1; 1; 0, 2; 0, 6; 0, 16, 8; 0, 40, 80; 0, 96, 528, 96; ...
Links
- Alois P. Heinz, Rows n = 0..200, flattened
- FindStat - Combinatorial Statistic Finder, The number of cyclic valleys and cyclic peaks of a permutation.
Crossrefs
Programs
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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 *)
Formula
T(n,k) = n*A008303(n-1, k-1) for n > 1. - Andrew Howroyd, May 13 2020
Extensions
More terms from Alois P. Heinz, Oct 26 2015
Comments