A275281 Number T(n,k) of set partitions of [n] with symmetric block size list of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 7, 0, 1, 0, 1, 10, 19, 13, 3, 1, 0, 1, 0, 56, 0, 22, 0, 1, 0, 1, 35, 160, 171, 86, 34, 4, 1, 0, 1, 0, 463, 0, 470, 0, 50, 0, 1, 0, 1, 126, 1337, 2306, 2066, 1035, 250, 70, 5, 1, 0, 1, 0, 3874, 0, 10299, 0, 2160, 0, 95, 0, 1

Offset: 0

Author

Alois P. Heinz, Jul 21 2016

Keywords

• nonn
• tabl

Examples

			T(4,2) = 3: 12|34, 13|24, 14|23.
T(5,3) = 7: 12|3|45, 13|2|45, 1|234|5, 1|235|4, 14|2|35, 1|245|3, 15|2|34.
T(6,4) = 13: 12|3|4|56, 13|2|4|56, 1|23|45|6, 1|23|46|5, 14|2|3|56, 1|24|35|6, 1|24|36|5, 1|25|34|6, 1|26|34|5, 15|2|3|46, 1|25|36|4, 1|26|35|4, 16|2|3|45.
T(7,5) = 22: 12|3|4|5|67, 13|2|4|5|67, 1|23|4|56|7, 1|23|4|57|6, 14|2|3|5|67, 1|24|3|56|7, 1|24|3|57|6, 1|2|345|6|7, 1|2|346|5|7, 1|2|347|5|6, 15|2|3|4|67, 1|25|3|46|7, 1|25|3|47|6, 1|2|356|4|7, 1|2|357|4|6, 1|26|3|45|7, 1|27|3|45|6, 16|2|3|4|57, 1|26|3|47|5, 1|2|367|4|5, 1|27|3|46|5, 17|2|3|4|56.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   0,    1;
  0, 1,   3,    2,    1;
  0, 1,   0,    7,    0,    1;
  0, 1,  10,   19,   13,    3,    1;
  0, 1,   0,   56,    0,   22,    0,   1;
  0, 1,  35,  160,  171,   86,   34,   4,  1;
  0, 1,   0,  463,    0,  470,    0,  50,  0, 1;
  0, 1, 126, 1337, 2306, 2066, 1035, 250, 70, 5, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• Wikipedia, Partition of a set

Crossrefs

Columns k=0-1,3,5,7,9 give: A000007, A000012 for n>0, A275289, A275290, A275291, A275292.

Bisections of columns k=2,4,6,8,10 give: A001700(n-1) for n>0, A275293, A275294, A275295, A275296.

Row sums give A275282.

T(n,A004525(n)) gives A305197.

T(2n,n) gives A275283.

T(2n+1,A109613(n)) gives A305198.

T(n,n) gives A000012.

T(n+3,n+1) gives A002623.

Programs

Maple

b:= proc(n, s) option remember; expand(`if`(n>s,
      binomial(n-1, n-s-1)*x, 1)+add(binomial(n-1, j-1)*
      b(n-j, s+j)*binomial(s+j-1, j-1), j=1..(n-s)/2)*x^2)
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..12);

Mathematica

b[n_, s_] := b[n, s] = Expand[If[n>s, Binomial[n-1, n-s-1]*x, 1] + Sum[ Binomial[n-1, j-1]*b[n-j, s+j]*Binomial[s+j-1, j-1], {j, 1, (n-s)/2} ]*x^2]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 03 2017, translated from Maple *)

Formula

T(n,k) = 0 if n is odd and k is even.