A261477 Number T(n,k) of set partitions of [n] into exactly k parts such that no part contains two elements with a circular distance less than three; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 7, 14, 7, 1, 0, 0, 0, 0, 7, 44, 42, 12, 1, 0, 0, 0, 1, 21, 138, 210, 102, 18, 1, 0, 0, 0, 0, 50, 458, 985, 720, 210, 25, 1, 0, 0, 0, 0, 77, 1397, 4400, 4587, 1980, 385, 33, 1

Offset

0,26

Comments

The circular distance of 1 and n is 1 (for n>1).

Links

Alois P. Heinz, Rows n = 0..100, flattened

Example

T(6,3) = 1: 14|25|36.
T(6,4) = 3: 14|25|3|6, 14|2|36|5, 1|25|36|4.
T(6,5) = 3: 14|2|3|5|6, 1|25|3|4|6, 1|2|36|4|5.
T(6,6) = 1: 1|2|3|4|5|6.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 0, 1;
  0, 0, 0, 0,  1;
  0, 0, 0, 0,  0,   1;
  0, 0, 0, 1,  3,   3,   1;
  0, 0, 0, 0,  7,  14,   7,   1;
  0, 0, 0, 0,  7,  44,  42,  12,   1;
  0, 0, 0, 1, 21, 138, 210, 102,  18,  1;
  0, 0, 0, 0, 50, 458, 985, 720, 210, 25, 1;

Maple

g:= proc(n, l, m, h) option remember;
      `if`(n=0, `if`(1 in [l, m] or l=2, `if`(h<3, x^h, 0), x^h),
       add(`if`(j in [l, m], 0, g(n-1, j, l, max(h, j))), j=1..h+1))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(g(n, 0$3)):
seq(T(n), n=0..14);

Mathematica

g[n_, l_, m_, h_] := g[n, l, m, h] = If[n == 0, If[l == 1 || m == 1 || l == 2, If[h < 3, x^h, 0], x^h], Sum[If[j == l || j == m, 0, g[n - 1, j, l, Max[h, j]]], {j, 1, h + 1}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][g[n, 0, 0, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A000007, A063524, A185012, A079978 (for n>0), A261480, A261481, A261482, A261483, A261484, A261485, A261486.

Row sums give A261478.

T(2n,n) give A261479.

Main diagonal gives A000012.

Sequence in context: A119969 A051343 A356326 * A205826 A131193 A350479

Adjacent sequences: A261474 A261475 A261476 * A261478 A261479 A261480

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 20 2015

Status

approved