A321296 Number T(n,k) of colored set partitions of [n] where colors of the elements of subsets are in (weakly) increasing order and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 3, 0, 5, 20, 16, 0, 15, 122, 237, 131, 0, 52, 774, 2751, 3524, 1496, 0, 203, 5247, 30470, 68000, 65055, 22482, 0, 877, 38198, 341244, 1181900, 1913465, 1462320, 426833, 0, 4140, 298139, 3949806, 19946654, 48636035, 61692855, 39282229, 9934563

Offset

0,5

Links

Alois P. Heinz, Rows n = 0..140, flattened

Wikipedia, Partition of a set

Example

T(3,2) = 20: 1a2a3b, 1a2b3b, 1a|2a3b, 1a|2b3b, 1b|2a3a, 1b|2a3b, 1a3b|2a, 1b3b|2a, 1a3a|2b, 1a3b|2b, 1a2b|3a, 1b2b|3a, 1a2a|3b, 1a2b|3b, 1a|2a|3b, 1a|2b|3a, 1b|2a|3a, 1a|2b|3b, 1b|2a|3b, 1b|2b|3a.
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,     3;
  0,   5,    20,     16;
  0,  15,   122,    237,     131;
  0,  52,   774,   2751,    3524,    1496;
  0, 203,  5247,  30470,   68000,   65055,   22482;
  0, 877, 38198, 341244, 1181900, 1913465, 1462320, 426833;
  ...

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*
      binomial(n-1, j-1)*binomial(k+j-1, j), j=1..n))
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[A[n-j, k] Binomial[n-1, j-1]* Binomial[k + j - 1, j], {j, n}]];
T[n_, k_] := Sum[A[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)

Crossrefs

Columns k=0-2 give: A000007, A000110 (for n>0), A325890.

Main diagonal gives A023998.

Row sums give A325888.

T(2n,n) gives A325889.

Cf. A322670.

Sequence in context: A344137 A350260 A358276 * A190902 A115562 A343069

Adjacent sequences: A321293 A321294 A321295 * A321297 A321298 A321299

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 29 2019

Status

approved