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A323128 Number T(n,k) of colored set partitions of [n] where elements of subsets have distinct colors and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 5
1, 0, 1, 0, 1, 4, 0, 1, 18, 30, 0, 1, 74, 360, 360, 0, 1, 310, 3450, 8880, 6240, 0, 1, 1382, 31770, 160080, 271800, 146160, 0, 1, 6510, 298662, 2635920, 8152200, 10190880, 4420080, 0, 1, 32398, 2918244, 42687960, 214527600, 468669600, 460474560, 166924800 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

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

Wikipedia, Partition of a set

EXAMPLE

T(3,2) = 18: 1a|2a3b, 1a|2b3a, 1b|2a3b, 1b|2b3a, 1a3b|2a, 1b3a|2a, 1a3b|2b, 1b3a|2b, 1a2b|3a, 1b2a|3a, 1a2b|3b, 1b2a|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, 1,    4;

  0, 1,   18,     30;

  0, 1,   74,    360,     360;

  0, 1,  310,   3450,    8880,    6240;

  0, 1, 1382,  31770,  160080,  271800,   146160;

  0, 1, 6510, 298662, 2635920, 8152200, 10190880, 4420080;

  ...

MAPLE

A:= proc(n, k) option remember; `if`(n=0, 1, add(k!/(k-j)!

      *binomial(n-1, j-1)*A(n-j, k), j=1..min(k, 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[k!/(k - j)! Binomial[n - 1, j - 1]* A[n - j, k], {j, Min[k, 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-Fran├žois Alcover, Apr 30 2020, after Alois P. Heinz *)

CROSSREFS

Columns k=0-1 give: A000007, A057427.

Row sums give A104600.

Main diagonal gives A137341.

T(2n,n) gives A324523.

Sequence in context: A221817 A054375 A195739 * A334703 A259938 A342202

Adjacent sequences:  A323125 A323126 A323127 * A323129 A323130 A323131

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 30 2019

STATUS

approved

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Last modified September 21 07:28 EDT 2021. Contains 347596 sequences. (Running on oeis4.)