A323128 - OEIS

%I #78 Apr 30 2020 11:33:04

%S 1,0,1,0,1,4,0,1,18,30,0,1,74,360,360,0,1,310,3450,8880,6240,0,1,1382,

%T 31770,160080,271800,146160,0,1,6510,298662,2635920,8152200,10190880,

%U 4420080,0,1,32398,2918244,42687960,214527600,468669600,460474560,166924800

%N 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.

%H Alois P. Heinz, <a href="/A323128/b323128.txt">Rows n = 0..140, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%e 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.

%e Triangle T(n,k) begins:

%e   1;

%e   0, 1;

%e   0, 1,    4;

%e   0, 1,   18,     30;

%e   0, 1,   74,    360,     360;

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

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

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

%e   ...

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

%p       *binomial(n-1, j-1)*A(n-j, k), j=1..min(k, n)))

%p     end:

%p T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):

%p seq(seq(T(n, k), k=0..n), n=0..10);

%t 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 T[n_, k_] := Sum[A[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 30 2020, after _Alois P. Heinz_ *)

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

%Y Row sums give A104600.

%Y Main diagonal gives A137341.

%Y T(2n,n) gives A324523.

%K nonn,tabl

%O 0,6

%A _Alois P. Heinz_, Aug 30 2019