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A322670 Number T(n,k) of colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A322670 #92 Apr 30 2020 06:04:23
%S A322670 1,0,1,0,1,3,0,1,12,16,0,1,41,156,131,0,1,140,1155,2460,1496,0,1,497,
%T A322670 8020,32600,47355,22482,0,1,1848,55629,385420,1004360,1098678,426833,
%U A322670 0,1,7191,394884,4396189,18304510,34625304,30259712,9934563
%N A322670 Number T(n,k) of colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A322670 Alois P. Heinz, <a href="/A322670/b322670.txt">Rows n = 0..140, flattened</a>
%H A322670 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A322670 Sum_{k=1..n} k * T(n,k) = A325930(n).
%e A322670 T(3,2) = 12: 1a|2a3b, 1b|2a3b, 1a3b|2a, 1a3b|2b, 1a2b|3a, 1a2b|3b, 1a|2a|3b, 1a|2b|3a, 1b|2a|3a, 1a|2b|3b, 1b|2a|3b, 1b|2b|3a.
%e A322670 Triangle T(n,k) begins:
%e A322670   1;
%e A322670   0, 1;
%e A322670   0, 1,    3;
%e A322670   0, 1,   12,    16;
%e A322670   0, 1,   41,   156,    131;
%e A322670   0, 1,  140,  1155,   2460,    1496;
%e A322670   0, 1,  497,  8020,  32600,   47355,   22482;
%e A322670   0, 1, 1848, 55629, 385420, 1004360, 1098678, 426833;
%e A322670   ...
%p A322670 A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*
%p A322670       binomial(n-1, j-1)*binomial(k, j), j=1..min(k, n)))
%p A322670     end:
%p A322670 T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p A322670 seq(seq(T(n, k), k=0..n), n=0..10);
%t A322670 A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[A[n-j, k] Binomial[n-1, j-1]* Binomial[k, j], {j, 1, Min[k, n]}]];
%t A322670 T[n_, k_] := Sum[A[n, k-i] (-1)^i Binomial[k, i], {i, 0, k}];
%t A322670 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 30 2020, after _Alois P. Heinz_ *)
%Y A322670 Columns k=0-2 give: A000007, A057427, A325482.
%Y A322670 Main diagonal gives A023998.
%Y A322670 Row sums give A325478.
%Y A322670 T(2n,n) gives A325481.
%Y A322670 Cf. A321296, A323128, A325930.
%K A322670 nonn,tabl
%O A322670 0,6
%A A322670 _Alois P. Heinz_, Aug 29 2019