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A322670
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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.
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7
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1, 0, 1, 0, 1, 3, 0, 1, 12, 16, 0, 1, 41, 156, 131, 0, 1, 140, 1155, 2460, 1496, 0, 1, 497, 8020, 32600, 47355, 22482, 0, 1, 1848, 55629, 385420, 1004360, 1098678, 426833, 0, 1, 7191, 394884, 4396189, 18304510, 34625304, 30259712, 9934563
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OFFSET
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0,6
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LINKS
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FORMULA
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Sum_{k=1..n} k * T(n,k) = A325930(n).
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EXAMPLE
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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.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 3;
0, 1, 12, 16;
0, 1, 41, 156, 131;
0, 1, 140, 1155, 2460, 1496;
0, 1, 497, 8020, 32600, 47355, 22482;
0, 1, 1848, 55629, 385420, 1004360, 1098678, 426833;
...
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MAPLE
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A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*
binomial(n-1, j-1)*binomial(k, j), 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);
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MATHEMATICA
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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[n_, k_] := Sum[A[n, k-i] (-1)^i Binomial[k, i], {i, 0, k}];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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