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A276891
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Number T(n,k) of ordered set partitions of [n] where k is minimal such that for each block b the smallest integer interval containing b has at most k elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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14
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1, 0, 1, 0, 2, 1, 0, 6, 4, 3, 0, 24, 20, 18, 13, 0, 120, 114, 118, 114, 75, 0, 720, 750, 878, 924, 870, 541, 0, 5040, 5616, 7224, 8152, 8760, 7818, 4683, 0, 40320, 47304, 65514, 79682, 90084, 94560, 81078, 47293, 0, 362880, 443400, 652446, 845874, 998560, 1135776, 1148016, 954474, 545835
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OFFSET
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0,5
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LINKS
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FORMULA
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EXAMPLE
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Triangle T(n,k) begins:
1;
0, 1;
0, 2, 1;
0, 6, 4, 3;
0, 24, 20, 18, 13;
0, 120, 114, 118, 114, 75;
0, 720, 750, 878, 924, 870, 541;
0, 5040, 5616, 7224, 8152, 8760, 7818, 4683;
0, 40320, 47304, 65514, 79682, 90084, 94560, 81078, 47293;
...
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MAPLE
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b:= proc(n, m, l) option remember; `if`(n=0, m!,
add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
`if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
end:
A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0),
`if`(k=1, n!, b(n, 0, [0$(k-1)]))):
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);
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MATHEMATICA
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b[n_, m_, l_List] := b[n, m, l] = If[n == 0, m!, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1] ~Complement~ {0}}]]; A[n_, k_] := If[k == 0, If[n == 0, 1, 0], If[k == 1, n!, b[n, 0, Array[0 &, k - 1]]]]; T [n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 04 2017, translated from Maple *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000142 (for n>0), A320615, A320616, A320617, A320618, A320619, A320620, A320621, A320622, A320623.
Main diagonal gives A000670(n-1) for n>0.
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KEYWORD
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AUTHOR
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STATUS
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approved
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