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A261959
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Number A(n,k) of ordered set partitions of {1,2,...,n} such that no part has the same size as any of its k immediate predecessors; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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10
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1, 1, 1, 1, 1, 3, 1, 1, 1, 13, 1, 1, 1, 7, 75, 1, 1, 1, 7, 21, 541, 1, 1, 1, 7, 9, 81, 4683, 1, 1, 1, 7, 9, 31, 793, 47293, 1, 1, 1, 7, 9, 31, 403, 4929, 545835, 1, 1, 1, 7, 9, 31, 403, 1597, 33029, 7087261, 1, 1, 1, 7, 9, 31, 403, 757, 7913, 388537, 102247563
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refs;
listen;
history;
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OFFSET
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0,6
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LINKS
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EXAMPLE
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A(3,1) = 7: 123, 1|23, 23|1, 2|13, 13|2, 3|12, 12|3.
A(4,1) = 21: 1234, 1|234, 234|1, 2|134, 134|2, 3|124, 124|3, 4|123, 123|4, 3|12|4, 4|12|3, 2|13|4, 4|13|2, 2|14|3, 3|14|2, 1|23|4, 4|23|1, 1|24|3, 3|24|1, 1|34|2, 2|34|1.
Square array A(n,k) begins:
: 1, 1, 1, 1, 1, 1, 1, ...
: 1, 1, 1, 1, 1, 1, 1, ...
: 3, 1, 1, 1, 1, 1, 1, ...
: 13, 7, 7, 7, 7, 7, 7, ...
: 75, 21, 9, 9, 9, 9, 9, ...
: 541, 81, 31, 31, 31, 31, 31, ...
: 4683, 793, 403, 403, 403, 403, 403, ...
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MAPLE
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b:= proc(n, l) option remember; `if`(n=0, 1,
add(`if`(j in l, 0, binomial(n, j)*b(n-j,
`if`(l=[], [], [subsop(1=NULL, l)[], j]))), j=1..n))
end:
A:= (n, k)-> b(n, [0$min(n, k)]):
seq(seq(A(n, d-n), n=0..d), d=0..10);
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MATHEMATICA
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b[n_, l_List] := b[n, l] = If[n == 0, 1, Sum[If[MemberQ[l, j], 0, Binomial[n, j]*b[n-j, If[l == {}, {}, Append[ReplacePart[l, 1 -> Nothing], j]]]], {j, 1, n}]]; A[n_, k_] := b[n, Array[0&, Min[n, k]]]; Table[A[n, d-n], {d, 0, 10} , {n, 0, d}] // Flatten (* Jean-François Alcover, Dec 17 2016, after Alois P. Heinz *)
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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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