Search: a287671 -id:a287671
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A287641
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Number A(n,k) of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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+10
14
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1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 5, 1, 1, 1, 2, 5, 14, 1, 1, 1, 2, 5, 15, 42, 1, 1, 1, 2, 5, 15, 51, 132, 1, 1, 1, 2, 5, 15, 52, 191, 429, 1, 1, 1, 2, 5, 15, 52, 202, 773, 1430, 1, 1, 1, 2, 5, 15, 52, 203, 861, 3336, 4862, 1, 1, 1, 2, 5, 15, 52, 203, 876, 3970, 15207, 16796, 1
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OFFSET
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0,9
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LINKS
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FORMULA
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A(n,k) = Sum_{j=0..k} A287640(n,j).
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EXAMPLE
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A(5,0) = 1: 12345.
A(5,1) = 42 = 52 - 10 = A000110(5) - 10 counts all set partitions of [5] except: 124|3|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 14|2|35, 14|2|3|5, 1|24|3|5, 134|2|5.
A(5,2) = 51 = 52 - 1 = A000110(5) - 1 counts all set partitions of [5] except: 134|2|5.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 2, 2, 2, 2, 2, 2, ...
1, 5, 5, 5, 5, 5, 5, 5, ...
1, 14, 15, 15, 15, 15, 15, 15, ...
1, 42, 51, 52, 52, 52, 52, 52, ...
1, 132, 191, 202, 203, 203, 203, 203, ...
1, 429, 773, 861, 876, 877, 877, 877, ...
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MAPLE
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b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
[seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
end:
A:= (n, k)-> `if`(k=0, 1, b(n, [0$k])):
seq(seq(A(n, d-n), n=0..d), d=0..12);
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MATHEMATICA
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b[0, _] = 1; b[n_, l_List] := b[n, l] = Sum[b[n - 1, Append[ Table[ Max[ l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}];
A[n_, k_] := If[k == 0, 1, b[n, Table[0, k]]];
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CROSSREFS
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Columns k=0-10 give: A000012, A000108, A275605, A287666, A287667, A287668, A287669, A287670, A287671, A287672, A287673.
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KEYWORD
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AUTHOR
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STATUS
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approved
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