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A287213
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Number T(n,k) of set partitions of [n] such that the maximal absolute difference between consecutive elements within a block equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows.
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15
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1, 1, 1, 1, 1, 3, 1, 1, 7, 5, 2, 1, 15, 18, 13, 5, 1, 31, 57, 61, 38, 15, 1, 63, 169, 248, 215, 129, 52, 1, 127, 482, 935, 1061, 836, 495, 203, 1, 255, 1341, 3368, 4835, 4789, 3573, 2108, 877, 1, 511, 3669, 11777, 20973, 25430, 22986, 16657, 9831, 4140
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OFFSET
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0,6
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COMMENTS
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The maximal absolute difference is assumed to be zero if there is no block with consecutive elements.
T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 if k>=n and k>0.
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LINKS
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FORMULA
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EXAMPLE
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T(4,0) = 1: 1|2|3|4.
T(4,1) = 7: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34.
T(4,2) = 5: 124|3, 134|2, 13|24, 13|2|4, 1|24|3.
T(4,3) = 2: 14|23, 14|2|3.
Triangle T(n,k) begins:
1;
1;
1, 1;
1, 3, 1;
1, 7, 5, 2;
1, 15, 18, 13, 5;
1, 31, 57, 61, 38, 15;
1, 63, 169, 248, 215, 129, 52;
1, 127, 482, 935, 1061, 836, 495, 203;
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MAPLE
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b:= proc(n, k, l) option remember; `if`(n=0, 1, b(n-1, k, map(x->
`if`(x-n>=k, [][], x), [l[], n]))+add(b(n-1, k, sort(map(x->
`if`(x-n>=k, [][], x), subsop(j=n, l)))), j=1..nops(l)))
end:
A:= (n, k)-> b(n, min(k, n-1), []):
T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..max(n-1, 0)), n=0..12);
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MATHEMATICA
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b[0, _, _] = 1; b[n_, k_, l_] := b[n, k, l] =b[n - 1, k, If[# - n >= k, Nothing, #]& /@ Append[l, n]] + Sum[b[n - 1, k, Sort[If[# - n >= k, Nothing, #]& /@ ReplacePart[l, j -> n]]], {j, 1, Length[l]}];
A[n_, k_] := b[n, Min[k, n - 1], {}];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
Table[Table[T[n, k], {k, 0, Max[n - 1, 0]}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)
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CROSSREFS
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Columns k=0-10 give: A000012, A000225(n-1), A258109, A294052, A294053, A294054, A294055, A294056, A294057, A294058, A294059.
Row sums and T(n+2,n+1) give A000110.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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