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A261477
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Number T(n,k) of set partitions of [n] into exactly k parts such that no part contains two elements with a circular distance less than three; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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10
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1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 7, 14, 7, 1, 0, 0, 0, 0, 7, 44, 42, 12, 1, 0, 0, 0, 1, 21, 138, 210, 102, 18, 1, 0, 0, 0, 0, 50, 458, 985, 720, 210, 25, 1, 0, 0, 0, 0, 77, 1397, 4400, 4587, 1980, 385, 33, 1
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OFFSET
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0,26
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COMMENTS
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The circular distance of 1 and n is 1 (for n>1).
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LINKS
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EXAMPLE
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T(6,3) = 1: 14|25|36.
T(6,4) = 3: 14|25|3|6, 14|2|36|5, 1|25|36|4.
T(6,5) = 3: 14|2|3|5|6, 1|25|3|4|6, 1|2|36|4|5.
T(6,6) = 1: 1|2|3|4|5|6.
Triangle T(n,k) begins:
1;
0, 1;
0, 0, 1;
0, 0, 0, 1;
0, 0, 0, 0, 1;
0, 0, 0, 0, 0, 1;
0, 0, 0, 1, 3, 3, 1;
0, 0, 0, 0, 7, 14, 7, 1;
0, 0, 0, 0, 7, 44, 42, 12, 1;
0, 0, 0, 1, 21, 138, 210, 102, 18, 1;
0, 0, 0, 0, 50, 458, 985, 720, 210, 25, 1;
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MAPLE
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g:= proc(n, l, m, h) option remember;
`if`(n=0, `if`(1 in [l, m] or l=2, `if`(h<3, x^h, 0), x^h),
add(`if`(j in [l, m], 0, g(n-1, j, l, max(h, j))), j=1..h+1))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(g(n, 0$3)):
seq(T(n), n=0..14);
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MATHEMATICA
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g[n_, l_, m_, h_] := g[n, l, m, h] = If[n == 0, If[l == 1 || m == 1 || l == 2, If[h < 3, x^h, 0], x^h], Sum[If[j == l || j == m, 0, g[n - 1, j, l, Max[h, j]]], {j, 1, h + 1}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][g[n, 0, 0, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)
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CROSSREFS
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Columns k=0-10 give: A000007, A063524, A185012, A079978 (for n>0), A261480, A261481, A261482, A261483, A261484, A261485, A261486.
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KEYWORD
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AUTHOR
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STATUS
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approved
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