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A275281
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Number T(n,k) of set partitions of [n] with symmetric block size list of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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13
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1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 7, 0, 1, 0, 1, 10, 19, 13, 3, 1, 0, 1, 0, 56, 0, 22, 0, 1, 0, 1, 35, 160, 171, 86, 34, 4, 1, 0, 1, 0, 463, 0, 470, 0, 50, 0, 1, 0, 1, 126, 1337, 2306, 2066, 1035, 250, 70, 5, 1, 0, 1, 0, 3874, 0, 10299, 0, 2160, 0, 95, 0, 1
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OFFSET
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0,13
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LINKS
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FORMULA
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T(n,k) = 0 if n is odd and k is even.
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EXAMPLE
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T(4,2) = 3: 12|34, 13|24, 14|23.
T(5,3) = 7: 12|3|45, 13|2|45, 1|234|5, 1|235|4, 14|2|35, 1|245|3, 15|2|34.
T(6,4) = 13: 12|3|4|56, 13|2|4|56, 1|23|45|6, 1|23|46|5, 14|2|3|56, 1|24|35|6, 1|24|36|5, 1|25|34|6, 1|26|34|5, 15|2|3|46, 1|25|36|4, 1|26|35|4, 16|2|3|45.
T(7,5) = 22: 12|3|4|5|67, 13|2|4|5|67, 1|23|4|56|7, 1|23|4|57|6, 14|2|3|5|67, 1|24|3|56|7, 1|24|3|57|6, 1|2|345|6|7, 1|2|346|5|7, 1|2|347|5|6, 15|2|3|4|67, 1|25|3|46|7, 1|25|3|47|6, 1|2|356|4|7, 1|2|357|4|6, 1|26|3|45|7, 1|27|3|45|6, 16|2|3|4|57, 1|26|3|47|5, 1|2|367|4|5, 1|27|3|46|5, 17|2|3|4|56.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 0, 1;
0, 1, 3, 2, 1;
0, 1, 0, 7, 0, 1;
0, 1, 10, 19, 13, 3, 1;
0, 1, 0, 56, 0, 22, 0, 1;
0, 1, 35, 160, 171, 86, 34, 4, 1;
0, 1, 0, 463, 0, 470, 0, 50, 0, 1;
0, 1, 126, 1337, 2306, 2066, 1035, 250, 70, 5, 1;
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MAPLE
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b:= proc(n, s) option remember; expand(`if`(n>s,
binomial(n-1, n-s-1)*x, 1)+add(binomial(n-1, j-1)*
b(n-j, s+j)*binomial(s+j-1, j-1), j=1..(n-s)/2)*x^2)
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..12);
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MATHEMATICA
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b[n_, s_] := b[n, s] = Expand[If[n>s, Binomial[n-1, n-s-1]*x, 1] + Sum[ Binomial[n-1, j-1]*b[n-j, s+j]*Binomial[s+j-1, j-1], {j, 1, (n-s)/2} ]*x^2]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 03 2017, translated from Maple *)
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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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