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A364971
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Number T(n,k) of partitions of [n] for which the difference between the longest and the shortest block size is k; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.
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2
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1, 1, 2, 2, 3, 5, 6, 4, 2, 35, 10, 5, 27, 60, 95, 15, 6, 2, 371, 315, 161, 21, 7, 142, 938, 2002, 770, 252, 28, 8, 282, 4005, 9744, 5313, 1386, 372, 36, 9, 1073, 16950, 50275, 33705, 11082, 2310, 525, 45, 10, 2, 74657, 283525, 217800, 78078, 20097, 3630, 715, 55, 11
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OFFSET
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0,3
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COMMENTS
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T(0,0) = 1 by convention.
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LINKS
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EXAMPLE
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T(4,0) = 5: 1|2|3|4, 12|34, 13|24, 14|23, 1234.
T(4,1) = 6: 1|2|34, 1|23|4, 1|24|3, 12|3|4, 13|2|4, 14|2|3.
T(4,2) = 4: 1|234, 123|4, 124|3, 134|2.
Triangle T(n,k) begins:
1;
1;
2;
2, 3;
5, 6, 4;
2, 35, 10, 5;
27, 60, 95, 15, 6;
2, 371, 315, 161, 21, 7;
142, 938, 2002, 770, 252, 28, 8;
282, 4005, 9744, 5313, 1386, 372, 36, 9;
1073, 16950, 50275, 33705, 11082, 2310, 525, 45, 10;
...
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MAPLE
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b:= proc(n, l, m) option remember; `if`(n=0, x^(m-l), add(
b(n-j, min(l, j), max(m, j))*binomial(n-1, j-1), j=1..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..12);
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MATHEMATICA
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b[n_, l_, m_] := b[n, l, m] = If[n == 0, x^(m - l), Sum[b[n - j, Min[l, j], Max[m, j]]*Binomial[n - 1, j - 1], {j, 1, n}]];
T[n_] := CoefficientList[b[n, n, 0], x];
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CROSSREFS
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Column k=0 gives A038041 (for n>=1).
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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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