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A350016
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Irregular triangle read by rows: T(n,k) is the number of n-permutations whose third-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-2).
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7
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1, 1, 2, 5, 1, 17, 1, 6, 74, 11, 15, 20, 394, 56, 60, 120, 90, 2484, 407, 525, 490, 630, 504, 18108, 3235, 4725, 2240, 4620, 4032, 3360, 149904, 29143, 40509, 27440, 26460, 33264, 30240, 25920, 1389456, 291394, 398790, 319760, 163800, 302400, 277200, 259200, 226800
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OFFSET
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0,3
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COMMENTS
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If the permutation has no third cycle, then its third-longest cycle is defined to have length 0.
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
[0] 1;
[1] 1;
[2] 2;
[3] 5, 1;
[4] 17, 1, 6;
[5] 74, 11, 15, 20;
[6] 394, 56, 60, 120, 90;
[7] 2484, 407, 525, 490, 630, 504;
[8] 18108, 3235, 4725, 2240, 4620, 4032, 3360;
[9] 149904, 29143, 40509, 27440, 26460, 33264, 30240, 25920;
...
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MAPLE
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m:= infinity:
b:= proc(n, l) option remember; `if`(n=0, x^`if`(l[3]=m,
0, l[3]), add(b(n-j, sort([l[], j])[1..3])
*binomial(n-1, j-1)*(j-1)!, j=1..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [m$3])):
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MATHEMATICA
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m = Infinity;
b[n_, l_] := b[n, l] = If[n == 0, x^If[l[[3]] == m, 0, l[[3]]], Sum[b[n-j, Sort[Append[l, j]][[1;; 3]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]];
T[n_] := With[{p = b[n, {m, m, m}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
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CROSSREFS
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Column 0 gives 1 together with A000774.
Column 1 gives the column 3 of A208956.
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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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