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A350273
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Irregular triangle read by rows: T(n,k) is the number of n-permutations whose fourth-longest cycle has length exactly k; n >= 0, 0 <= k <= floor(n/4).
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6
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1, 1, 2, 6, 23, 1, 109, 11, 619, 101, 4108, 932, 31240, 8975, 105, 268028, 91387, 3465, 2562156, 991674, 74970, 27011016, 11514394, 1391390, 311378616, 143188574, 24188010, 246400, 3897004032, 1905067958, 412136010, 12812800, 52626496896, 27059601596, 7053834788, 438357920
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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 fourth cycle, then its fourth-longest cycle is defined to have length 0.
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LINKS
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FORMULA
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Sum_{k=0..floor(n/4)} k * T(n,k) = A332853(n) for n >= 4.
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EXAMPLE
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Triangle begins:
[0] 1;
[1] 1;
[2] 2;
[3] 6;
[4] 23, 1;
[5] 109, 11;
[6] 619, 101;
[7] 4108, 932;
[8] 31240, 8975, 105;
[9] 268028, 91387, 3465;
...
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MAPLE
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b:= proc(n, l) option remember; `if`(n=0, x^l[1], add((j-1)!*
b(n-j, sort([l[], j])[2..5])*binomial(n-1, j-1), j=1..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$4])):
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MATHEMATICA
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b[n_, l_] := b[n, l] = If[n == 0, x^l[[1]], Sum[(j - 1)!*b[n - j, Sort[ Append[l, j]][[2 ;; 5]]]*Binomial[n - 1, j - 1], {j, 1, n}]];
T[n_] := With[{p = b[n, {0, 0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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