OFFSET
0,3
COMMENTS
If the permutation has no fourth cycle, then its fourth-longest cycle is defined to have length 0.
LINKS
Alois P. Heinz, Rows n = 0..100, flattened
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.
FORMULA
Sum_{k=0..floor(n/4)} k * T(n,k) = A332853(n) for n >= 4.
EXAMPLE
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;
...
MAPLE
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])):
seq(T(n), n=0..14); # Alois P. Heinz, Dec 22 2021
MATHEMATICA
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]}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 29 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Steven Finch, Dec 22 2021
EXTENSIONS
More terms from Alois P. Heinz, Dec 22 2021
STATUS
approved