A350273 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).

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

Offset: 0

Author

Steven Finch, Dec 22 2021

Keywords

• nonn
• tabf

Comments

If the permutation has no fourth cycle, then its fourth-longest cycle is defined to have length 0.

Examples

			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;
    ...
		

Links

• Alois P. Heinz, Rows n = 0..100, flattened
• Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Crossrefs

Row sums give A000142(n).

Cf. A126074, A145877, A332853, A349979, A349980, A350015, A350016, A350274.

Programs

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 *)

Formula

Sum_{k=0..floor(n/4)} k * T(n,k) = A332853(n) for n >= 4.

Extensions

More terms from Alois P. Heinz, Dec 22 2021