A350274 Triangle read by rows: T(n,k) is the number of n-permutations whose fourth-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-3).

1, 1, 2, 6, 23, 1, 109, 1, 10, 619, 16, 45, 40, 4108, 92, 210, 420, 210, 31240, 771, 1645, 2800, 2520, 1344, 268028, 6883, 17325, 15960, 26460, 18144, 10080, 2562156, 68914, 173250, 148400, 226800, 211680, 151200, 86400, 27011016, 757934, 1854930, 1798720, 1801800, 2494800, 1940400, 1425600, 831600

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,    1,    10;
[6]    619,   16,    45,    40;
[7]   4108,   92,   210,   420,   210;
[8]  31240,  771,  1645,  2800,  2520,  1344;
[9] 268028, 6883, 17325, 15960, 26460, 18144, 10080;
    ...
		

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

Column 0 is 1 for n=0, together with A000142(n) - A122105(n-1) for n>=1.

Row sums give A000142.

Cf. A126074, A145877, A332908, A349979, A349980, A350015, A350016, A350273.

Programs

Maple

m:= infinity:
b:= proc(n, l) option remember; `if`(n=0, x^`if`(l[4]=m,
      0, l[4]), add(b(n-j, sort([l[], j])[1..4])
               *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$4])):
seq(T(n), n=0..11);  # Alois P. Heinz, Dec 22 2021

Mathematica

m = Infinity;
b[n_, l_] := b[n, l] = If[n == 0, x^If[l[[4]] == m, 0, l[[4]]], Sum[b[n-j, Sort[Append[l, j]][[1 ;; 4]]]*Binomial[n-1, j-1]*(j-1)!, {j, 1, n}]];
T[n_] := With[{p = b[n, {m, m, m, m}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Dec 29 2021, after Alois P. Heinz *)

Formula

Sum_{k=0..n-3} k * T(n,k) = A332908(n) for n >= 4.

Extensions

More terms from Alois P. Heinz, Dec 22 2021