A349980 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose second-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-1).

1, 1, 1, 1, 2, 1, 3, 6, 7, 3, 8, 24, 31, 15, 20, 30, 120, 191, 135, 40, 90, 144, 720, 1331, 945, 280, 420, 504, 840, 5040, 10655, 7077, 4480, 1260, 2688, 3360, 5760, 40320, 95887, 64197, 41552, 11340, 18144, 20160, 25920, 45360, 362880, 958879, 646965, 395360, 238140, 72576, 151200, 172800, 226800, 403200

Offset: 0

Author

Steven Finch, Dec 07 2021

Keywords

• nonn
• tabf

Comments

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

Examples

			Triangle begins:
[0]     1;
[1]     1;
[2]     1,     1;
[3]     2,     1,     3;
[4]     6,     7,     3,     8;
[5]    24,    31,    15,    20,    30;
[6]   120,   191,   135,    40,    90,   144;
[7]   720,  1331,   945,   280,   420,   504,   840;
[8]  5040, 10655,  7077,  4480,  1260,  2688,  3360,  5760;
[9] 40320, 95887, 64197, 41552, 11340, 18144, 20160, 25920, 45360;
    ...
		

Links

• Alois P. Heinz, Rows n = 0..141, flattened
• Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Crossrefs

Column 0 gives 1 together with A000142.

Column 1 gives the nonzero terms of A155521.

Row sums give A000142.

T(n,n-1) gives A059171(n) for n>=1.

Cf. A126074, A145877, A332906, A349979, A350015, A350016, A350273, A350274.

Programs

Maple

m:= infinity:
b:= proc(n, l) option remember; `if`(n=0, x^`if`(l[2]=m,
      0, l[2]), add(b(n-j, sort([l[], j])[1..2])
               *binomial(n-1, j-1)*(j-1)!, j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, n-1)))(b(n, [m$2])):
seq(T(n), n=0..10);  # Alois P. Heinz, Dec 07 2021

Mathematica

m = Infinity;
b[n_, l_] := b[n, l] = If[n == 0, x^If[l[[2]] == m, 0, l[[2]]], Sum[b[n-j, Sort[Append[l, j]][[1;;2]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]];
T[n_] := With[{p = b[n, {m, m}]}, Table[Coefficient[p, x, i], {i, 0, Max[0, n - 1]}]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

Formula

Sum_{k=0..max(0,n-1)} k * T(n,k) = A332906(n). - Alois P. Heinz, Dec 07 2021

Extensions

More terms from Alois P. Heinz, Dec 07 2021