A350016 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose third-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-2).

1, 1, 2, 5, 1, 17, 1, 6, 74, 11, 15, 20, 394, 56, 60, 120, 90, 2484, 407, 525, 490, 630, 504, 18108, 3235, 4725, 2240, 4620, 4032, 3360, 149904, 29143, 40509, 27440, 26460, 33264, 30240, 25920, 1389456, 291394, 398790, 319760, 163800, 302400, 277200, 259200, 226800

Offset: 0

Author

Steven Finch, Dec 08 2021

Keywords

• nonn
• tabf

Comments

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

Examples

			Triangle begins:
[0]      1;
[1]      1;
[2]      2;
[3]      5,     1;
[4]     17,     1,     6;
[5]     74,    11,    15,    20;
[6]    394,    56,    60,   120,    90;
[7]   2484,   407,   525,   490,   630,   504;
[8]  18108,  3235,  4725,  2240,  4620,  4032,  3360;
[9] 149904, 29143, 40509, 27440, 26460, 33264, 30240, 25920;
...
		

Links

• Alois P. Heinz, Rows n = 0..142, flattened
• Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

Crossrefs

Column 0 gives 1 together with A000774.

Column 1 gives the column 3 of A208956.

Row sums give A000142.

Cf. A126074, A145877, A332907, A349979, A349980, A350015, A350273, A350274.

Programs

Maple

m:= infinity:
b:= proc(n, l) option remember; `if`(n=0, x^`if`(l[3]=m,
      0, l[3]), add(b(n-j, sort([l[], j])[1..3])
               *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$3])):
seq(T(n), n=0..10);  # Alois P. Heinz, Dec 11 2021

Mathematica

m = Infinity;
b[n_, l_] := b[n, l] = If[n == 0, x^If[l[[3]] == m, 0, l[[3]]], Sum[b[n-j, Sort[Append[l, j]][[1;;3]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]];
T[n_] := With[{p = b[n, {m, m, m}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

Formula

Sum_{k=0..n-2} k * T(n,k) = A332907(n) for n >= 3. - Alois P. Heinz, Dec 12 2021