A349979 - OEIS

A349979

Irregular triangle read by rows: T(n,k) is the number of n-permutations whose second-longest cycle has length exactly k; n>=0, 0<=k<=floor(n/2).

1, 1, 1, 1, 2, 4, 6, 15, 3, 24, 61, 35, 120, 290, 270, 40, 720, 1646, 1974, 700, 5040, 11025, 14707, 8288, 1260, 40320, 85345, 117459, 90272, 29484, 362880, 749194, 1023390, 974720, 446040, 72576, 3628800, 7347374, 9813210, 10666480, 6332040, 2128896

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OFFSET

0,5

COMMENTS

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

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.

FORMULA

Sum_{k=0..floor(n/2)} k * T(n,k) = A332851(n). - Alois P. Heinz, Dec 07 2021

EXAMPLE

Triangle begins:

[0] 1;

[1] 1;

[2] 1, 1;

[3] 2, 4;

[4] 6, 15, 3;

[5] 24, 61, 35;

[6] 120, 290, 270, 40;

[7] 720, 1646, 1974, 700;

[8] 5040, 11025, 14707, 8288, 1260;

[9] 40320, 85345, 117459, 90272, 29484;

...

MAPLE

b:= proc(n, l) option remember; `if`(n=0, x^l[1], add((j-1)!*

b(n-j, sort([l[], j])[2..3])*binomial(n-1, j-1), j=1..n))

end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n, [0$2])):

seq(T(n), n=0..12); # Alois P. Heinz, Dec 07 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 ;; 3]]]*Binomial[n - 1, j - 1], {j, 1, n}]];

T[n_] := With[{p = b[n, {0, 0}]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];

Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

CROSSREFS

Column 0 gives 1 together with A000142.

Column 1 gives 1 - (n-1)! + A006231(n).

Row sums give A000142.

T(2n,n) gives A110468(n-1) for n>=1.

Cf. A126074, A145877, A332851, A349980, A350015, A350016, A350273, A350274.

Sequence in context: A077639 A039791 A275664 * A192536 A227902 A217525

Adjacent sequences: A349976 A349977 A349978 * A349980 A349981 A349982

KEYWORD

nonn,tabf

AUTHOR

Steven Finch, Dec 07 2021

STATUS

approved