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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). 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
If the permutation has no second cycle, then its second-longest cycle is defined to have length 0.
LINKS
Steven Finch, Second best, Third worst, Fourth in line, arxiv:2202.07621 [math.CO], 2022.
FORMULA
Sum_{k=0..max(0,n-1)} k * T(n,k) = A332906(n). - Alois P. Heinz, Dec 07 2021
EXAMPLE
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;
...
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 *)
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.
Sequence in context: A118287 A024930 A121966 * A021472 A258047 A053225
KEYWORD
nonn,tabf
AUTHOR
Steven Finch, Dec 07 2021
EXTENSIONS
More terms from Alois P. Heinz, Dec 07 2021
STATUS
approved

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Last modified March 29 04:59 EDT 2024. Contains 371264 sequences. (Running on oeis4.)