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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).
7
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
0,3
COMMENTS
If the permutation has no third cycle, then its third-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..n-2} k * T(n,k) = A332907(n) for n >= 3. - Alois P. Heinz, Dec 12 2021
EXAMPLE
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;
...
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 *)
CROSSREFS
Column 0 gives 1 together with A000774.
Column 1 gives the column 3 of A208956.
Row sums give A000142.
Sequence in context: A214733 A106852 A352010 * A162975 A350015 A187244
KEYWORD
nonn,tabf
AUTHOR
Steven Finch, Dec 08 2021
STATUS
approved