login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A350273
Irregular triangle read by rows: T(n,k) is the number of n-permutations whose fourth-longest cycle has length exactly k; n >= 0, 0 <= k <= floor(n/4).
6
1, 1, 2, 6, 23, 1, 109, 11, 619, 101, 4108, 932, 31240, 8975, 105, 268028, 91387, 3465, 2562156, 991674, 74970, 27011016, 11514394, 1391390, 311378616, 143188574, 24188010, 246400, 3897004032, 1905067958, 412136010, 12812800, 52626496896, 27059601596, 7053834788, 438357920
OFFSET
0,3
COMMENTS
If the permutation has no fourth cycle, then its fourth-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..floor(n/4)} k * T(n,k) = A332853(n) for n >= 4.
EXAMPLE
Triangle begins:
[0] 1;
[1] 1;
[2] 2;
[3] 6;
[4] 23, 1;
[5] 109, 11;
[6] 619, 101;
[7] 4108, 932;
[8] 31240, 8975, 105;
[9] 268028, 91387, 3465;
...
MAPLE
b:= proc(n, l) option remember; `if`(n=0, x^l[1], add((j-1)!*
b(n-j, sort([l[], j])[2..5])*binomial(n-1, j-1), j=1..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$4])):
seq(T(n), n=0..14); # Alois P. Heinz, Dec 22 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 ;; 5]]]*Binomial[n - 1, j - 1], {j, 1, n}]];
T[n_] := With[{p = b[n, {0, 0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 29 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Steven Finch, Dec 22 2021
EXTENSIONS
More terms from Alois P. Heinz, Dec 22 2021
STATUS
approved