A350273 - OEIS

%I #20 Feb 17 2022 13:48:36

%S 1,1,2,6,23,1,109,11,619,101,4108,932,31240,8975,105,268028,91387,

%T 3465,2562156,991674,74970,27011016,11514394,1391390,311378616,

%U 143188574,24188010,246400,3897004032,1905067958,412136010,12812800,52626496896,27059601596,7053834788,438357920

%N 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).

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

%H Alois P. Heinz, <a href="/A350273/b350273.txt">Rows n = 0..100, flattened</a>

%H Steven Finch, <a href="http://arxiv.org/abs/2202.07621">Second best, Third worst, Fourth in line</a>, arxiv:2202.07621 [math.CO], 2022.

%F Sum_{k=0..floor(n/4)} k * T(n,k) = A332853(n) for n >= 4.

%e Triangle begins:

%e [0]      1;

%e [1]      1;

%e [2]      2;

%e [3]      6;

%e [4]     23,     1;

%e [5]    109,    11;

%e [6]    619,   101;

%e [7]   4108,   932;

%e [8]  31240,  8975,  105;

%e [9] 268028, 91387, 3465;

%e     ...

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

%p       b(n-j, sort([l[], j])[2..5])*binomial(n-1, j-1), j=1..n))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$4])):

%p seq(T(n), n=0..14);  # _Alois P. Heinz_, Dec 22 2021

%t 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 T[n_] := With[{p = b[n, {0, 0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];

%t Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 29 2021, after _Alois P. Heinz_ *)

%Y Row sums give A000142(n).

%Y Cf. A126074, A145877, A332853, A349979, A349980, A350015, A350016, A350274.

%K nonn,tabf

%O 0,3

%A _Steven Finch_, Dec 22 2021

%E More terms from _Alois P. Heinz_, Dec 22 2021