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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).

%I A350273 #20 Feb 17 2022 13:48:36
%S A350273 1,1,2,6,23,1,109,11,619,101,4108,932,31240,8975,105,268028,91387,
%T A350273 3465,2562156,991674,74970,27011016,11514394,1391390,311378616,
%U A350273 143188574,24188010,246400,3897004032,1905067958,412136010,12812800,52626496896,27059601596,7053834788,438357920
%N 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).
%C A350273 If the permutation has no fourth cycle, then its fourth-longest cycle is defined to have length 0.
%H A350273 Alois P. Heinz, <a href="/A350273/b350273.txt">Rows n = 0..100, flattened</a>
%H A350273 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 A350273 Sum_{k=0..floor(n/4)} k * T(n,k) = A332853(n) for n >= 4.
%e A350273 Triangle begins:
%e A350273 [0]      1;
%e A350273 [1]      1;
%e A350273 [2]      2;
%e A350273 [3]      6;
%e A350273 [4]     23,     1;
%e A350273 [5]    109,    11;
%e A350273 [6]    619,   101;
%e A350273 [7]   4108,   932;
%e A350273 [8]  31240,  8975,  105;
%e A350273 [9] 268028, 91387, 3465;
%e A350273     ...
%p A350273 b:= proc(n, l) option remember; `if`(n=0, x^l[1], add((j-1)!*
%p A350273       b(n-j, sort([l[], j])[2..5])*binomial(n-1, j-1), j=1..n))
%p A350273     end:
%p A350273 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$4])):
%p A350273 seq(T(n), n=0..14);  # _Alois P. Heinz_, Dec 22 2021
%t A350273 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 A350273 T[n_] := With[{p = b[n, {0, 0, 0, 0}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
%t A350273 Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 29 2021, after _Alois P. Heinz_ *)
%Y A350273 Row sums give A000142(n).
%Y A350273 Cf. A126074, A145877, A332853, A349979, A349980, A350015, A350016, A350274.
%K A350273 nonn,tabf
%O A350273 0,3
%A A350273 _Steven Finch_, Dec 22 2021
%E A350273 More terms from _Alois P. Heinz_, Dec 22 2021