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%I #29 Feb 17 2022 13:20:06
%S 1,1,2,6,23,1,109,1,10,619,16,45,40,4108,92,210,420,210,31240,771,
%T 1645,2800,2520,1344,268028,6883,17325,15960,26460,18144,10080,
%U 2562156,68914,173250,148400,226800,211680,151200,86400,27011016,757934,1854930,1798720,1801800,2494800,1940400,1425600,831600
%N Triangle read by rows: T(n,k) is the number of n-permutations whose fourth-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-3).
%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="/A350274/b350274.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..n-3} k * T(n,k) = A332908(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, 1, 10;
%e [6] 619, 16, 45, 40;
%e [7] 4108, 92, 210, 420, 210;
%e [8] 31240, 771, 1645, 2800, 2520, 1344;
%e [9] 268028, 6883, 17325, 15960, 26460, 18144, 10080;
%e ...
%p m:= infinity:
%p b:= proc(n, l) option remember; `if`(n=0, x^`if`(l[4]=m,
%p 0, l[4]), add(b(n-j, sort([l[], j])[1..4])
%p *binomial(n-1, j-1)*(j-1)!, j=1..n))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [m$4])):
%p seq(T(n), n=0..11); # _Alois P. Heinz_, Dec 22 2021
%t m = Infinity;
%t b[n_, l_] := b[n, l] = If[n == 0, x^If[l[[4]] == m, 0, l[[4]]], Sum[b[n-j, Sort[Append[l, j]][[1 ;; 4]]]*Binomial[n-1, j-1]*(j-1)!, {j, 1, n}]];
%t T[n_] := With[{p = b[n, {m, m, m, m}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
%t Table[T[n], {n, 0, 11}] // Flatten (* _Jean-François Alcover_, Dec 29 2021, after _Alois P. Heinz_ *)
%Y Column 0 is 1 for n=0, together with A000142(n) - A122105(n-1) for n>=1.
%Y Row sums give A000142.
%Y Cf. A126074, A145877, A332908, A349979, A349980, A350015, A350016, A350273.
%K nonn,tabf
%O 0,3
%A _Steven Finch_, Dec 22 2021
%E More terms from _Alois P. Heinz_, Dec 22 2021
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