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A350274 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).

%I A350274 #29 Feb 17 2022 13:20:06
%S A350274 1,1,2,6,23,1,109,1,10,619,16,45,40,4108,92,210,420,210,31240,771,
%T A350274 1645,2800,2520,1344,268028,6883,17325,15960,26460,18144,10080,
%U A350274 2562156,68914,173250,148400,226800,211680,151200,86400,27011016,757934,1854930,1798720,1801800,2494800,1940400,1425600,831600
%N A350274 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 A350274 If the permutation has no fourth cycle, then its fourth-longest cycle is defined to have length 0.
%H A350274 Alois P. Heinz, <a href="/A350274/b350274.txt">Rows n = 0..100, flattened</a>
%H A350274 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 A350274 Sum_{k=0..n-3} k * T(n,k) = A332908(n) for n >= 4.
%e A350274 Triangle begins:
%e A350274 [0]      1;
%e A350274 [1]      1;
%e A350274 [2]      2;
%e A350274 [3]      6;
%e A350274 [4]     23,    1;
%e A350274 [5]    109,    1,    10;
%e A350274 [6]    619,   16,    45,    40;
%e A350274 [7]   4108,   92,   210,   420,   210;
%e A350274 [8]  31240,  771,  1645,  2800,  2520,  1344;
%e A350274 [9] 268028, 6883, 17325, 15960, 26460, 18144, 10080;
%e A350274     ...
%p A350274 m:= infinity:
%p A350274 b:= proc(n, l) option remember; `if`(n=0, x^`if`(l[4]=m,
%p A350274       0, l[4]), add(b(n-j, sort([l[], j])[1..4])
%p A350274                *binomial(n-1, j-1)*(j-1)!, j=1..n))
%p A350274     end:
%p A350274 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [m$4])):
%p A350274 seq(T(n), n=0..11);  # _Alois P. Heinz_, Dec 22 2021
%t A350274 m = Infinity;
%t A350274 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 A350274 T[n_] := With[{p = b[n, {m, m, m, m}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
%t A350274 Table[T[n], {n, 0, 11}] // Flatten (* _Jean-François Alcover_, Dec 29 2021, after _Alois P. Heinz_ *)
%Y A350274 Column 0 is 1 for n=0, together with A000142(n) - A122105(n-1) for n>=1.
%Y A350274 Row sums give A000142.
%Y A350274 Cf. A126074, A145877, A332908, A349979, A349980, A350015, A350016, A350273.
%K A350274 nonn,tabf
%O A350274 0,3
%A A350274 _Steven Finch_, Dec 22 2021
%E A350274 More terms from _Alois P. Heinz_, Dec 22 2021