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A349980 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose second-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-1).

%I A349980 #29 Feb 17 2022 13:19:40
%S A349980 1,1,1,1,2,1,3,6,7,3,8,24,31,15,20,30,120,191,135,40,90,144,720,1331,
%T A349980 945,280,420,504,840,5040,10655,7077,4480,1260,2688,3360,5760,40320,
%U A349980 95887,64197,41552,11340,18144,20160,25920,45360,362880,958879,646965,395360,238140,72576,151200,172800,226800,403200
%N A349980 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose second-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-1).
%C A349980 If the permutation has no second cycle, then its second-longest cycle is defined to have length 0.
%H A349980 Alois P. Heinz, <a href="/A349980/b349980.txt">Rows n = 0..141, flattened</a>
%H A349980 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 A349980 Sum_{k=0..max(0,n-1)} k * T(n,k) = A332906(n). - _Alois P. Heinz_, Dec 07 2021
%e A349980 Triangle begins:
%e A349980 [0]     1;
%e A349980 [1]     1;
%e A349980 [2]     1,     1;
%e A349980 [3]     2,     1,     3;
%e A349980 [4]     6,     7,     3,     8;
%e A349980 [5]    24,    31,    15,    20,    30;
%e A349980 [6]   120,   191,   135,    40,    90,   144;
%e A349980 [7]   720,  1331,   945,   280,   420,   504,   840;
%e A349980 [8]  5040, 10655,  7077,  4480,  1260,  2688,  3360,  5760;
%e A349980 [9] 40320, 95887, 64197, 41552, 11340, 18144, 20160, 25920, 45360;
%e A349980     ...
%p A349980 m:= infinity:
%p A349980 b:= proc(n, l) option remember; `if`(n=0, x^`if`(l[2]=m,
%p A349980       0, l[2]), add(b(n-j, sort([l[], j])[1..2])
%p A349980                *binomial(n-1, j-1)*(j-1)!, j=1..n))
%p A349980     end:
%p A349980 T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, n-1)))(b(n, [m$2])):
%p A349980 seq(T(n), n=0..10);  # _Alois P. Heinz_, Dec 07 2021
%t A349980 m = Infinity;
%t A349980 b[n_, l_] := b[n, l] = If[n == 0, x^If[l[[2]] == m, 0, l[[2]]], Sum[b[n-j, Sort[Append[l, j]][[1;;2]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]];
%t A349980 T[n_] := With[{p = b[n, {m, m}]}, Table[Coefficient[p, x, i], {i, 0, Max[0, n - 1]}]];
%t A349980 Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Dec 28 2021, after _Alois P. Heinz_ *)
%Y A349980 Column 0 gives 1 together with A000142.
%Y A349980 Column 1 gives the nonzero terms of A155521.
%Y A349980 Row sums give A000142.
%Y A349980 T(n,n-1) gives A059171(n) for n>=1.
%Y A349980 Cf. A126074, A145877, A332906, A349979, A350015, A350016, A350273, A350274.
%K A349980 nonn,tabf
%O A349980 0,5
%A A349980 _Steven Finch_, Dec 07 2021
%E A349980 More terms from _Alois P. Heinz_, Dec 07 2021