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A350016 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose third-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-2).

%I A350016 #32 Feb 17 2022 13:20:54
%S A350016 1,1,2,5,1,17,1,6,74,11,15,20,394,56,60,120,90,2484,407,525,490,630,
%T A350016 504,18108,3235,4725,2240,4620,4032,3360,149904,29143,40509,27440,
%U A350016 26460,33264,30240,25920,1389456,291394,398790,319760,163800,302400,277200,259200,226800
%N A350016 Irregular triangle read by rows: T(n,k) is the number of n-permutations whose third-shortest cycle has length exactly k; n >= 0, 0 <= k <= max(0,n-2).
%C A350016 If the permutation has no third cycle, then its third-longest cycle is defined to have length 0.
%H A350016 Alois P. Heinz, <a href="/A350016/b350016.txt">Rows n = 0..142, flattened</a>
%H A350016 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 A350016 Sum_{k=0..n-2} k * T(n,k) = A332907(n) for n >= 3. - _Alois P. Heinz_, Dec 12 2021
%e A350016 Triangle begins:
%e A350016 [0]      1;
%e A350016 [1]      1;
%e A350016 [2]      2;
%e A350016 [3]      5,     1;
%e A350016 [4]     17,     1,     6;
%e A350016 [5]     74,    11,    15,    20;
%e A350016 [6]    394,    56,    60,   120,    90;
%e A350016 [7]   2484,   407,   525,   490,   630,   504;
%e A350016 [8]  18108,  3235,  4725,  2240,  4620,  4032,  3360;
%e A350016 [9] 149904, 29143, 40509, 27440, 26460, 33264, 30240, 25920;
%e A350016 ...
%p A350016 m:= infinity:
%p A350016 b:= proc(n, l) option remember; `if`(n=0, x^`if`(l[3]=m,
%p A350016       0, l[3]), add(b(n-j, sort([l[], j])[1..3])
%p A350016                *binomial(n-1, j-1)*(j-1)!, j=1..n))
%p A350016     end:
%p A350016 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [m$3])):
%p A350016 seq(T(n), n=0..10);  # _Alois P. Heinz_, Dec 11 2021
%t A350016 m = Infinity;
%t A350016 b[n_, l_] := b[n, l] = If[n == 0, x^If[l[[3]] == m, 0, l[[3]]], Sum[b[n-j, Sort[Append[l, j]][[1;;3]]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]];
%t A350016 T[n_] := With[{p = b[n, {m, m, m}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
%t A350016 Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Dec 28 2021, after _Alois P. Heinz_ *)
%Y A350016 Column 0 gives 1 together with A000774.
%Y A350016 Column 1 gives the column 3 of A208956.
%Y A350016 Row sums give A000142.
%Y A350016 Cf. A126074, A145877, A332907, A349979, A349980, A350015, A350273, A350274.
%K A350016 nonn,tabf
%O A350016 0,3
%A A350016 _Steven Finch_, Dec 08 2021