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A185105 Number T(n,k) of entries in the k-th cycles of all permutations of {1,2,..,n}; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

%I A185105 #39 Jun 02 2021 05:16:21
%S A185105 1,3,1,12,5,1,60,27,8,1,360,168,59,12,1,2520,1200,463,119,17,1,20160,
%T A185105 9720,3978,1177,221,23,1,181440,88200,37566,12217,2724,382,30,1,
%U A185105 1814400,887040,388728,135302,34009,5780,622,38,1,19958400,9797760,4385592,1606446,441383,86029,11378,964,47,1
%N A185105 Number T(n,k) of entries in the k-th cycles of all permutations of {1,2,..,n}; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
%C A185105 Row sums are n!*n = A001563(n) (see example).
%C A185105 For fixed k>=1, A185105(n,k) ~ n!*n/2^k. - _Vaclav Kotesovec_, Apr 25 2017
%H A185105 Alois P. Heinz, <a href="/A185105/b185105.txt">Rows n = 1..141, flattened</a>
%H A185105 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%e A185105 The six permutations of n=3 in ordered cycle form are:
%e A185105 { {1}, {2}, {3}    }
%e A185105 { {1}, {2, 3}, {}  }
%e A185105 { {1, 2}, {3}, {}  }
%e A185105 { {1, 2, 3}, {}, {}}
%e A185105 { {1, 3, 2}, {}, {}}
%e A185105 { {1, 3}, {2}, {}  }
%e A185105 .
%e A185105 The lengths of the cycles in position k=1 sum to 12, those of the cycles in position k=2 sum to 5 and those of the cycles in position k=3 sum to 1.
%e A185105 Triangle begins:
%e A185105        1;
%e A185105        3,     1;
%e A185105       12,     5,     1;
%e A185105       60,    27,     8,     1;
%e A185105      360,   168,    59,    12,    1;
%e A185105     2520,  1200,   463,   119,   17,   1;
%e A185105    20160,  9720,  3978,  1177,  221,  23,  1;
%e A185105   181440, 88200, 37566, 12217, 2724, 382, 30, 1;
%e A185105   ...
%p A185105 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A185105       add((p-> p+coeff(p, x, 0)*j*x^i)(b(n-j, i+1))*
%p A185105        binomial(n-1, j-1)*(j-1)!, j=1..n)))
%p A185105     end:
%p A185105 T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 1)):
%p A185105 seq(T(n), n=1..12);  # _Alois P. Heinz_, Apr 15 2017
%t A185105 Table[it = Join[RotateRight /@ ToCycles[#], Table[{}, {k}]] & /@ Permutations[Range[n]]; Tr[Length[Part[#, k]]& /@ it], {n, 7}, {k, n}]
%t A185105 (* Second program: *)
%t A185105 b[n_, i_] := b[n, i] = Expand[If[n==0, 1, Sum[Function[p, p + Coefficient[ p, x, 0]*j*x^i][b[n-j, i+1]]*Binomial[n-1, j-1]*(j-1)!, {j, 1, n}]]];
%t A185105 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 1]];
%t A185105 Array[T, 12] // Flatten (* _Jean-François Alcover_, May 30 2018, after _Alois P. Heinz_ *)
%Y A185105 Columns k=1-10 give: A001710(n+1), A138772, A159324(n-1)/2 or A285231, A285232, A285233, A285234, A285235, A285236, A285237, A285238.
%Y A185105 T(2n,n) gives A285239.
%Y A185105 Cf. A000142, A001563, A270236, A284816, A285439.
%K A185105 nonn,tabl
%O A185105 1,2
%A A185105 _Wouter Meeussen_, Dec 26 2012
%E A185105 More terms from _Alois P. Heinz_, Apr 15 2017