%I #31 May 09 2018 09:56:57

%S 1,1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,1,3,2,2,1,2,1,3,2,2,1,3,1,1,1,1,4,

%T 2,4,1,5,1,1,1,1,1,1,4,3,4,1,7,1,1,1,2,2,1,1,1,1,5,3,6,2,9,1,2,1,3,4,

%U 1,1,1,1,1,1,5,3,6,2,12,1,2,1,4,1,6,2,2,1,2,1,1,1,2

%N Number T(n,k) of cycle types of degree-n permutations having the k-th smallest possible order; triangle T(n,k), n>=0, 1<=k<=A009490(n), read by rows.

%C Sum_{k>=0} A256553(n,k)*T(n,k) = A181844(n).

%H Alois P. Heinz, <a href="/A256554/b256554.txt">Rows n = 0..60, flattened</a>

%e Triangle T(n,k) begins:

%e 1;

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 2, 1, 1;

%e 1, 2, 1, 1, 1, 1;

%e 1, 3, 2, 2, 1, 2;

%e 1, 3, 2, 2, 1, 3, 1, 1, 1;

%e 1, 4, 2, 4, 1, 5, 1, 1, 1, 1, 1;

%e 1, 4, 3, 4, 1, 7, 1, 1, 1, 2, 2, 1, 1, 1;

%e 1, 5, 3, 6, 2, 9, 1, 2, 1, 3, 4, 1, 1, 1, 1, 1;

%p b:= proc(n, i) option remember; `if`(n=0 or i=1, x,

%p b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),

%p t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))

%p end:

%p T:= n->(p->seq((h->`if`(h=0, [][], h))(coeff(p, x, i))

%p , i=1..degree(p)))(b(n$2)):

%p seq(T(n), n=0..12);

%t b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x, b[n, i - 1] + Function[p, Sum[Coefficient[p, x, t]*x^LCM[t, i], {t, 1, Exponent[p, x]}]][Sum[b[n - i*j, i - 1], {j, 1, n/i}]]]; T[n_] := Function[p, Table[Function[h, If[h == 0, {{}, {}}, h]][Coefficient[p, x, i]], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jan 23 2017, translated from Maple *)

%Y Row sums give A000041.

%Y Row lengths give A009490.

%Y Columns k=1-9 give: A000012, A004526, A002264, A008642(n-4), A002266, A074752, A132270, A008643(n-8) for n>7, A008649(n-9) for n>8.

%Y Last elements of rows give A074064.

%Y Main diagonal gives A074761.

%Y Cf. A181844, A256067, A256553.

%K nonn,look,tabf

%O 0,9

%A _Alois P. Heinz_, Apr 01 2015