A256554 - OEIS

A256554

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.

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, 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, 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

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OFFSET

0,9

COMMENTS

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

LINKS

Alois P. Heinz, Rows n = 0..60, flattened

EXAMPLE

Triangle T(n,k) begins:

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, 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, 1, 1, 1, 1, 1;

MAPLE

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

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

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

end:

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

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

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

MATHEMATICA

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 *)

CROSSREFS

Row sums give A000041.

Row lengths give A009490.

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.

Last elements of rows give A074064.

Main diagonal gives A074761.

Cf. A181844, A256067, A256553.

Sequence in context: A287170 A216784 A256067 * A321649 A003650 A059233

Adjacent sequences: A256551 A256552 A256553 * A256555 A256556 A256557

KEYWORD

nonn,look,tabf

AUTHOR

Alois P. Heinz, Apr 01 2015

STATUS

approved