A074064 Number of cycle types of degree-n permutations having the maximum possible order.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4

Offset

0,7

Links

Alois P. Heinz, Table of n, a(n) for n = 0..180

Formula

Coefficient of x^n in expansion of Sum_{i divides A000793(n)} mu(A000793(n)/i)*1/Product_{j divides i} (1-x^j).

Example

For n = 22 we have 4 such cycle types: [1, 1, 1, 3, 4, 5, 7], [1, 2, 3, 4, 5, 7], [3, 3, 4, 5, 7], [4, 5, 6, 7].

Maple

A000793 := proc(n) option remember; local l, p, i ; l := 1: p := combinat[partition](n): for i from 1 to combinat[numbpart](n) do if ilcm( p[i][j] $ j=1..nops(p[i])) > l then l := ilcm( p[i][j] $ j=1..nops(p[i])) ; fi: od: RETURN(l) ; end proc:
taylInv := proc(i, n) local resul, j, idiv, k ; resul := 1 ; idiv := numtheory[divisors](i) ; for k from 1 to nops(idiv) do j := op(k, idiv) ; resul := resul*taylor(1/(1-x^j), x=0, n+1) ; resul := convert(taylor(resul, x=0, n+1), polynom) ; od ; coeftayl(resul, x=0, n) ; end proc:
A074064 := proc(n) local resul, a793, dvs, i, k ; resul := 0: a793 := A000793(n) ; dvs := numtheory[divisors](a793) ; for k from 1 to nops(dvs) do i := op(k, dvs) ; resul := resul+numtheory[mobius](a793/i)*taylInv(i, n) ; od : RETURN(resul) ; end proc: # R. J. Mathar, Mar 30 2007

Mathematica

b[n_, i_] := b[n, i] = Module[{p}, p = If[i < 1, 1, Prime[i]]; If[n == 0 || i < 1, 1, Max[b[n, i-1], Table[p^j*b[n-p^j, i-1], {j, 1, Log[p, n] // Floor}]]]];
g[n_] := g[n] = b[n, If[n < 8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]];
a[n_] := a[n] = SeriesCoefficient[Sum[MoebiusMu[g[n]/i]/Product[1-x^j, {j, Divisors[i]}], {i, Divisors[g[n]]}] + O[x]^(n+1), n];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 25 2017, after Alois P. Heinz *)

Crossrefs

Cf. A000793, A074859, A256067, A256554.

Sequence in context: A334184 A249545 A296081 * A275215 A304886 A352080

Adjacent sequences: A074061 A074062 A074063 * A074065 A074066 A074067

Keyword

easy,nonn

Author

Vladeta Jovovic, Sep 15 2002

Extensions

More terms from R. J. Mathar, Mar 30 2007

More terms from Sean A. Irvine, Oct 04 2011

More terms from Alois P. Heinz, Mar 29 2015

Status

approved