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A079128 Number of degree-n permutations with (mutually) relatively prime cycle lengths. 4
1, 1, 4, 15, 96, 455, 4320, 29295, 300160, 2663199, 36288000, 348523175, 5748019200, 68027248575, 1116542242816, 16813959537375, 334764638208000, 4954072089341375, 115242726703104000, 1966765155600364119, 45415699475660800000, 930312555383281809375 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(p) = p!-(p-1)! for prime p. Conjecture: a(n) is divisible by n^2-1 for n>3.

Conjecture: gcd(a(n),n)=1. - Vladeta Jovovic, Jan 25 2003

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..450

MAPLE

with(combinat):

b:= proc(n, i, g) option remember; `if`(n=0, `if`(g>=2, 1, 0),

      `if`(i<2, 0, b(n, i-1, g) +`if`(igcd(g, i)<2, 0,

       add((i-1)!^j/j! *multinomial(n, i$j, n-i*j)*

         b(n-i*j, i-1, igcd(i, g)), j=1..n/i))))

    end:

a:= n-> n!-b(n, n, 0):

seq(a(n), n=1..25);  # Alois P. Heinz, Jun 06 2013

MATHEMATICA

f[list_] :=

Total[list]!/Apply[Times, Table[list[[i]], {i, 1, Length[list]}]]/

  Apply[Times, Select[Table[Count[list, i], {i, 1, Total[list]}], # > 0 &]!];

Table[Total[Map[f, Select[IntegerPartitions[n], Apply[GCD, #] == 1 &]]], {n, 1, 25}] (* Geoffrey Critzer, Jun 06 2013 *)

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, g_] := b[n, i, g] = If[n==0, If[g >= 2, 1, 0], If[i<2, 0, b[n, i-1, g] + If[GCD[g, i]<2, 0, Sum[(i-1)!^j/j!*multinomial[n, Append[Array[i&, j], n-i*j]]*b[n-i*j, i-1, GCD[i, g]], {j, 1, n/i}]]]]; a[n_] := n! - b[n, n, 0]; Table[a[n], {n, 1, 25}] (* Jean-Fran├žois Alcover, Jan 08 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A079129, A000837, A079129, A226388.

Sequence in context: A040025 A208991 A109365 * A289489 A221095 A081548

Adjacent sequences:  A079125 A079126 A079127 * A079129 A079130 A079131

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Vladimir Baltic, Dec 27 2002

STATUS

approved

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Last modified August 14 09:14 EDT 2020. Contains 336480 sequences. (Running on oeis4.)