A056665 Number of equivalence classes of n-valued Post functions of 1 variable under action of complementing group C(1,n).

1, 3, 11, 70, 629, 7826, 117655, 2097684, 43046889, 1000010044, 25937424611, 743008623292, 23298085122493, 793714780783770, 29192926025492783, 1152921504875290696, 48661191875666868497, 2185911559749720272442, 104127350297911241532859

Offset: 1

Author

Vladeta Jovovic, Aug 09 2000

Keywords

• easy
• nonn

Comments

Diagonal of arrays defined in A054630 and A054631.

Given n colors, a(n) = number of necklaces with n beads and 1 up to n colors effectively assigned to them (super_labeled: which also generates n different monochrome necklaces). - Wouter Meeussen, Aug 09 2002

Number of endofunctions on a set with n objects up to cyclic permutation (rotation). E.g., for n = 3, the 11 endofunctions are 1,1,1; 2,2,2; 3,3,3; 1,1,2; 1,2,2; 1,1,3; 1,3,3; 2,2,3; 2,3,3; 1,2,3; and 1,3,2. - Franklin T. Adams-Watters, Jan 17 2007

Also number of pre-necklaces in Sigma(n,n) (see Ruskey and others). - Peter Luschny, Aug 12 2012

From Olivier Gérard, Aug 01 2016: (Start)

Decomposition of the endofunctions by class size.

.

n | 1 2 3 4 5 6 7

--+----------------------------------

1 | 1

2 | 2 1

3 | 3 0 8

4 | 4 6 0 60

5 | 5 0 0 0 624

6 | 6 15 70 0 0 7735

7 | 7 0 0 0 0 0 117648

.

The right diagonal gives the number of Lyndon Words or aperiodic necklaces, A075147. By multiplying each column by the corresponding size and summing, one gets A000312.

(End)

Examples

			The 11 necklaces for n=3 are (grouped by partition of 3): (RRR,GGG,BBB),(RRG,RGG, RRB,RBB, GGB,GBB), (RGB,RBG).
		

References

• D. E. Knuth. Generating All Tuples and Permutations. The Art of Computer Programming, Vol. 4, Fascicle 2, 7.2.1.1. Addison-Wesley, 2005.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200
• M. A. Harrison and R. G. High, On the cycle index of a product of permutation groups, J. Combin. Theory, 4 (1968), 277-299.
• F. Ruskey, C. Savage, and T. M. Y. Wang, Generating necklaces, Journal of Algorithms, 13(3), 414 - 430, 1992.
• Index entries for sequences related to groups

Crossrefs

Cf. A001323, A001324, A001372, A000312.

Diagonal of arrays defined in A054630, A054631 and A075195.

Cf. A075147 Aperiodic necklaces, a subset of this sequence.

Cf. A000169 Classes under translation mod n

Cf. A168658 Classes under complement to n+1

Cf. A130293 Classes under translation and rotation

Cf. A081721 Classes under rotation and reversal

Cf. A275549 Classes under reversal

Cf. A275550 Classes under reversal and complement

Cf. A275551 Classes under translation and reversal

Cf. A275552 Classes under translation and complement

Cf. A275553 Classes under translation, complement and reversal

Cf. A275554 Classes under translation, rotation and complement

Cf. A275555 Classes under translation, rotation and reversal

Cf. A275556 Classes under translation, rotation, complement and reversal

Cf. A275557 Classes under rotation and complement

Cf. A275558 Classes under rotation, complement and reversal

Cf. A228640.

Programs

Maple

with(numtheory):
a:= n-> add(phi(d)*n^(n/d), d=divisors(n))/n:
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 18 2013

Mathematica

Table[Fold[ #1+EulerPhi[ #2] n^(n/#2)&, 0, Divisors[n]]/n, {n, 7}]

PARI

a(n) = sum(k=1,n,n^gcd(k,n)) / n; \\ Joerg Arndt, Mar 19 2017

Sage

# This algorithm counts all n-ary n-tuples (a_1,..,a_n) such that the string a_1...a_n is preprime. It is algorithm F in Knuth 7.2.1.1.
def A056665_list(n):
    C = []
    for m in (1..n):
        a = [0]*(n+1); a[0]=-1;
        j = 1; count = 0
        while(true):
            if m%j == 0 : count += 1;
            j = n
            while a[j] >= m-1 : j -= 1
            if j == 0 : break
            a[j] += 1
            for k in (j+1..n): a[k] = a[k-j]
        C.append(count)
    return C

Sage

def A056665(n): return sum(euler_phi(d)*n^(n//d)//n for d in divisors(n))
[A056665(n) for n in (1..18)] # Peter Luschny, Aug 12 2012

Formula

a(n) = Sum_{d|n} phi(d)*n^(n/d)/n.

a(n) ~ n^(n-1). - Vaclav Kotesovec, Sep 11 2014

a(n) = (1/n) * Sum_{k=1..n} n^gcd(k,n). - Joerg Arndt, Mar 19 2017

a(n) = [x^n] -Sum_{k>=1} phi(k)*log(1 - n*x^k)/k. - Ilya Gutkovskiy, Mar 21 2018

From Richard L. Ollerton, May 07 2021: (Start)

a(n) = (1/n)*Sum_{k=1..n} n^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)).

a(n) = (1/n)*A228640(n). (End)