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A056665
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Number of equivalence classes of n-valued Post functions of 1 variable under action of complementing group C(1,n).
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21
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1, 3, 11, 70, 629, 7826, 117655, 2097684, 43046889, 1000010044, 25937424611, 743008623292, 23298085122493, 793714780783770, 29192926025492783, 1152921504875290696, 48661191875666868497, 2185911559749720272442, 104127350297911241532859
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OFFSET
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1,2
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COMMENTS
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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; 1,1,2; 1,1,3; 1,2,1; 1,2,3; 1,3,1; 1,3,2; 2,1,1; 2,1,2; 2,3,1; and 3,1,2. - 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.
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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)
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REFERENCES
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D. E. Knuth. Generating All Tuples and Permutations. The Art of Computer Programming, Vol. 4, Fascicle 2, 7.2.1.1. Addison-Wesley, 2005.
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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The 11 necklaces for n=3 are (grouped by partition of 3): (RRR,GGG,BBB),(RRG,RGG, RRB,RBB, GGB,GBB), (RGB,RBG).
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MAPLE
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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
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MATHEMATICA
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Table[Fold[ #1+EulerPhi[ #2] n^(n/#2)&, 0, Divisors[n]]/n, {n, 7}]
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PROG
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(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
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
(PARI) a(n) = sum(k=1, n, n^gcd(k, n)) / n; \\ Joerg Arndt, Mar 19 2017
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CROSSREFS
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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
Sequence in context: A009025 A009103 A018192 * A127716 A035378 A258387
Adjacent sequences: A056662 A056663 A056664 * A056666 A056667 A056668
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KEYWORD
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easy,nonn
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AUTHOR
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Vladeta Jovovic, Aug 09 2000
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STATUS
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approved
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