A320748 -id:A320748 - cp's OEIS frontend

A000011 Number of n-bead necklaces (turning over is allowed) where complements are equivalent.

1, 1, 2, 2, 4, 4, 8, 9, 18, 23, 44, 63, 122, 190, 362, 612, 1162, 2056, 3914, 7155, 13648, 25482, 48734, 92205, 176906, 337594, 649532, 1246863, 2405236, 4636390, 8964800, 17334801, 33588234, 65108062, 126390032, 245492244, 477353376, 928772650, 1808676326, 3524337980

Offset: 0

N. J. A. Sloane

a(n) is also the number of minimal fibrations of a bidirectional n-cycle over the 2-bouquet up to precompositions with automorphisms of the n-cycle and postcomposition with automorphisms of the 2-bouquet. (Boldi et al.) - Sebastiano Vigna, Jan 08 2018

For n >= 3, also the number of distinct planar embeddings of the n-sunlet graph. - Eric W. Weisstein, May 21 2024

			From Jason Orendorff (jason.orendorff(AT)gmail.com), Jan 09 2009: (Start)
The binary bracelets for small n are:
  n: bracelets
  0: (the empty bracelet)
  1: 0
  2: 00, 01
  3: 000, 001
  4: 0000, 0001, 0011, 0101
  5: 00000, 00001, 00011, 00101
  6: 000000, 000001, 000011, 000101, 000111, 001001, 001011, 010101
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..3335 (first 201 terms from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook)
Paolo Boldi and Sebastiano Vigna, Fibrations of Graphs, Discrete Math., 243 (2002), 21-66.
H. Bottomley, Initial terms of A000011 and A000013
Aharon Davidson, From Planck Area to Graph Theory: Topologically Distinct Black Hole Microstates, arXiv:1907.03090 [gr-qc], 2019.
Daniel T. Eatough and Keith A. Seffen, Calculating the Fold Angles of Any Vertex Roof Using a Spherical Image Technique, J. Mechanisms Robotics (2020) Vol. 12, No. 3, 031004.
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
W. D. Hoskins and Anne Penfold Street, Twills on a given number of harnesses, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
W. D. Hoskins and A. P. Street, Twills on a given number of harnesses, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)
Yi Hu, Numerical Transfer Matrix Method of Next-nearest-neighbor Ising Models, Master's Thesis, Duke Univ. (2021).
Yi Hu and Patrick Charbonneau, Numerical transfer matrix study of frustrated next-nearest-neighbor Ising models on square lattices, arXiv:2106.08442 [cond-mat.stat-mech], 2021.
Karyn McLellan, Periodic coefficients and random Fibonacci sequences, Electronic Journal of Combinatorics, 20(4), 2013, #P32.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
A. P. Street, Letter to N. J. A. Sloane, N.D.
Zhe Sun, T. Suenaga, P. Sarkar, S. Sato, M. Kotani, and H. Isobe, Stereoisomerism, crystal structures, and dynamics of belt-shaped cyclonaphthylenes, Proc. Nat. Acad. Sci. USA, vol. 113 no. 29, pp. 8109-8114, doi: 10.1073/pnas.1606530113.
Eric Weisstein's World of Mathematics, Planar Embedding.
Eric Weisstein's World of Mathematics, Sunlet Graph.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264; doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text).
Index entries for sequences related to necklaces
Index entries for sequences related to bracelets

Column 2 of A320748.
Cf. A000013. Bisections give A000117 and A092668.
The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.

with(numtheory): A000011 := proc(n) local s,d; if n = 0 then RETURN(1) else s := 2^(floor(n/2)); for d in divisors(n) do s := s+(phi(2*d)*2^(n/d))/(2*n); od; RETURN(s/2); fi; end;

a[n_] := Fold[ #1 + EulerPhi[2#2]2^(n/#2)/(2n) &, 2^Floor[n/2], Divisors[n]]/2
a[ n_] := If[ n < 1, Boole[n == 0], 2^Quotient[n, 2] / 2 + DivisorSum[ n, EulerPhi[2 #] 2^(n/#) &] / (4 n)]; (* Michael Somos, Dec 19 2014 *)

{a(n) = if( n<1, n==0, 2^(n\2) / 2 + sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (4*n))}; /* Michael Somos, Jun 03 2002 */

a(n) = (A000013(n) + 2^floor(n/2))/2.

Better description from Christian G. Bower

More terms from David W. Wilson, Jan 13 2000

A320742 Array read by antidiagonals: T(n,k) is the number of chiral pairs of color patterns (set partitions) in a cycle of length n using k or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 6, 13, 2, 0, 0, 0, 0, 0, 0, 6, 30, 46, 7, 0, 0, 0, 0, 0, 0, 6, 34, 130, 144, 12, 0, 0, 0, 0, 0, 0, 6, 34, 181, 532, 420, 31, 0, 0, 0, 0, 0, 0, 6, 34, 190, 871, 2006, 1221, 58, 0, 0, 0, 0, 0, 0, 6, 34, 190, 996, 4016, 7626, 3474, 126, 0, 0, 0, 0, 0, 0, 6, 34, 190, 1011, 5070, 18526, 28401, 9856, 234, 0

Offset: 1

Robert A. Russell, Oct 21 2018

Two color patterns are equivalent if the colors are permuted.

Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.

			Array begins with T(1,1):
0  0    0     0     0      0      0      0      0      0      0      0 ...
0  0    0     0     0      0      0      0      0      0      0      0 ...
0  0    0     0     0      0      0      0      0      0      0      0 ...
0  0    0     0     0      0      0      0      0      0      0      0 ...
0  0    0     0     0      0      0      0      0      0      0      0 ...
0  0    4     6     6      6      6      6      6      6      6      6 ...
0  1   13    30    34     34     34     34     34     34     34     34 ...
0  2   46   130   181    190    190    190    190    190    190    190 ...
0  7  144   532   871    996   1011   1011   1011   1011   1011   1011 ...
0 12  420  2006  4016   5070   5328   5352   5352   5352   5352   5352 ...
0 31 1221  7626 18526  26454  29215  29705  29740  29740  29740  29740 ...
0 58 3474 28401 85101 139484 165164 171556 172415 172466 172466 172466 ...
For T(6,4)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD and AABCBD-AABCDC.

Andrew Howroyd, Table of n, a(n) for n = 1..1275
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Partial row sums of A320647.
Columns 1-6 are A000004, A059053, A320743, A320744, A320745, A320746
For increasing k, columns converge to A320749.
Cf. A320747 (oriented), A320748 (unoriented), A305749 (achiral).

Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d, Adnk[d,n-1,k-#]&], Boole[n == 0 && k == 0]]
Ach[n_,k_] := Ach[n,k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
Table[Sum[(DivisorSum[n, EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j,k-n+1}], {k,15}, {n,k}] // Flatten

\\ Ach is A304972 and R is A152175 as square matrices.
Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
R(n)={Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
T(n)={my(M=(R(n) - Ach(n))/2); for(i=2, n, M[,i] += M[,i-1]); M}
{ my(A=T(12)); for(n=1, #A, print(A[n, ])) } \\ Andrew Howroyd, Nov 03 2019

T(n,k) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k)+Ach(n-2,k-1)+Ach(n-2,k-2)) and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).

T(n,k) = (A320747(n,k) - A305749(n,k)) / 2 = A320747(n,k) - A320748(n,k)= A320748(n,k) - A305749(n,k).

A324802 T(n,k) is the number of non-equivalent distinguishing partitions of the cycle on n vertices with exactly k parts. Regular triangle read by rows, n >= 1, 1 <= k <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 1, 12, 17, 4, 0, 0, 0, 2, 43, 82, 49, 9, 0, 0, 0, 7, 137, 388, 339, 125, 15, 0, 0, 0, 12, 404, 1572, 1994, 1044, 254, 24, 0, 0, 0, 31, 1190, 6405, 10900, 7928, 2761, 490, 35, 0, 0, 0, 57, 3394, 24853, 56586, 54272, 25609, 6365, 850, 51, 0, 0

Offset: 1

Bahman Ahmadi, Sep 04 2019

The cycle graph is defined for n>=3; extended to n=1,2 using the closed form.
Two partitions P1 and P2 of a the vertex set of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. A distinguishing partition is a partition of the vertex set of G such that no nontrivial automorphism of G can preserve it. Here T(n,k)=xi_k(C_n), the number of non-equivalent distinguishing partitions of the cycle on n vertices, with exactly k parts.
Number of n-bead bracelet structures using exactly k different colored beads that are not self-equivalent under either a nonzero rotation or reversal (turning over bracelet). Comparable sequences for unoriented (reversible) strings and necklaces (cyclic group) are A320525 and A327693. - Andrew Howroyd, Sep 23 2019

			Triangle begins:
  0;
  0,  0;
  0,  0,   0;
  0,  0,   0,    0;
  0,  0,   0,    0,    0;
  0,  0,   4,    2,    0,    0;
  0,  1,  12,   17,    4,    0,   0;
  0,  2,  43,   82,   49,    9,   0,  0;
  0,  7, 137,  388,  339,  125,  15,  0, 0;
  0, 12, 404, 1572, 1994, 1044, 254, 24, 0, 0;
  ...
For n=7, we can partition the vertices of the cycle C_7 with exactly 3 parts, in 12 ways, such that all these partitions are distinguishing for C_7 and that all the 12 partitions are non-equivalent. The partitions are as follows:
    { { 1 }, { 2, 3 }, { 4, 5, 6, 7 } },
    { { 1 }, { 2, 3, 4, 6 }, { 5, 7 } },
    { { 1 }, { 2, 3, 4, 7 }, { 5, 6 } },
    { { 1 }, { 2, 3, 5, 6 }, { 4, 7 } },
    { { 1 }, { 2, 3, 5, 7 }, { 4, 6 } },
    { { 1 }, { 2, 3, 6 }, { 4, 5, 7 } },
    { { 1 }, { 2, 3, 7 }, { 4, 5, 6 } },
    { { 1 }, { 2, 4, 5, 6 }, { 3, 7 } },
    { { 1 }, { 2, 4, 7 }, { 3, 5, 6 } },
    { { 1, 2 }, { 3, 4, 6 }, { 5, 7 } },
    { { 1, 2 }, { 3, 5, 6 }, { 4, 7 } },
    { { 1, 2, 4 }, { 3, 6 }, { 5, 7 } }.
From _Andrew Howroyd_, Sep 23 2019: (Start)
For n=6, k=4 the partitions are:
    { { 1, 2, 4 }, { 3 }, { 5 }, { 6 } },
    { { 1, 2 }, { 3, 5 }, { 4 }, { 6 } }.
These correspond to the bracelet structures AABACD and AABCBD.
(End)

Bahman Ahmadi, GAP Program
B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.

Column k=2 is A327734.
Row sums are A327740.
Cf. A309784, A309785, A320525, A320748, A152176, A320647, A324803, A327693.

T(n,k) = A324803(n,k) - A324803(n,k-1).

a(56)-a(78) from Andrew Howroyd, Sep 23 2019

A320747 Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an oriented cycle of length n using k or fewer colors (subsets).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 1, 1, 2, 3, 6, 4, 1, 1, 2, 3, 7, 9, 8, 1, 1, 2, 3, 7, 11, 26, 10, 1, 1, 2, 3, 7, 12, 39, 53, 20, 1, 1, 2, 3, 7, 12, 42, 103, 146, 30, 1, 1, 2, 3, 7, 12, 43, 123, 367, 369, 56, 1, 1, 2, 3, 7, 12, 43, 126, 503, 1235, 1002, 94, 1, 1, 2, 3, 7, 12, 43, 127, 539, 2008, 4439, 2685, 180, 1, 1, 2, 3, 7, 12, 43, 127, 543, 2304, 8720, 15935, 7434, 316, 1, 1, 2, 3, 7, 12, 43, 127, 544, 2356, 11023, 38365, 58509, 20441, 596, 1

Offset: 1

Robert A. Russell, Oct 21 2018

Two color patterns are equivalent if the colors are permuted. An oriented cycle counts each chiral pair as two.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
In other words, the number of n-bead necklace structures using a maximum of k different colored beads. - Andrew Howroyd, Oct 30 2019

			Array begins with T(1,1):
1   1    1     1      1      1      1      1      1      1      1      1 ...
1   2    2     2      2      2      2      2      2      2      2      2 ...
1   2    3     3      3      3      3      3      3      3      3      3 ...
1   4    6     7      7      7      7      7      7      7      7      7 ...
1   4    9    11     12     12     12     12     12     12     12     12 ...
1   8   26    39     42     43     43     43     43     43     43     43 ...
1  10   53   103    123    126    127    127    127    127    127    127 ...
1  20  146   367    503    539    543    544    544    544    544    544 ...
1  30  369  1235   2008   2304   2356   2360   2361   2361   2361   2361 ...
1  56 1002  4439   8720  11023  11619  11697  11702  11703  11703  11703 ...
1  94 2685 15935  38365  54682  60499  61579  61684  61689  61690  61690 ...
1 180 7434 58509 173609 284071 336447 349746 351619 351766 351772 351773 ...
For T(4,2)=4, the patterns are AAAA, AAAB, AABB, and ABAB.
For T(4,3)=6, the patterns are the above four, AABC and ABAC.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Partial row sums of A152175.
Columns 1-6 are A057427, A000013, A002076, A056292, A056293, A056294.
For increasing k, columns converge to A084423.
Cf. A320748 (unoriented), A320742 (chiral), A305749 (achiral).

Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d, Adnk[d,n-1,k-#] &], Boole[n==0 && k==0]]
Table[Sum[DivisorSum[n, EulerPhi[#] Adnk[#,n/#,j] &], {j,k-n+1}]/n, {k,15}, {n,k}] // Flatten

\\ R is A152175 as square matrix
R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
T(n)={my(M=R(n)); for(i=2, n, M[,i] += M[,i-1]); M}
{ my(A=T(12)); for(n=1, #A, print(A[n, ])) } \\ Andrew Howroyd, Nov 03 2019

T(n,k) = (1/n)*Sum_{j=1..k} Sum_{d|n} phi(d)*A(d,n/d,j), where A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).

T(n,k) = A320748(n,k) + A320742(n,k) = 2*A320748(n,k) - A305749(n,k) = A305749(n,k) + 2*A320742(n,k).

A324803 T(n,k) is the number of non-equivalent distinguishing partitions of the cycle on n vertices with at most k part. Square array read by descending antidiagonals, n >= 1, k >= 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 6, 13, 2, 0, 0, 0, 0, 0, 0, 6, 30, 45, 7, 0, 0, 0, 0, 0, 0, 6, 34, 127, 144, 12, 0, 0, 0, 0, 0, 0, 6, 34, 176, 532, 416, 31, 0, 0, 0, 0, 0, 0, 6, 34, 185, 871, 1988, 1221, 57, 0, 0, 0, 0, 0, 0, 6, 34, 185, 996, 3982

Offset: 1

Bahman Ahmadi, Sep 04 2019

The cycle graph is defined for n >= 3; extended to n=1,2 using the closed form.

Two partitions P1 and P2 of a the vertex set of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. A distinguishing partition is a partition of the vertex set of G such that no nontrivial automorphism of G can preserve it. Here T(n,k)=Xi_k(C_n), the number of non-equivalent distinguishing partitions of the cycle on n vertices, with at most k parts.

			Table begins:
=================================================================
  n/k | 1   2    3     4     5     6     7     8     9    10
------+----------------------------------------------------------
    1 | 0,  0,   0,    0,    0,    0,    0,    0,    0,    0, ...
    2 | 0,  0,   0,    0,    0,    0,    0,    0,    0,    0, ...
    3 | 0,  0,   0,    0,    0,    0,    0,    0,    0,    0, ...
    4 | 0,  0,   0,    0,    0,    0,    0,    0,    0,    0, ...
    5 | 0,  0,   0,    0,    0,    0,    0,    0,    0,    0, ...
    6 | 0,  0,   4,    6,    6,    6,    6,    6,    6,    6, ...
    7 | 0,  1,  13,   30,   34,   34,   34,   34,   34,   34, ...
    8 | 0,  2,  45,  127,  176,  185,  185,  185,  185,  185, ...
    9 | 0,  7, 144,  532,  871,  996, 1011, 1011, 1011, 1011, ...
   10 | 0, 12, 416, 1988, 3982, 5026, 5280, 5304, 5304, 5304, ...
  ...
For n=7, we can partition the vertices of the cycle C_7 with at most 3 parts, in 13 ways, such that all these partitions are distinguishing for C_7 and that all the 13 partitions are non-equivalent. The partitions are as follows:
    { { 1 }, { 2, 3 }, { 4, 5, 6, 7 } },
    { { 1 }, { 2, 3, 4, 6 }, { 5, 7 } },
    { { 1 }, { 2, 3, 4, 7 }, { 5, 6 } },
    { { 1 }, { 2, 3, 5, 6 }, { 4, 7 } },
    { { 1 }, { 2, 3, 5, 7 }, { 4, 6 } },
    { { 1 }, { 2, 3, 6 }, { 4, 5, 7 } },
    { { 1 }, { 2, 3, 7 }, { 4, 5, 6 } },
    { { 1 }, { 2, 4, 5, 6 }, { 3, 7 } },
    { { 1 }, { 2, 4, 7 }, { 3, 5, 6 } },
    { { 1, 2 }, { 3, 4, 6 }, { 5, 7 } },
    { { 1, 2 }, { 3, 5, 6 }, { 4, 7 } },
    { { 1, 2, 4 }, { 3, 6 }, { 5, 7 } },
    { { 1, 2, 3, 5 }, { 4, 6, 7 } }.

Bahman Ahmadi, GAP Program
B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.

Column k=2 is A327734.

Cf. A309784, A309785, A320748, A152176, A320647, A324802.

T(n,k) = Sum_{i<=k} A324802(n,i).

cp's OEIS Frontend

A000011 Number of n-bead necklaces (turning over is allowed) where complements are equivalent.

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A320742 Array read by antidiagonals: T(n,k) is the number of chiral pairs of color patterns (set partitions) in a cycle of length n using k or fewer colors (subsets).

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A324802 T(n,k) is the number of non-equivalent distinguishing partitions of the cycle on n vertices with exactly k parts. Regular triangle read by rows, n >= 1, 1 <= k <= n.

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A320747 Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an oriented cycle of length n using k or fewer colors (subsets).

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A324803 T(n,k) is the number of non-equivalent distinguishing partitions of the cycle on n vertices with at most k part. Square array read by descending antidiagonals, n >= 1, k >= 1.

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