A058163 -id:A058163 - cp's OEIS frontend

A058162 Number of labeled Abelian groups with a fixed identity.

1, 1, 1, 4, 6, 60, 120, 1920, 7560, 90720, 362880, 13305600, 39916800, 1037836800, 10897286400, 265686220800, 1307674368000, 66691392768000, 355687428096000, 20274183401472000, 202741834014720000

Offset: 1

Christian G. Bower, Nov 15 2000, Mar 12 2008

nonn

The distinction here between labeled and unlabeled Abelian groups is analogous to the distinction between unlabeled rooted trees (A000081) and labeled rooted trees (A000169).
That is, the number of Cayley tables. - Artur Jasinski, Mar 12 2008
Number of Latin squares in dimension n with first row and first column 1,2,3 ..., n which are associative and commutative (Abelian). Each of these squares is isomorphic with the Cayley table of one of the existed Abelian group in dimension n. - Artur Jasinski, Nov 02 2005. Cf. A111341.

			The 2 unlabeled Abelian groups of order 4 are C4 and C2^2. The 4 labeled Abelian groups whose identity is "0" consist of 3 of type C4 (where the nongenerator can be "2", "3", or "4") and 1 of type C2^2.

Max Alekseyev, Table of n, a(n) for n = 1..100
C. J. Hillar, D. Rhea. Automorphisms of finite Abelian groups. American Mathematical Monthly 114:10 (2007), 917-923. Preprint arXiv:math/0605185 [math.GR]
Index entries for sequences related to groups

Cf. A000688, A058160, A058161, A058163.

a(n) = A034382(n) / n. Formula for A034382 is based on the fundamental theorem of finite Abelian groups and the formula given by Hillar and Rhea (2007).

a(16) and a(21) corrected by Max Alekseyev, Sep 12 2019

A058161 Number of labeled cyclic groups with a fixed identity.

Original entry on oeis.org

1, 1, 1, 3, 6, 60, 120, 1260, 6720, 90720, 362880, 9979200, 39916800, 1037836800, 10897286400, 163459296000, 1307674368000, 59281238016000, 355687428096000, 15205637551104000, 202741834014720000, 5109094217170944000, 51090942171709440000

Offset: 1

Christian G. Bower, Nov 14 2000

nonn

Degree of Lagrange resolvent of polynomial of n-th degree. Equals degree of symmetric group of order n divided by order of metacyclic group of order n. - Artur Jasinski, Jan 22 2008

			a(4)=3 because we have: <(1234)> = <(1432)>,  <(1243)> = <(1342)>,  <(1324)> = <(1423)>. - _Geoffrey Critzer_, Sep 07 2015

J. L. Lagrange, Oeuvres, Vol. III Paris 1869.

Index entries for sequences related to groups

a(n) = A000142(n-1)/A000010(n) = A034381(n)/n.
Cf. A058162, A058163.
Cf. A002618.

Table[n!/(n EulerPhi[n]), {n, 1, 20}] (* Artur Jasinski, Jan 22 2008 *)

a(n) = (n-1)!/phi(n).

a(n) = n!/A002618(n) - Artur Jasinski, Jan 22 2008

A034383 Number of labeled groups.

Original entry on oeis.org

1, 2, 3, 16, 30, 480, 840, 22080, 68040, 1088640, 3991680, 259459200, 518918400, 16605388800, 163459296000, 10353459916800, 22230464256000, 1867358997504000, 6758061133824000, 648773868847104000, 5474029518397440000, 122618261212102656000

Offset: 1

Christian G. Bower

nonn

From Jianing Song, Mar 02 2024: (Start)
In other words, number of ways to define a group structure on a set of n elements. Note that for a group G, a group structure on the set G is given by mapping (x,y) to sigma^(-1)(sigma(x)*sigma(y)), where sigma is a bijection on the set G; sigma and sigma' give the same structure if and only if sigma' is the composition of a group automorphism of G and sigma.
By definition, a(n) = A034381(n) if n in A003277, otherwise a(n) > A034381(n). The indices of records of a(n)/A034381(n) among the known terms are 1, 4, 8, 16, 24, 32, 48, 64, 96, 128, 192, with a(192)/A034381(192) = 122774329/1640520 ~ 74.8.
Also by definition, a(n) >= A000001(n)*n!/A059773(n). If the conjecture A059773(2^r) = A002884(r) is true, then A059773(2^r) <= 2^(r^2), while A000001(2^r) >= 2^((2/27)*r^2*(r-6)) (see the Math Stack Exchange link below), so a(2^r)/A034381(2^r) tends to infinity quickly as r tends to infinity.
The sequence is strictly increasing for the first 256 terms (a(256) > A034381(256) > A034381(255) = a(255) since 255 is in A003277). On the other hand, assuming that A059773(2^r) = A002884(r), then a(2^20)/(2^20)! >= A000001(2^20)/A002884(20) > 99798.4, while a(2^20+1)/(2^20)! = A034381(2^20+1)/(2^20)! = (2^20+1)/phi(2^20+1) since 2^20+1 = 17*61681 is in A003277, so we would have a(2^20) > a(2^20+1). It is conjectured a(2^r) > a(2^r+1) for all sufficiently large r. (End)

Stephen A. Silver, Table of n, a(n) for n = 1..255
H. U. Besche, The Small Groups library
Math Stack Exchange, Known bounds for the number of groups of a given order.
Index entries for sequences related to groups

Cf. A000001, A034381, A034382, A058157, A058163.

A034383 := function(n) local fn, sum, k; fn := Factorial(n); sum := 0; for k in [1 .. NrSmallGroups(n)] do sum := sum + fn / Size(AutomorphismGroup(SmallGroup(n,k))); od; return sum; end; # Stephen A. Silver, Feb 10 2013

a(n) = n * A058163(n).

a(n) = Sum n!/|Aut(G)|, where the sum is taken over the different products G of cyclic groups with |G| = n.

More terms from Stephen A. Silver, Feb 10 2013

A123234 Number of n X n Latin squares up to row and column permutation (or "RC-equivalence").

Original entry on oeis.org

1, 1, 1, 4, 16, 1868, 2420400, 66915816462

Offset: 1

Dan Eilers, Oct 06 2006

Brendan McKay writes: (Start)
"It would be possible to find the counts for n=9 and n=10 using the method of my paper in JCD [see link below]. For n=10 it is probably a 24-digit number. I'll explain the method I used. See the paper above for terminology.
"Is(L) is the autotopism group. Also define the group RC(L) of all autotopisms for which the symbols component is the identity. For any Latin square L we have:
"The isotopy class containing L contains (n!)^3/|Is(L)| squares.
"The RC-equivalence class containing L contains (n!)^2/|RC(L)| squares.
"If L and L' are isotopic then |RC(L)| = |RC(L')|. Therefore the number of RC-equivalence classes in the isotopy class of L is n!*|RC(L)|/|Is(L)|. I modified an existing program slightly to find |RC(L)|/|Is(L)|. and applied it to one square from each isotopy class. The sum of n!*|RC(L)|/|Is(L)| is the total number of RC-equivalence classes. " (End)

			01234 => 20413 => 01234
13042 => 01234 => 14320
24310 => 32041 => 20413
30421 => 43102 => 32041
42103 => 14320 => 43102
The first square is transformed by permuting columns; the 2nd square is transformed by permuting rows.
Both the first and 3rd square are in reduced form, so are considered equivalent by row/col permutation.

Dan R. Eilers, Phil A. Sallee, The number of Latin squares up to row and column permutation, Poster Session, Harvey Mudd College Mathematics Conference on Enumerative Combinatorics (2006) (for terms 1 to 7)
Brendan D. McKay, private communication (2006) (for term 8)

B. D. McKay, A. Meynert, W. Myrvold, (2007), Small latin squares, quasigroups, and loops, J. Combin. Designs, 15 (2007), 98-119. doi:10.1002/jcd.20105
Index entries for sequences related to Latin squares and rectangles

Cf. A000315, A002724, A058163.

A058158 Triangle read by rows: T(n,k) is the number of labeled monoids of order n with k idempotents and a fixed identity.

Original entry on oeis.org

1, 1, 1, 1, 6, 4, 4, 45, 72, 35, 6, 528, 1308, 1676, 604, 80, 19935, 39700, 70170, 62060, 16727, 120, 3599436, 1969470, 3829000, 5167800, 3260382, 681232, 2760, 6085914205, 281840664, 294812385, 481221020, 482447637, 228315640, 38187291

Offset: 1

Christian G. Bower, Nov 14 2000

			Triangle begins:
   1;
   1,     1;
   1,     6,     4;
   4,    45,    72,    35;
   6,   528,  1308,  1676,   604;
  80, 19935, 39700, 70170, 62060, 16727;
  ...

Index entries for sequences related to monoids

Row sums give A058154.
Column 1: A058163.
Main diagonal is A351730(n-1).
Cf. A058137 (isomorphism classes), A058157, A058160 (commutative), A058166.

T(n,k) = A058157(n,k)/n.

a(30)-a(36) from Andrew Howroyd, Feb 15 2022

A336412 Number of labeled dihedral groups with a fixed identity.

Original entry on oeis.org

1, 1, 20, 630, 18144, 3326400, 148262400, 40864824000, 6586804224000, 3041127510220800, 464463110651904000, 538583682060103680000, 99430833611096064000000, 129629398219266097152000000, 73681349947830849621196800000, 64240926985765022013480960000000

Offset: 1

Dan Eilers, Jul 20 2020

nonn

a(n) is the number of dihedral groups of order 2n with a fixed identity, or equivalently the number of reduced Latin squares of order 2n that can be viewed as the Cayley table of D_{2n}, by adding a border that matches the first row and column. The reduced Latin squares differ from each other by a permutation of their symbols. Two such Latin squares that differ by a permutation of their symbols have been called isoplanar by Bailey (1984), cited by Nilrat and Praeger (1988), cited by Denes and Keedwell (1991). Latin squares based on dihedral groups are of interest in the stable marriage problem, where Benjamin et al. (1995) exhibited such squares having many stable matchings when viewed as ranking matrices. Two isoplanar Latin squares generally produce a different number of stable matchings, so there is motivation to generate all symbol permutations to find the ones with the most stable matchings.

See comments in A002618 regarding automorphisms of dihedral groups by Ola Veshta and Yaghoub Sharifi. - Dan Eilers, Jun 08 2024

			For n=3 the a(3)=20 isoplanar reduced Latin squares based on the dihedral group of order 6, in lexicographical order, are:
1)             2)             3)             4)             5)
1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6
2 1 4 3 6 5    2 1 4 3 6 5    2 1 4 3 6 5    2 1 4 3 6 5    2 1 5 6 3 4
3 5 1 6 2 4    3 5 6 2 4 1    3 6 1 5 4 2    3 6 5 2 1 4    3 4 1 2 6 5
4 6 2 5 1 3    4 6 5 1 3 2    4 5 2 6 3 1    4 5 6 1 2 3    4 3 6 5 1 2
5 3 6 1 4 2    5 3 2 6 1 4    5 4 6 2 1 3    5 4 1 6 3 2    5 6 2 1 4 3
6 4 5 2 3 1    6 4 1 5 2 3    6 3 5 1 2 4    6 3 2 5 4 1    6 5 4 3 2 1
6)             7)             8)             9)             10)
1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6
2 1 5 6 3 4    2 1 5 6 3 4    2 1 5 6 3 4    2 1 6 5 4 3    2 1 6 5 4 3
3 4 6 5 2 1    3 6 1 5 4 2    3 6 4 1 2 5    3 4 1 2 6 5    3 4 5 6 1 2
4 3 2 1 6 5    4 5 6 1 2 3    4 5 1 3 6 2    4 3 5 6 2 1    4 3 2 1 6 5
5 6 4 3 1 2    5 4 2 3 6 1    5 4 6 2 1 3    5 6 4 3 1 2    5 6 1 2 3 4
6 5 1 2 4 3    6 3 4 2 1 5    6 3 2 5 4 1    6 5 2 1 3 4    6 5 4 3 2 1
11)            12)            13)            14)            15)
1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6
2 1 6 5 4 3    2 1 6 5 4 3    2 3 1 5 6 4    2 3 1 6 4 5    2 4 5 1 6 3
3 5 1 6 2 4    3 5 4 1 6 2    3 1 2 6 4 5    3 1 2 5 6 4    3 6 1 5 4 2
4 6 5 1 3 2    4 6 1 3 2 5    4 6 5 1 3 2    4 5 6 1 2 3    4 1 6 2 3 5
5 3 4 2 6 1    5 3 2 6 1 4    5 4 6 2 1 3    5 6 4 3 1 2    5 3 2 6 1 4
6 4 2 3 1 5    6 4 5 2 3 1    6 5 4 3 2 1    6 4 5 2 3 1    6 5 4 3 2 1
16)            17)            18)            19)            20)
1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6
2 4 6 1 3 5    2 5 4 6 1 3    2 5 6 3 1 4    2 6 4 5 3 1    2 6 5 3 4 1
3 5 1 6 2 4    3 6 1 5 4 2    3 4 1 2 6 5    3 5 1 6 2 4    3 4 1 2 6 5
4 1 5 2 6 3    4 3 2 1 6 5    4 6 5 1 3 2    4 3 2 1 6 5    4 5 6 1 2 3
5 6 4 3 1 2    5 1 6 3 2 4    5 1 4 6 2 3    5 4 6 2 1 3    5 3 2 6 1 4
6 3 2 5 4 1    6 4 5 2 3 1    6 3 2 5 4 1    6 1 5 3 4 2    6 1 4 5 3 2

Denes, J. and Keedwell, A. D. (1991) Latin Squares New Developments in the Theory and Applications. p. 98.

R. A. Bailey, Quasi-Complete Latin Squares: Construction and Randomization, Journal of the Royal Statistical Society. Series B (Methodological) 46, no. 2 (1984): 330, 323-34.
A. T. Benjamin, C. Converse, and H. A. Krieger, Note. How do I marry thee? Let me count the ways, Discrete Appl. Math. 59 (1995) 285-292.
C. K. Nilrat and C. E. Prager, Complete latin squares: terraces for groups, Ars Combinatoria 24 (1988), 17-29.
Yaghoub Sharifi, Automorphisms of dihedral groups.
E. G. Thurber, Concerning the maximum number of stable matchings in the stable marriage problem, Discrete Mathematics Volume 248, Issue 1-3, 6 April 2002, 195-219.

Cf. A058163 (all groups), A058162 (Abelian groups), A058161 (cyclic groups), A069156 (stable matchings), A002618 (n*phi(n)).

A336412:=List([1..16], n->Factorial(2*n-1)/Size(AutomorphismGroup(DihedralGroup(2*n)))); # Dan Eilers, Jun 08 2024

a(1) = a(2) = 1; a(n>2) = (2*n-1)! / A002618(n). - Dan Eilers, Jun 08 2024

a(8)-a(16) and edited by Dan Eilers, Jun 08 2024

cp's OEIS Frontend

A058162 Number of labeled Abelian groups with a fixed identity.

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A058161 Number of labeled cyclic groups with a fixed identity.

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A034383 Number of labeled groups.

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A123234 Number of n X n Latin squares up to row and column permutation (or "RC-equivalence").

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A058158 Triangle read by rows: T(n,k) is the number of labeled monoids of order n with k idempotents and a fixed identity.

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A336412 Number of labeled dihedral groups with a fixed identity.

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