A058162 -id:A058162 - cp's OEIS frontend

A034382 Number of labeled Abelian groups of order n.

1, 2, 3, 16, 30, 360, 840, 15360, 68040, 907200, 3991680, 159667200, 518918400, 14529715200, 163459296000, 4250979532800, 22230464256000, 1200445069824000, 6758061133824000, 405483668029440000

Offset: 1

Christian G. Bower

nonn

Max Alekseyev, Table of n, a(n) for n = 1..100
Hy Ginsberg, Totally Symmetric Quasigroups of Order 16, arXiv:2211.13204 [math.CO], 2022.
C. J. Hillar and D. Rhea. Automorphisms of finite Abelian groups. American Mathematical Monthly 114:10 (2007), 917-923. Preprint arXiv:math/0605185 [math.GR], 2006.
Sugarknri et al., Number of labeled Abelian groups of order n, Mathematics Stack Exchange, 2019.
Index entries for sequences related to groups

Cf. A000688, A034381, A034383, A058159.

a(n) = A058162(n) * n.

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

A058160 Triangle read by rows: T(n,k) is the number of labeled commutative monoids of order n with k idempotents and a fixed identity.

Original entry on oeis.org

1, 1, 1, 1, 6, 2, 4, 45, 36, 9, 6, 408, 648, 348, 76, 60, 7195, 13500, 11790, 5280, 1065, 120, 200016, 369930, 425280, 293640, 118890, 22566, 1920, 13811077, 14716716, 18035745, 16347660, 9953265, 3750516, 674611, 7560, 3405271616, 1024787568, 971651856, 999666920, 794967600, 445001424, 157962168, 27019896

Offset: 1

Christian G. Bower, Nov 14 2000

			Triangle begins:
   1;
   1,    1;,
   1,    6,     2;
   4,   45,    36,     9;
   6,  408,   648,   348,   76;
  60, 7195, 13500, 11790, 5280, 1065;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..55 (rows 1..10)
Index entries for sequences related to monoids

Row sums give A058156.
Column 1: A058162.
Main diagonal A058164.
Cf. A058142 (isomorphism classes), A058158, A058159.

T(n, k) = A058159(n, k)/n.

Terms a(30) and beyond from Andrew Howroyd, Feb 15 2022

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

A058163 Number of labeled groups with a fixed identity.

Original entry on oeis.org

1, 1, 1, 4, 6, 80, 120, 2760, 7560, 108864, 362880, 21621600, 39916800, 1186099200, 10897286400, 647091244800, 1307674368000, 103742166528000, 355687428096000, 32438693442355200, 260668072304640000, 5573557327822848000, 51090942171709440000

Offset: 1

Christian G. Bower, Nov 15 2000

nonn

Stephen A. Silver, Table of n, a(n) for n = 1..255
Index entries for sequences related to groups

Cf. A000688, A034383, A058158, A058162, A058163.

A058163 := function(n) local fn1, sum, k; fn1 := Factorial(n-1); sum := 0; for k in [1 .. NrSmallGroups(n)] do sum := sum + fn1 / Size(AutomorphismGroup(SmallGroup(n,k))); od; return sum; end; # Stephen A. Silver, Feb 10 2013

a(n) = A034383(n)/n.

More terms from Stephen A. Silver, Feb 10 2013

A111341 Number of Latin squares in dimension n with first row and first column 1,2,3,...,n that are non-associative but commutative (element ij = element ji).

Original entry on oeis.org

0, 0, 0, 0, 0, 396, 6120, 10934400, 1225559160, 130025295822240, 252282619805005440, 2209617218725251390961920, 98758655816833727741298666240

Offset: 1

Artur Jasinski, Nov 06 2005

At order 6 there are commutative but non-associative Latin squares. - Artur Jasinski, Dec 08 2006

a(n) = A035481(n) - A058162(n).

a(5) corrected and a(8)-a(13) added by Max Alekseyev, Jul 23 2025

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

A138286 a(n) = A058162(n) / A000688(n).

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A034382 Number of labeled Abelian groups of order n.

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A058160 Triangle read by rows: T(n,k) is the number of labeled commutative monoids of order n with k idempotents and a fixed identity.

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A058161 Number of labeled cyclic groups with a fixed identity.

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A058163 Number of labeled groups with a fixed identity.

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A111341 Number of Latin squares in dimension n with first row and first column 1,2,3,...,n that are non-associative but commutative (element ij = element ji).

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A336412 Number of labeled dihedral groups with a fixed identity.

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