A058160 -id:A058160 - cp's OEIS frontend

A058159 Triangle read by rows: T(n,k) is the number of labeled commutative monoids of order n with k idempotents.

1, 2, 2, 3, 18, 6, 16, 180, 144, 36, 30, 2040, 3240, 1740, 380, 360, 43170, 81000, 70740, 31680, 6390, 840, 1400112, 2589510, 2976960, 2055480, 832230, 157962, 15360, 110488616, 117733728, 144285960, 130781280, 79626120, 30004128, 5396888, 68040, 30647444544, 9223088112, 8744866704, 8997002280, 7154708400, 4005012816, 1421659512, 243179064

Offset: 1

Christian G. Bower, Nov 14 2000

			Triangle begins:
    1;
    2,     2;
    3,    18,     6;
   16,   180,   144,    36;
   30,  2040,  3240,  1740,   380;
  360, 43170, 81000, 70740, 31680, 6390;
  ...

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

Row sums give A058155.
Column 1: A034382.
Main diagonal: A055512.
Cf. A058142 (isomorphism classes), A058157, A058160.

T(n, k) = A058160(n, k)*n.

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

A058162 Number of labeled Abelian groups with a fixed identity.

Original entry on oeis.org

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

A058142 Triangle read by rows: number of commutative monoids of order n with k idempotents.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 2, 9, 6, 2, 1, 26, 30, 16, 5, 1, 98, 142, 111, 54, 15, 1, 455, 718, 713, 482, 215, 53, 3, 4018, 4277, 4637, 3919, 2414, 996, 222, 2, 101910, 36124, 32937, 31308, 24047, 13758, 5294, 1078, 1, 10054303, 708355, 290278, 260758, 229411, 164216, 88178, 31867, 5994

Offset: 1

Christian G. Bower, Nov 14 2000

			Triangle begins:
  1;
  1,  1;
  1,  3,   1;
  2,  9,   6,   2;
  1, 26,  30,  16, 5;
  1, 98, 142, 111, 54, 15;
  ...

Index entries for sequences related to monoids

Row sums give A058131.
Main diagonal: A006966.
Columns 1..2: A000688, A058143.
Cf. A058116, A058137, A058150, A058159, A058160.

a(30)-a(55) from Andrew Howroyd, Feb 15 2022

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

A058164 Number of labeled lattices with a fixed bottom.

Original entry on oeis.org

1, 1, 2, 9, 76, 1065, 22566, 674611, 27019896, 1393871121, 89805306250, 7051089328371, 661327038073428, 72882388902988561, 9308502142507505406, 1361778362223282167235, 225917426273413368357616, 42135305039539420956486369, 8768279861975723002123310226

Offset: 1

Christian G. Bower, Nov 15 2000

Main diagonal of A058160.

Cf. A055512, A058165.

A055512 = Cases[Import["https://oeis.org/A055512/b055512.txt", "Table"], {, }][[All, 2]];
a[n_] := A055512[[n + 1]]/n;
a /@ Range[18] (* Jean-François Alcover, Jan 02 2020 *)

a(n) = A055512(n)/n = A058165(n)*(n-1).

a(18) from Jean-François Alcover, Jan 02 2020

a(19) from the data at A055512 added by Amiram Eldar, Jul 22 2025

A058156 Number of labeled commutative monoids of order n with a fixed identity.

Original entry on oeis.org

1, 2, 9, 94, 1486, 38890, 1430442, 77291510, 7826336608, 3904851153672

Offset: 1

Christian G. Bower, Nov 14 2000

Index entries for sequences related to monoids

Row sums of A058160.

Cf. A058155.

a(n) = A058155(n)/n.

a(8)-a(9) from Alex Meiburg, Oct 20 2021

a(10) from Andrew Howroyd, Feb 15 2022

cp's OEIS Frontend

A058159 Triangle read by rows: T(n,k) is the number of labeled commutative monoids of order n with k idempotents.

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A058162 Number of labeled Abelian groups with a fixed identity.

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A058142 Triangle read by rows: number of commutative monoids of order n with k idempotents.

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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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A058164 Number of labeled lattices with a fixed bottom.

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Mathematica

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A058156 Number of labeled commutative monoids of order n with a fixed identity.

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