A058129 -id:A058129 - cp's OEIS frontend

A118601 Partial sums of A058129.

Original entry on oeis.org

1, 3, 10, 45, 273, 2510, 34069, 1703066

Offset: 1

Jonathan Vos Post, May 08 2006

Cf. A001329, A001423, A001426, A023814, A027851, A029851, A058108, A058132, A058133, A063756, A079173, A118581.

a(n) = SUM[i=1..n] A058129(i). a(n) = SUM[i=1..n] (2*A058133(i) - A058132(i)).

One more term from Jonathan Vos Post, Jul 20 2009

Edited by N. J. A. Sloane, Jul 25 2009

A058133 Number of monoids (semigroups with identity) of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

0, 1, 2, 6, 27, 156, 1373, 17730, 858977, 1844075697, 52991253973742

Offset: 0

Christian G. Bower, Nov 13 2000

Eric Postpischil Posting to sci.math newsgroup, May 21 1990
Andreas Distler and Tom Kelsey, The Monoids of Order Eight and Nine. Lecture Notes in Computer Science 5144 (2008), 61-76.
Andreas Distler and Tom Kelsey, The Monoids of Orders Eight, Nine & Ten. Annals of Mathematics and Artificial Intelligence 56 (2009), 3-25.
Index entries for sequences related to monoids

Cf. A001423, A058129, A058132, A151823.

a(n)=(A058129(n)+A058132(n))/2.

a(8) and a(9) from Distler & Kelsey (2008), added by N. J. A. Sloane, Jul 10 2009
a(10) from Distler & Kelsey (2009), added by Max Alekseyev, Jul 13 2009
a(0) prepended by Jianing Song, Oct 26 2019

A058137 Triangle read by rows: monoids of order n with k idempotents.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 2, 9, 14, 10, 1, 32, 63, 86, 46, 2, 219, 406, 694, 665, 251, 1, 5585, 3331, 6343, 8582, 6035, 1682, 5

Offset: 1

Christian G. Bower, Nov 14 2000

			1; 1,1; 1,3,3; 2,9,14,10; 1,32,63,86,46; ...

Index entries for sequences related to monoids

Row sums give A058129. Main diagonal: A058112(n-1). Columns 1-3: A000001, A058138, A058139.

A125696 Number of categories with n morphisms.

Original entry on oeis.org

1, 1, 3, 11, 55, 329, 2858, 36440, 1723286, 3687822810

Offset: 0

Franklin T. Adams-Watters and Christian G. Bower, Jan 05 2007

			The 11 categories with 3 morphisms consist of:
* 7=A058129(3) categories with 1 object (monoids),
* 3 categories with 2 objects, consisting of: 2=A058129(2) disconnected combinations of a 2-element monoid and a 1-element monoid, and the category with 2 objects and a single morphism between the two objects,
* 1 category with 3 objects (3 separate 1-element monoids).

Geoff Cruttwell, Counting Finite Categories, presentation, (2018).
Eric Weisstein's World of Mathematics, Category

Cf. A058129, A125697, A125698, A125720.

Euler transform of A125698.
G.f.: Product_{i>=1} 1/(1-x^i)^A125698(i).
From Elijah Beregovsky, May 13 2025 (Start)
a(n) >= A058129(n).
Conjecture: a(n) = A058129(n)*(1+o(n)). See Cruttwell presentation in Links. (End)

a(0) and a(7)-a(9) from Thomas Anton, from the work of G. Cruttwell and R. Leblanc, Jan 25 2019

A125697 Table, T(n,k) is the number of categories with n morphisms and k objects.

Original entry on oeis.org

1, 2, 1, 7, 3, 1, 35, 16, 3, 1, 228, 77, 20, 3, 1, 2237, 485, 111, 21, 3, 1, 31559, 4013, 716, 127, 21, 3, 1, 1668997, 47648, 5623, 862, 131, 21, 3, 1, 3685886630, 1868157, 60201, 6739, 926, 132, 21, 3, 1

Offset: 1

Franklin T. Adams-Watters and Christian G. Bower, Jan 05 2007

This is a two-dimensional Euler transform of A125699.

			The table starts:
     1;
     2,   1;
     7,   3,   1;
    35,  16,   3,  1;
   228,  77,  20,  3, 1;
  2237, 485, 111, 21, 3, 1;
  ...

Geoff Cruttwell, Counting Finite Categories, presentation, (2018).
Ben Spitz, SmallCategories
Eric Weisstein's World of Mathematics, Category

Cf. A125696 (row sums), A058129 (column 1), A125699, A125701.

G.f.: Product_{i>=1} Product_{j=1..ceiling(i/2)} 1/(1 - x^i y^j)^A125699(i,j).
T(n,k) = A125701(n-k) when k >= (2/3)*n.
From Ben Spitz, Aug 30 2023: (Start)
T(3n,2n) = T(3n-1,2n-1) + 1 when n >= 1.
T(3n-1,2n-1) = T(3n-2,2n-2) + 4 when n >= 2.
T(3n-2,2n-2) = T(3n-3,2n-3) + 19 when n >= 4.
(End)

a(23)-a(29) from Ben Spitz, Jul 17 2023
a(30)-a(36) from Ben Spitz, Aug 29 2023
a(37)-a(45) from Elijah Beregovsky, after the work of Cruttwell and Leblanc, May 20 2025

A058132 Number of self-converse monoids (semigroups with identity) of order n.

Original entry on oeis.org

0, 1, 2, 5, 19, 84, 509, 3901, 48957

Offset: 0

Christian G. Bower, Nov 13 2000

Index entries for sequences related to monoids

Cf. A058129, A058133.

a(n) = A058133(n)*2 - A058129(n).

a(8) from Max Alekseyev, Jul 13 2009

a(0) prepended by Jianing Song, Oct 26 2019

A125699 Table, T(n,k) is the number of connected categories with n morphisms and k objects.

Original entry on oeis.org

1, 2, 7, 1, 35, 6, 228, 28, 2, 2237, 159, 11, 31559, 1075, 77, 3, 1668997, 9389, 497, 24

Offset: 1

Franklin T. Adams-Watters and Christian G. Bower, Jan 05 2007

Connected in the sense that if the morphism direction and composition is ignored, resulting in a multigraph, that multigraph is connected.

			The table starts:
        1;
        2;
        7,    1;
       35,    6;
      228,   28,   2;
     2237,  159,  11;
    31559, 1075,  77,  3;
  1668997, 9389, 497, 24;
  ...

Ben Spitz, SmallCategories
Eric Weisstein's World of Mathematics, Category

Cf. A125696, A125698 (row sums), A110654 (row lengths), A058129 (column 1), A125700 (diagonal sums), A125702 (T(2n-1, n)).

a(14)-a(20) from Ben Spitz, Sep 02 2023

A253950 Number of finite, negative, totally ordered monoids of size n (semigroups with a neutral element that is also the top element).

Original entry on oeis.org

1, 1, 2, 8, 44, 308, 2641, 27120, 332507, 5035455

Offset: 1

Milan Petrík, Jan 20 2015

The terms have been computed using the algorithm described in the referenced papers.

Eric Babson, Moon Duchin, Annina Iseli, Pietro Poggi-Corradini, Dylan Thurston, and Jamie Tucker-Foltz, Models of random spanning trees, arXiv:2407.20226 [cs.DM], 2024. See p. 9.
M. Petrík, GitLab repository with an implementation of the algorithm in Python 3
M. Petrík, Many-Valued Conjunctions. Habilitation thesis, Czech Technical University in Prague, Faculty of Electrical Engineering, Prague, Czech Republic. Submitted in 2020. Available at Czech Technical University Digital Library.
M. Petrík and Th. Vetterlein, Rees coextensions of finite tomonoids and free pomonoids. Semigroup Forum 99 (2019) 345-367. DOI: 10.1007/s00233-018-9972-z.
M. Petrík and Th. Vetterlein, Rees coextensions of finite, negative tomonoids. Journal of Logic and Computation 27 (2017) 337-356. DOI: 10.1093/logcom/exv047.
M. Petrík and Th. Vetterlein, Algorithm to generate finite negative totally ordered monoids. In: IPMU 2016: 16th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems. Eindhoven, Netherlands, June 20-24, 2016.
M. Petrík and Th. Vetterlein, Algorithm to generate the Archimedean, finite, negative tomonoids. In: Joint 7th International Conference on Soft Computing and Intelligent Systems and 15th International Symposium on Advanced Intelligent Systems. Kitakyushu, Japan, Dec. 3-6, 2014. DOI: 10.1109/SCIS-ISIS.2014.7044822.
Index entries for sequences related to monoids

Cf. A058129, A030453, A253948, A253949, A374293.

a(10) from Milan Petrík, May 09 2021

A058131 Number of isomorphism classes of commutative monoids (commutative semigroups with identity) of order n.

Original entry on oeis.org

0, 1, 2, 5, 19, 78, 421, 2637, 20486, 246458, 11833361

Offset: 0

Christian G. Bower, Nov 13 2000

Remigiusz Durka and Kamil Grela, On the number of possible resonant algebras, arXiv:1911.12814 [hep-th], 2019.
Eric Postpischil, Posting to sci.math newsgroup, May 21 1990
Eric Weisstein's World of Mathematics, Commutative Monoid
Eric Weisstein's World of Mathematics, Additive Cellular Automaton
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 952.
Index entries for sequences related to monoids

Row sums of A058142.

Cf. A001426, A058129, A058155 (labeled).

a(0) prepended by Jianing Song, Oct 26 2019
a(8) and a(9) from Alex Meiburg, Oct 20 2021
a(10) from Andrew Howroyd, Feb 15 2022

A253948 Number of finite, negative, Archimedean, commutative, totally ordered monoids of size n (semi-groups with a neutral element that is also the top element).

Original entry on oeis.org

1, 1, 1, 2, 6, 22, 95, 471, 2670, 17387, 131753, 1184059, 12896589

Offset: 1

Milan Petrík, Jan 20 2015

Also number of Archimedean triangular norms on an n-chain.

The terms have been computed using the algorithm described in the referenced papers.

M. Petrík, GitLab repository with an implementation of the algorithm in Python 3
M. Petrík, Many-Valued Conjunctions. Habilitation thesis, Czech Technical University in Prague, Faculty of Electrical Engineering, Prague, Czech Republic. Submitted in 2020. Available at Czech Technical University Digital Library.
M. Petrík and Th. Vetterlein, Rees coextensions of finite tomonoids and free pomonoids. Semigroup Forum 99 (2019) 345-367. DOI: 10.1007/s00233-018-9972-z.
M. Petrík and Th. Vetterlein, Rees coextensions of finite, negative tomonoids. Journal of Logic and Computation 27 (2017) 337-356. DOI: 10.1093/logcom/exv047.
M. Petrík and Th. Vetterlein, Algorithm to generate finite negative totally ordered monoids. In: IPMU 2016: 16th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems. Eindhoven, Netherlands, June 20-24, 2016.
M. Petrík and Th. Vetterlein, Algorithm to generate the Archimedean, finite, negative tomonoids. In: Joint 7th International Conference on Soft Computing and Intelligent Systems and 15th International Symposium on Advanced Intelligent Systems. Kitakyushu, Japan, Dec. 3-6, 2014. DOI: 10.1109/SCIS-ISIS.2014.7044822.
Index entries for sequences related to monoids

Cf. A058129, A030453, A253949, A253950.

a(13) from Milan Petrík, May 09 2021

cp's OEIS Frontend

A118601 Partial sums of A058129.

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A058133 Number of monoids (semigroups with identity) of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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A058137 Triangle read by rows: monoids of order n with k idempotents.

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A125696 Number of categories with n morphisms.

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A125697 Table, T(n,k) is the number of categories with n morphisms and k objects.

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A058132 Number of self-converse monoids (semigroups with identity) of order n.

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A125699 Table, T(n,k) is the number of connected categories with n morphisms and k objects.

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A253950 Number of finite, negative, totally ordered monoids of size n (semigroups with a neutral element that is also the top element).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A058131 Number of isomorphism classes of commutative monoids (commutative semigroups with identity) of order n.

Views

Author

Keywords

Links

Crossrefs

Extensions

A253948 Number of finite, negative, Archimedean, commutative, totally ordered monoids of size n (semi-groups with a neutral element that is also the top element).

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Author

Keywords

Comments

Links