cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Christian G. Bower, Nov 14 2000

Keywords

Examples

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

Links

Crossrefs

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

Formula

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

Extensions

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

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

Views

Author

Christian G. Bower, Nov 15 2000

Keywords

Links

Crossrefs

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

Programs

  • GAP
    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

Formula

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

Extensions

More terms from Stephen A. Silver, Feb 10 2013

A058157 Triangle read by rows: T(n,k) is the number of labeled monoids of order n with k idempotents.

Original entry on oeis.org

1, 2, 2, 3, 18, 12, 16, 180, 288, 140, 30, 2640, 6540, 8380, 3020, 480, 119610, 238200, 421020, 372360, 100362, 840, 25196052, 13786290, 26803000, 36174600, 22822674, 4768624, 22080, 48687313640, 2254725312, 2358499080, 3849768160, 3859581096, 1826525120, 305498328
Offset: 1

Views

Author

Christian G. Bower, Nov 14 2000

Keywords

Examples

			Triangle begins:
   1;
   2,    2;
   3,   18,   12;
  16,  180,  288,  140;
  30, 2640, 6540, 8380, 3020;
  ...

Links

Crossrefs

Row sums give A058153.
Column 1: A034383.
Main diagonal is A351731.
Cf. A058137 (isomorphism classes), A058158, A058159 (commutative), A058166.

Formula

T(n,k) = A058158(n,k)*n.

Extensions

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

A351730 Number of labeled idempotent semigroups of order n.

Original entry on oeis.org

1, 1, 4, 35, 604, 16727, 681232, 38187291, 2810370122, 261999605819
Offset: 0

Views

Author

Andrew Howroyd, Feb 17 2022

Keywords

Comments

a(n) is also the number of labeled idempotent monoids of order n+1 with a fixed identity.

Crossrefs

Main diagonal of A058166 and A058158.
Cf. A023814, A058112 (isomorphism classes), A351731.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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