A058164 -id:A058164 - cp's OEIS frontend

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.

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

A058167 Triangle read by rows: T(n,k) is the number of labeled commutative semigroups of order n with k idempotents.

Original entry on oeis.org

1, 4, 2, 24, 30, 9, 260, 492, 312, 76, 4805, 10060, 9900, 4900, 1065, 157956, 284130, 348420, 259500, 112500, 22566, 12277440, 11892846, 14768775, 14093380, 9063600, 3592554, 674611, 3287166928, 896150920, 812261856, 854806120, 707722680, 413149464, 152565280, 27019896

Offset: 1

Christian G. Bower, Nov 15 2000

			Triangle begins:
     1;
     4,     2;
    24,    30,    9;
   260,   492,  312,   76;
  4805, 10060, 9900, 4900, 1065;
  ...

Index entries for sequences related to semigroups

Row sums give A023815.
Main diagonal is A058164(n+1).
Cf. A058116 (isomorphism classes).

a(29)-a(36) from Andrew Howroyd, Jan 27 2022

A058804 Number of labeled vertically indecomposable lattices with a fixed bottom.

Original entry on oeis.org

1, 1, 1, 0, 3, 28, 485, 11886, 396277, 17214856, 944152281, 63752343130, 5192082859601, 501340949329980, 56567812054113373, 7366464172601396422, 1095342925750005427965, 184246192159690980312976

Offset: 0

Christian G. Bower, Dec 28 2000

a(n)=A058803(n)/n=A058805(n)*(n-1).

E.g.f. of a(n+1): A(x)=(B(x)-1)/B(x) where B(x) is e.g.f. of A058164(n+1).

A058165 Number of labeled lattices with a fixed bottom and top.

Original entry on oeis.org

1, 1, 3, 19, 213, 3761, 96373, 3377487, 154874569, 8980530625, 641008120761, 55110586506119, 5606337607922197, 664893010179107529, 90785224148218811149, 14119839142088335522351, 2478547355267024762146257

Offset: 2

Christian G. Bower, Nov 15 2000

Cf. A055512, A058164.

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

a(n) = A055512(n)/(n*(n-1)) = A058164(n)/(n-1).

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

cp's OEIS Frontend

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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A058167 Triangle read by rows: T(n,k) is the number of labeled commutative semigroups of order n with k idempotents.

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Examples

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Extensions

A058804 Number of labeled vertically indecomposable lattices with a fixed bottom.

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A058165 Number of labeled lattices with a fixed bottom and top.

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