A003425 -id:A003425 - cp's OEIS frontend

A354279 Number of regular elements in the semigroup of all binary relations on [n].

1, 2, 16, 470, 40408, 8683982

Offset: 0

Geoffrey Critzer, May 22 2022

Let S be a semigroup. An element A in S is regular iff A = A*B*A for some B in S. An element in the semigroup of all binary relations is regular iff its row space forms a distributive lattice under set inclusion.

K. K.-H. Butler and G. Markowsky, The Number of Maximal Subgroups of the Semigroup of Binary Relations, Kyungpook Math. J. Vol 12, June 1972.

Cf. A003425.

A363036 Triangular array read by rows. T(n,k) is the number of regular elements in the semigroup of all binary relations on [n] that have rank k, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 9, 6, 1, 49, 306, 114, 1, 225, 8550, 26376, 5256, 1, 961, 194850, 3311250, 4669200, 507720

Offset: 0

Geoffrey Critzer, May 17 2023

			 1;
 1,   1;
 1,   9,      6;
 1,  49,    306,     114;
 1, 225,   8550,   26376,    5256;
 1, 961, 194850, 3311250, 4669200, 507720;
 ...

K. K.-H. Butler and G. Markowsky, The Number of Maximal Subgroups of the Semigroup of Binary Relations, Kyungpook Math. J. Vol 12, June 1972.

Cf. A354279 (row sums), A003425 (main diagonal), A060867 (column k=1), A354741.

A363911 n! times the number of posets with n unlabeled elements.

Original entry on oeis.org

1, 1, 4, 30, 384, 7560, 228960, 10306800, 685399680, 66490865280, 9316160179200, 1866087527673600, 529244914160793600, 210621677079215001600, 116661392964364363315200, 89281569344544938769408000, 93799600948326479830880256000

Offset: 0

Geoffrey Critzer, Jun 27 2023

nonn

Let H be Green's H relation on the semigroup of binary relations on [n]. Then a(n) is the number of elements that are H-related to a poset.

There are A000112(n) D-classes containing the nonsingular relations. There are A001035(n) L-classes in these D-classes. Each such L-class contains exactly one idempotent relation (which is necessarily a poset).

Wikipedia, Green's relations

Cf. A003425, A000142, A001035, A000112.

nn = 10; A000112 = Cases[Import["https://oeis.org/A000112/b000112.txt",
    "Table"], {, }][[All, 2]];Range[0, 16]! Table[A000112[[i]], {i, 1, 17}]

a(n) = A000142(n)*A000112(n).

cp's OEIS Frontend

A354279 Number of regular elements in the semigroup of all binary relations on [n].

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A363036 Triangular array read by rows. T(n,k) is the number of regular elements in the semigroup of all binary relations on [n] that have rank k, n>=0, 0<=k<=n.

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A363911 n! times the number of posets with n unlabeled elements.

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