A055080 -id:A055080 - cp's OEIS frontend

A005747 Number of n-covers of an unlabeled 6-set.

1, 16, 196, 2197, 22952, 223034, 2004975, 16642937, 127654604, 907349654, 6000728764, 37093282121, 215296646264, 1178514299094, 6108871834312, 30098215339608, 141433252447082, 635816643771438, 2742163498527536, 11374522802412498, 45482770393453638

Offset: 1

N. J. A. Sloane, Simon Plouffe

nonn

Number of n X 6 binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Vladeta Jovovic, Binary matrices up to row and column permutations

A diagonal of A055080.

First differences give A055084.

Vec((G(6, x) - G(5, x)) + O(x^30)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

Corrected and extended by Vladeta Jovovic, Jun 13 2000

Terms a(20) and beyond from Andrew Howroyd, Feb 28 2023

A005771 Number of n-covers of an unlabeled 5-set.

Original entry on oeis.org

1, 12, 103, 736, 4571, 25326, 127415, 588687, 2518997, 10053739, 37656707, 133084998, 445949359, 1422934989, 4340110439, 12697803333, 35744330644, 97081519369, 255032046536, 649459943602, 1606518048420, 3867119228081, 9073566868140, 20783186834063

Offset: 1

N. J. A. Sloane, Simon Plouffe

nonn

Number of n X 5 binary matrices with at least one 1 in every column up to row and column permutations. - Andrew Howroyd, Feb 28 2023

R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Vladeta Jovovic, Binary matrices up to row and column permutations

A diagonal of A055080.
First differences give A055083.
Cf. A006148, A052264.

Vec(G(5, x) - G(4, x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

a(n) = A052264(n) - A006148(n). - Andrew Howroyd, Feb 28 2023

More terms from Vladeta Jovovic, Jun 13 2000

Terms a(21) and beyond from Andrew Howroyd, Feb 28 2023

A055127 Triangle T(n,k) of numbers of proper k-covers of an unlabeled n-set, k=1..2^n-2.

Original entry on oeis.org

0, 1, 0, 2, 5, 4, 2, 1, 0, 4, 19, 58, 113, 168, 193, 171, 119, 68, 29, 10, 3, 1, 0, 6, 53, 325, 1551, 6007, 19533, 54119, 128936, 266085, 478223, 751487, 1035609, 1254303, 1336855, 1254307, 1035622, 751526, 478320, 266272, 129226, 54484, 19898, 6297

Offset: 2

Vladeta Jovovic, Jun 14 2000

			[0, 1], [0, 2, 5, 4, 2, 1], [0, 4, 19, 58, 113, 168, 193, 171, 119, 68, 29, 10, 3, 1], ...; There are 113 proper 5-covers of an unlabeled 4-set.

Eric Weisstein's World of Mathematics, Proper covers

Cf. A052265, A055080, A007537. Row sums give A055152.

A347699 Triangle read by rows: For n >= 1, 0 <= k <= n-1, T(n,k) = 0 if k=0, otherwise the number of inequivalent k X (n-k) 0,1 matrices having at least one 1 in each column.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 9, 4, 1, 1, 9, 23, 17, 5

Offset: 1

N. J. A. Sloane, Oct 13 2021

It appears that this is essentially the same as the triangle in A055080, except that the 1's at the ends of the rows in A055080 have been moved to the start of the rows. (This is at present only a conjecture.)
Even if the conjecture turns out to be true, this triangle deserves its own entry because the arrays look so different.
The conjecture would imply the row sums are given by A048194.

			Triangle begins:
  1;
  1,  1;
  1,  1,  2;
  1,  1,  4,  3;
  1,  1,  6,  9,  4;
  1,  1,  9, 23, 17,  5;
...

H. Sharp, Jr., Enumeration of vacuously transitive relations, Discrete Math. 4 (1973), 185-196.

Cf. A048194, A055080, A347700.

cp's OEIS Frontend

A005747 Number of n-covers of an unlabeled 6-set.

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A005771 Number of n-covers of an unlabeled 5-set.

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A055127 Triangle T(n,k) of numbers of proper k-covers of an unlabeled n-set, k=1..2^n-2.

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A347699 Triangle read by rows: For n >= 1, 0 <= k <= n-1, T(n,k) = 0 if k=0, otherwise the number of inequivalent k X (n-k) 0,1 matrices having at least one 1 in each column.

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