A347700 -id:A347700 - cp's OEIS frontend

A347701 Rightmost diagonal of A347700.

Original entry on oeis.org

2, 4, 14, 32, 60

Offset: 1

N. J. A. Sloane, Oct 13 2021

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

Cf. A003041, A347700.

A003041 Number of vacuously transitive relations on n nodes up to isomorphism.

Original entry on oeis.org

2, 7, 24, 92, 388

Offset: 1

N. J. A. Sloane

A transitive relation is vacuously transitive if it does not contain any transitive triple, that is, three distinct ordered pairs (a,b), (b,c), (a,c). - Jukka Kohonen, Sep 17 2021

			a(2)=7: The seven relations are {}, {(1,1)}, {(1,1),(2,2)}, {(2,1)}, {(1,1),(2,1)}, {(1,1),(2,1),(2,2)} and {(2,1),(2,2)}. - _Jukka Kohonen_, Sep 17 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

Cf. A347700, A348240.

Clarified and offset corrected by Jukka Kohonen, Sep 17 2021

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

A347701 Rightmost diagonal of A347700.

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A003041 Number of vacuously transitive relations on n nodes up to isomorphism.

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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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Examples

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Crossrefs