A331039 -id:A331039 - cp's OEIS frontend

A331509 Array read by antidiagonals: A(n,k) is the number of nonisomorphic T_0 n-regular set-systems on a k-set.

1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 3, 0, 0, 1, 0, 1, 6, 3, 0, 0, 1, 0, 1, 15, 19, 1, 0, 0, 1, 0, 1, 42, 141, 29, 0, 0, 0, 1, 0, 1, 109, 1571, 769, 23, 0, 0, 0, 1, 0, 1, 320

Offset: 0

Andrew Howroyd, Jan 18 2020

An n-regular set-system is a finite set of nonempty sets in which each element appears in n blocks.
A set-system is T_0 if for every two distinct elements there exists a block containing one but not the other element.
A(n,k) is the number of nonequivalent binary matrices with k distinct columns and any number of distinct nonzero rows with n ones in every column up to permutation of rows and columns.

			Array begins:
=================================
n\k | 0 1 2 3  4   5    6   7
----+----------------------------
  0 | 1 1 0 0  0   0    0   0 ...
  1 | 1 1 1 1  1   1    1   1 ...
  2 | 1 0 1 3  6  15   42 109 ...
  3 | 1 0 0 3 19 141 1571 ...
  4 | 1 0 0 1 29 769 ...
  5 | 1 0 0 0 23 ...
  ...
The A(2,3) = 3 matrices are:
  [1 1 1]    [1 1 0]    [1 1 0]
  [1 0 0]    [1 0 1]    [1 0 1]
  [0 1 0]    [0 1 0]    [0 1 1]
  [0 0 1]    [0 0 1]

Row 2 appears to be A005368. Row 3 is A331716.

Cf. A330942, A331039, A331461, A331508, A331510.

A331655 Number of binary matrices with n distinct columns and any number of distinct nonzero rows with 4 ones in every column and rows in decreasing lexicographic order.

Original entry on oeis.org

1, 0, 0, 1, 272, 64453, 23553340, 13241130441, 11008118941631, 13027230343637042, 21234181599255320655, 46357847997267210103060, 132373322228662190671151849, 484443861947038578745971380703, 2232754658868099948336222687731941, 12763566506391999019612414249332466653

Offset: 0

Andrew Howroyd, Jan 24 2020

nonn

The condition that the rows be in decreasing order is equivalent to considering nonequivalent matrices with distinct rows up to permutation of rows.

			The a(3) = 1 matrix is:
  [1 1 1]
  [1 1 0]
  [1 0 1]
  [1 0 0]
  [0 1 1]
  [0 1 0]
  [0 0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..50

Row n=4 of A331039.

Cf. A188446.

a(n) = Sum_{k=0..n} Stirling1(n,k)*A188446(k).

A331654 Number of binary matrices with a total of n ones, distinct columns each with the same number of ones and distinct nonzero rows in decreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 44, 6, 519, 1, 8363, 1, 163357, 9427, 3988615, 1, 117148318, 1, 3986012464, 84012192, 157783127674, 1, 7143740399835, 248686, 364166073164915, 2479642897110, 20827974319925302, 1, 1324585467847848929, 1, 92917902002561639120, 190678639438170503

Offset: 1

Andrew Howroyd, Jan 24 2020

nonn

The condition that the rows be in decreasing order is equivalent to considering nonequivalent matrices with distinct rows up to permutation of rows.

			The a(6) = 6 matrices are:
  [1 0 0 0 0 0]  [1 1 1]  [1 1 0]  [1 1 0]  [1 0 1]  [1 1 0]
  [0 1 0 0 0 0]  [1 0 0]  [1 0 1]  [1 0 0]  [1 0 0]  [1 0 1]
  [0 0 1 0 0 0]  [0 1 0]  [0 1 0]  [0 1 1]  [0 1 1]  [0 1 1]
  [0 0 0 1 0 0]  [0 0 1]  [0 0 1]  [0 0 1]  [0 1 0]
  [0 0 0 0 1 0]
  [0 0 0 0 0 1]

Cf. A331039.

a(n) = Sum_{d|n} A331039(n/d, d).

cp's OEIS Frontend

A331509 Array read by antidiagonals: A(n,k) is the number of nonisomorphic T_0 n-regular set-systems on a k-set.

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A331655 Number of binary matrices with n distinct columns and any number of distinct nonzero rows with 4 ones in every column and rows in decreasing lexicographic order.

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A331654 Number of binary matrices with a total of n ones, distinct columns each with the same number of ones and distinct nonzero rows in decreasing lexicographic order.

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Author

Keywords

Comments

Examples

Crossrefs

Formula