A331510 -id:A331510 - cp's OEIS frontend

A330942 Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of nonzero rows with n ones in every column and columns in nonincreasing lexicographic order.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 7, 1, 1, 1, 8, 75, 32, 1, 1, 1, 16, 1105, 2712, 161, 1, 1, 1, 32, 20821, 449102, 116681, 842, 1, 1, 1, 64, 478439, 122886128, 231522891, 5366384, 4495, 1, 1, 1, 128, 12977815, 50225389432, 975712562347, 131163390878, 256461703, 24320, 1, 1

Offset: 0

Andrew Howroyd, Jan 13 2020

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.

A(n,k) is the number of labeled n-uniform hypergraphs with multiple edges allowed and with k edges and no isolated vertices. When n=2 these objects are multigraphs.

			Array begins:
============================================================
n\k | 0 1    2         3              4                5
----+-------------------------------------------------------
  0 | 1 1    1         1              1                1 ...
  1 | 1 1    2         4              8               16 ...
  2 | 1 1    7        75           1105            20821 ...
  3 | 1 1   32      2712         449102        122886128 ...
  4 | 1 1  161    116681      231522891     975712562347 ...
  5 | 1 1  842   5366384   131163390878 8756434117294432 ...
  6 | 1 1 4495 256461703 78650129124911 ...
  ...
The A(2,2) = 7 matrices are:
   [1 0]  [1 0]  [1 0]  [1 1]  [1 0]  [1 0]  [1 1]
   [1 0]  [0 1]  [0 1]  [1 0]  [1 1]  [0 1]  [1 1]
   [0 1]  [1 0]  [0 1]  [0 1]  [0 1]  [1 1]
   [0 1]  [0 1]  [1 0]

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Rows n=1..3 are A000012, A121316, A136246.
Columns k=0..3 are A000012, A000012, A226994, A137220.
The version with nonnegative integer entries is A331315.
Other variations considering distinct rows and columns and equivalence under different combinations of permutations of rows and columns are:
All solutions: A262809 (all), A331567 (distinct rows).
Up to row permutation: A188392, A188445, A331126, A331039.
Up to column permutation: this sequence, A331571, A331277, A331569.
Nonisomorphic: A331461, A331510, A331508, A331509.
Cf. A331638.

T[n_, k_] := With[{m = n k}, Sum[Binomial[Binomial[j, n] + k - 1, k] Sum[ (-1)^(i - j) Binomial[i, j], {i, j, m}], {j, 0, m}]];
Table[T[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Apr 10 2020, from PARI *)

T(n, k)={my(m=n*k); sum(j=0, m, binomial(binomial(j, n)+k-1, k)*sum(i=j, m, (-1)^(i-j)*binomial(i, j)))}

A(n,k) = Sum_{j=0..n*k} binomial(binomial(j,n)+k-1, k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).
A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A262809(n, j)/k!.
A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331277(n, j).
A331638(n) = Sum_{d|n} A(n/d, d).

A331461 Array read by antidiagonals: A(n,k) is the number of nonequivalent binary matrices with k columns and any number of nonzero rows with n ones in every column up to permutation of rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 5, 8, 4, 1, 1, 1, 7, 23, 16, 5, 1, 1, 1, 11, 66, 93, 30, 6, 1, 1, 1, 15, 212, 652, 332, 50, 7, 1, 1, 1, 22, 686, 6369, 6414, 1062, 80, 8, 1, 1, 1, 30, 2389, 79568, 226041, 56712, 3117, 120, 9, 1, 1, 1, 42, 8682, 1256425, 12848128, 7295812, 441881, 8399, 175, 10, 1, 1

Offset: 0

Andrew Howroyd, Jan 18 2020

A(n,k) is the number of non-isomorphic set multipartitions (multiset of sets) with k parts each part has size n.

			Array begins:
===========================================================
n\k | 0 1 2   3    4       5          6              7
----+-----------------------------------------------------
  0 | 1 1 1   1    1       1          1              1 ...
  1 | 1 1 2   3    5       7         11             15 ...
  2 | 1 1 3   8   23      66        212            686 ...
  3 | 1 1 4  16   93     652       6369          79568 ...
  4 | 1 1 5  30  332    6414     226041       12848128 ...
  5 | 1 1 6  50 1062   56712    7295812     1817321457 ...
  6 | 1 1 7  80 3117  441881  195486906   200065951078 ...
  7 | 1 1 8 120 8399 3006771 4298181107 17131523059493 ...
  ...
The A(2,3) = 8 matrices are:
  [1 0 0]  [1 1 0]  [1 1 1]  [1 1 0]  [1 1 0]  [1 1 1]  [1 1 0]  [1 1 1]
  [1 0 0]  [1 0 0]  [1 0 0]  [1 1 0]  [1 0 1]  [1 1 0]  [1 0 1]  [1 1 1]
  [0 1 0]  [0 1 0]  [0 1 0]  [0 0 1]  [0 1 0]  [0 0 1]  [0 1 1]
  [0 1 0]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]
  [0 0 1]  [0 0 1]
  [0 0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..152

Rows n=0..6 are A000012, A000041, A050535, A050913, A058783, A058784, A058785.
Columns k=0..4 are A000012, A000012, A000027(n+1), A002624, A331720.
Cf. A188392, A262809, A304942, A306018, A330942, A331485, A331508, A331509, A331510.

\\ See A304942 for Blocks
T(n,k)={Blocks(k, n*k, n)}
{ for(n=0, 7, for(k=0, 6, print1(T(n,k), ", ")); print) }

A306018(n) = Sum_{d|n} A(n/d, d).

A331508 Array read by antidiagonals: A(n,k) is the number of nonisomorphic T_0 n-regular set multipartitions (multisets of sets) on a k-set.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 5, 3, 1, 1, 0, 1, 11, 12, 4, 1, 1, 0, 1, 26, 66, 25, 5, 1, 1, 0, 1, 68, 445, 278, 44, 6, 1, 1, 0, 1, 177, 4279, 5532, 966, 73, 7, 1, 1, 0, 1, 497, 53340, 200589, 53535, 2957, 112, 8, 1, 1, 0, 1, 1476, 846254, 11662671, 7043925, 431805, 8149, 166, 9, 1, 1

Offset: 0

Andrew Howroyd, Jan 18 2020

An n-regular set multipartition is a finite multiset of nonempty sets in which each element appears in n blocks.
A set multipartition 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 nonzero rows with n ones in every column up to permutation of rows and columns.
A(n,k) is the number of non-isomorphic set-systems with k parts each of size n.

			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 1 2  5   11     26      68      177 ...
  3 | 1 1 3 12   66    445    4279    53340 ...
  4 | 1 1 4 25  278   5532  200589 11662671 ...
  5 | 1 1 5 44  966  53535 7043925 ...
  6 | 1 1 6 73 2957 431805 ...
  ...
The A(2,3) = 5 matrices are:
  [1 0 0]  [1 1 0]  [1 1 1]  [1 1 0]  [1 1 0]
  [1 0 0]  [1 0 0]  [1 0 0]  [1 0 1]  [1 0 1]
  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 0]  [0 1 1]
  [0 1 0]  [0 0 1]  [0 0 1]  [0 0 1]
  [0 0 1]  [0 0 1]
  [0 0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..152 (first 17 antidiagonals)

Rows n=1..6 are A000012, A000664, A058790, A058791, A058792, A058793.

Cf. A283877, A306019, A330942, A331126, A331277, A331461, A331509, A331510.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))[k]}
T(n,k)={my(m=n*k, s=0); if(m==0, k<=1, forpart(q=m, my(g=sum(t=1, k, K(q, t, n)*x^t/t) + O(x*x^k)); s+=permcount(q)*polcoef(exp(g - subst(g,x,x^2)), k)); s/m!)}
{ for(n=0, 6, for(k=0, 5, print1(T(n, k), ", ")); print) } \\ Andrew Howroyd, Jan 16 2024

A306019(n) = Sum_{d|n} A(n/d, d).

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

Original entry on oeis.org

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.

A331717 Number of nonequivalent n-column binary matrices with 2 ones in every column and any number of distinct nonzero rows up to permutation of rows and columns.

Original entry on oeis.org

1, 0, 1, 4, 12, 36, 124, 412, 1500

Offset: 0

Andrew Howroyd, Feb 02 2020

			The a(3) = 4 matrices are:
  [1 1 1]  [1 1 0]  [1 1 1]  [1 1 0]
  [1 0 0]  [1 0 1]  [1 1 0]  [1 0 1]
  [0 1 0]  [0 1 0]  [0 0 1]  [0 1 1]
  [0 0 1]  [0 0 1]

Row n=2 of A331510.

Cf. A331718.

A331718 Number of nonequivalent n-column binary matrices with 3 ones in every column and any number of distinct nonzero rows up to permutation of rows and columns.

Original entry on oeis.org

1, 0, 0, 3, 23, 191, 2203, 31313

Offset: 0

Andrew Howroyd, Feb 02 2020

			The a(3) = 3 matrices are:
  [1, 1, 0]  [1, 1, 1]  [1, 1, 1]
  [1, 0, 1]  [1, 1, 0]  [1, 1, 0]
  [1, 0, 0]  [1, 0, 1]  [1, 0, 1]
  [0, 1, 1]  [0, 1, 0]  [0, 1, 1]
  [0, 1, 0]  [0, 0, 1]
  [0, 0, 1]

Row n=3 of A331510.

Cf. A331717.

A331719 Number of nonequivalent 5-column binary matrices with n ones in every column and any number of distinct nonzero rows up to permutation of rows and columns.

Original entry on oeis.org

1, 7, 36, 191, 837, 2688, 6184, 10199, 12054, 10199, 6184, 2688, 837, 191, 36, 7, 1

Offset: 0

Andrew Howroyd, Feb 02 2020

Column k=5 of A331510.

Cf. A331127.

cp's OEIS Frontend

A330942 Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of nonzero rows with n ones in every column and columns in nonincreasing lexicographic order.

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A331461 Array read by antidiagonals: A(n,k) is the number of nonequivalent binary matrices with k columns and any number of nonzero rows with n ones in every column up to permutation of rows and columns.

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A331508 Array read by antidiagonals: A(n,k) is the number of nonisomorphic T_0 n-regular set multipartitions (multisets of sets) on a k-set.

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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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A331717 Number of nonequivalent n-column binary matrices with 2 ones in every column and any number of distinct nonzero rows up to permutation of rows and columns.

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A331718 Number of nonequivalent n-column binary matrices with 3 ones in every column and any number of distinct nonzero rows up to permutation of rows and columns.

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A331719 Number of nonequivalent 5-column binary matrices with n ones in every column and any number of distinct nonzero rows up to permutation of rows and columns.

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