cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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.

Original entry on oeis.org

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

Views

Author

Andrew Howroyd, Jan 13 2020

Keywords

Comments

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.

Examples

			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]

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • 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)))}

Formula

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).

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

Views

Author

Andrew Howroyd, Jan 18 2020

Keywords

Comments

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.

Examples

			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]

Links

Crossrefs

Rows n=1..6 are A000012, A000664, A058790, A058791, A058792, A058793.
Cf. A283877, A306019, A330942, A331126, A331277, A331461, A331509, A331510.

Programs

  • PARI
    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

Formula

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

A331278 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of nonzero rows with column sums n and columns in decreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 12, 4, 1, 0, 1, 124, 124, 8, 1, 0, 1, 1800, 10596, 1280, 16, 1, 0, 1, 33648, 1764244, 930880, 13456, 32, 1, 0, 1, 769336, 484423460, 1849386640, 85835216, 143808, 64, 1, 0, 1, 20796960, 198461691404, 7798297361808, 2098356708016, 8206486848, 1556416, 128, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 13 2020

Keywords

Comments

The condition that the columns be in decreasing order is equivalent to considering nonequivalent matrices with distinct columns up to permutation of columns.
A(n,k) is the number of n-uniform k-block sets of multisets.

Examples

			Array begins:
====================================================================
n\k | 0  1      2          3                4                   5
----+---------------------------------------------------------------
  0 | 1  1      0          0                0                   0 ...
  1 | 1  1      1          1                1                   1 ...
  2 | 1  2     12        124             1800               33648 ...
  3 | 1  4    124      10596          1764244           484423460 ...
  4 | 1  8   1280     930880       1849386640       7798297361808 ...
  5 | 1 16  13456   85835216    2098356708016  140094551934813712 ...
  6 | 1 32 143808 8206486848 2516779512105152 ...
  ...
The A(2,2) matrices are:
  [1 0]  [1 0]  [1 0]  [2 0]  [1 1]  [1 0]
  [1 0]  [0 1]  [0 1]  [0 1]  [1 0]  [1 1]
  [0 1]  [1 0]  [0 1]  [0 1]  [0 1]  [0 1]
  [0 1]  [0 1]  [1 0]
.
  [1 0]  [1 0]  [1 0]  [2 1]  [2 0]  [1 0]
  [1 0]  [0 2]  [0 1]  [0 1]  [0 2]  [1 2]
  [0 2]  [1 0]  [1 1]

Links

Crossrefs

Rows n=1..2 are A000012, A173219.
Columns k=0..2 are A000012, A011782, A331396.
The version with binary entries is A331277.
The version with not necessarily distinct columns is A331315.
Cf. A316674 (unrestricted version), A330942, A331395.

Programs

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

Formula

A(n,k) = Sum_{j=0..n*k} binomial(binomial(j+n-1,n),k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).
A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A316674(n, j)/k!.
A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A331315(n, j).
A(n, k) = A011782(n) * A331277(n, k) for k > 0.
A331395(n) = Sum_{d|n} A(n/d, d).

A331639 Number of binary matrices with nonzero rows, a total of n ones and distinct columns each with the same number of ones and columns in decreasing lexicographic order.

Original entry on oeis.org

1, 2, 2, 8, 2, 95, 2, 1062, 2651, 17667, 2, 946585, 2, 10422801, 126470568, 555727036, 2, 61345560608, 2, 1559456567421, 28383861400820, 19815939349521, 2, 30118264353296169, 8755909495925859, 49334805652369611, 21097628287362414244, 98053701052228556867, 2, 27303813269345643163251
Offset: 1

Views

Author

Andrew Howroyd, Jan 23 2020

Keywords

Comments

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

Links

Crossrefs

Cf. A331277, A331638.

Formula

a(n) = Sum_{d|n} A331277(n/d, d).
a(p) = 2 for prime p.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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