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 1-8 of 8 results.

A331039 Array read by antidiagonals: A(n,k) is the number of 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, 5, 0, 0, 1, 0, 1, 43, 5, 0, 0, 1, 0, 1, 518, 175, 1, 0, 0, 1, 0, 1, 8186, 9426, 272, 0, 0, 0, 1, 0, 1, 163356, 751365, 64453, 205, 0, 0, 0, 1, 0, 1, 3988342, 84012191, 23553340, 248685, 80, 0, 0, 0, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 08 2020

Keywords

Comments

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

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 0 1 5  43     518        8186           163356 ...
  3 | 1 0 0 5 175    9426      751365         84012191 ...
  4 | 1 0 0 1 272   64453    23553340      13241130441 ...
  5 | 1 0 0 0 205  248685   421934358    1176014951129 ...
  6 | 1 0 0 0  80  620548  5055634889   69754280936418 ...
  7 | 1 0 0 0  15 1057989 43402628681 2972156676325398 ...
  ...
The A(2,3) = 5 matrices are:
  [1 1 1]    [1 1 0]    [1 1 0]    [1 0 1]    [1 1 0]
  [1 0 0]    [1 0 1]    [1 0 0]    [1 0 0]    [1 0 1]
  [0 1 0]    [0 1 0]    [0 1 1]    [0 1 1]    [0 1 1]
  [0 0 1]    [0 0 1]    [0 0 1]    [0 1 0]
The corresponding set-systems are:
  {{1,2,3}, {1}, {2}, {3}},
  {{1,2}, {1,3}, {2,3}},
  {{1,2}, {1,3}, {2}, {3}},
  {{1,2}, {1}, {2,3}, {3}},
  {{1,3}, {1}, {2,3}, {2}}.

Links

Crossrefs

Rows n=1..4 are A000012, A060053, A060070, A331655.
Cf. A188445, A330942, A330964, A331126, A331160, A331161, A331569, A331654.

Programs

  • PARI
    WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(WeighT(v)[n], k)*k!/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, k<=1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)}

Formula

A(n, k) = Sum_{j=1..k} Stirling1(k, j)*A188445(n, j) for n, k >= 1.
A(n, k) = 0 for k >= 1, n > 2^(k-1).
A331654(n) = Sum_{d|n} A(n/d, d).

A331315 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k columns and any number of nonzero rows with column sums n and columns in nonincreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 14, 4, 1, 1, 8, 150, 128, 8, 1, 1, 16, 2210, 10848, 1288, 16, 1, 1, 32, 41642, 1796408, 933448, 13472, 32, 1, 1, 64, 956878, 491544512, 1852183128, 85862144, 143840, 64, 1, 1, 128, 25955630, 200901557728, 7805700498776, 2098614254048, 8206774496, 1556480, 128, 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 n-uniform k-block multisets of multisets.

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  2     14        150             2210              41642 ...
  3 | 1  4    128      10848          1796408          491544512 ...
  4 | 1  8   1288     933448       1852183128      7805700498776 ...
  5 | 1 16  13472   85862144    2098614254048 140102945876710912 ...
  6 | 1 32 143840 8206774496 2516804131997152 ...
     ...
The A(2,2) = 14 matrices are:
  [1 0]  [1 0]  [1 0]  [2 0]  [1 1]  [1 0]  [1 0]
  [1 0]  [0 1]  [0 1]  [0 1]  [1 0]  [1 1]  [1 0]
  [0 1]  [1 0]  [0 1]  [0 1]  [0 1]  [0 1]  [0 2]
  [0 1]  [0 1]  [1 0]
.
  [1 0]  [1 0]  [2 1]  [2 0]  [1 1]  [1 0]  [2 2]
  [0 2]  [0 1]  [0 1]  [0 2]  [1 1]  [1 2]
  [1 0]  [1 1]

Links

Crossrefs

Rows n=1..2 are A000012, A121227.
Columns k=0..2 are A000012, A011782, A331397.
The version with binary entries is A330942.
The version with distinct columns is A331278.
Other variations considering distinct rows and columns and equivalence under different combinations of permutations of rows and columns are:
All solutions: A316674 (all), A331568 (distinct rows).
Up to row permutation: A219727, A219585, A331161, A331160.
Up to column permutation: this sequence, A331572, A331278, A331570.
Nonisomorphic: A331485.
Cf. A317583.

Programs

  • PARI
    T(n, k)={my(m=n*k); sum(j=0, m, binomial(binomial(j+n-1, 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-1,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))*A316674(n, j)/k!.
A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331278(n, j).
A(n, k) = A011782(n) * A330942(n, k) for k > 0.
A317583(n) = Sum_{d|n} A(n/d, d).

A331126 Array read by antidiagonals: A(n,k) is the number of 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, 9, 3, 1, 1, 0, 1, 70, 29, 4, 1, 1, 0, 1, 794, 666, 68, 5, 1, 1, 0, 1, 12055, 28344, 3642, 134, 6, 1, 1, 0, 1, 233238, 1935054, 469368, 14951, 237, 7, 1, 1, 0, 1, 5556725, 193926796, 119843417, 5289611, 50985, 388, 8, 1, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 10 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 binary matrices with k distinct columns and any number of nonzero rows with n ones in every column and rows in nonincreasing lexicographic order.

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   9     70       794         12055            233238 ...
  3 | 1 1 3  29    666     28344       1935054         193926796 ...
  4 | 1 1 4  68   3642    469368     119843417       53059346010 ...
  5 | 1 1 5 134  14951   5289611    4681749424     8639480647842 ...
  6 | 1 1 6 237  50985  46241343  134332244907   989821806791367 ...
  7 | 1 1 7 388 151901 333750928 3032595328876 85801167516707734 ...
     ...
The A(2,2) = 2 matrices are:
   [1 1]   [1 0]
   [1 0]   [1 0]
   [0 1]   [0 1]
           [0 1]
The corresponding set multipartitions are:
    {{1,2}, {1}, {2}},
    {{1}, {1}, {2}, {2}}.

Links

Crossrefs

Rows n=1..3 are A000012, A014500, A331389.
Columns k=0..3 are A000012, A000012, A001477, A331390.
Cf. A188392, A188445, A330942, A331039, A331160, A331161, A331391.

Programs

  • PARI
    WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(WeighT(v)[n], k)*k!/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k, q=Vec(exp(O(x*x^m) + intformal((x^n-1)/(1-x)))/(1-x))); if(n==0, k<=1, sum(j=0, m, my(s=0); forpart(p=j, s+=D(p, n, k), [1, n]); s*q[#q-j]))}

Formula

A(n, k) = Sum_{j=1..k} Stirling1(k, j)*A188392(n, j) for n, k >= 1.
A331391(n) = Sum_{d|n} A(n/d, d).

A331160 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of distinct nonzero rows with column sums n and rows in decreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 4, 2, 1, 0, 1, 27, 15, 2, 1, 0, 1, 266, 317, 44, 3, 1, 0, 1, 3599, 12586, 2763, 120, 4, 1, 0, 1, 62941, 803764, 390399, 21006, 319, 5, 1, 0, 1, 1372117, 75603729, 103678954, 10074052, 147296, 804, 6, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 10 2020

Keywords

Comments

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

Examples

			Array begins:
===================================================================
n\k | 0 1   2      3          4              5                6
----+--------------------------------------------------------------
  0 | 1 1   0      0          0              0                0 ...
  1 | 1 1   1      1          1              1                1 ...
  2 | 1 1   4     27        266           3599            62941 ...
  3 | 1 2  15    317      12586         803764         75603729 ...
  4 | 1 2  44   2763     390399      103678954      46278915417 ...
  5 | 1 3 120  21006   10074052    10679934500   21806685647346 ...
  6 | 1 4 319 147296  232165926   956594630508 8717423133548684 ...
  7 | 1 5 804 967829 4903530137 76812482919237 ...
      ...
The A(2,2) = 4 matrices are:
   [2 1]   [2 0]   [1 2]   [1 1]
   [0 1]   [0 2]   [1 0]   [1 0]
                           [0 1]

Links

Crossrefs

Rows n=1..3 are A000012, A331316, A331344
Columns k=0..2 are A000012, A000009, A331317.
Cf. A219585, A331039, A331126, A331161, A331315, A331318.

Programs

  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(EulerT(v)[n], k)*k!/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, k<=1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)}

Formula

A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A219585(n, j).
A331318(n) = Sum_{d|n} A(n/d, d).

A014501 Number of graphs with loops, having unlabeled (non-isolated) nodes and n labeled edges.

Original entry on oeis.org

1, 2, 7, 43, 403, 5245, 89132, 1898630, 49209846, 1517275859, 54669946851, 2269075206395, 107199678164289, 5707320919486026, 339510756324234931, 22400182888853554291, 1628654713107465602783, 129754625253841669625051
Offset: 0

Views

Author

Simon Plouffe, Gilbert Labelle (gilbert(AT)lacim.uqam.ca)

Keywords

References

  • G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Links

Crossrefs

Row n=2 of A331161.
Cf. A020554, A020555, A014500.

Formula

E.g.f.: exp(-1+x/2)*Sum((1+x)^binomial(n+1, 2)/n!, n=0..infinity) [probably in the Labelle paper]. - Vladeta Jovovic, Apr 27 2004

A331196 Number of nonnegative integer matrices with n distinct columns and any number of nonzero rows with each column sum being 3 and rows in nonincreasing lexicographic order.

Original entry on oeis.org

1, 3, 28, 599, 23243, 1440532, 131530132, 16720208200, 2837752812927, 622570020892599, 172077041175850521, 58679982298020226625, 24262822372018694983540, 11986886218243164848742812, 6987708088810202717378639087, 4754544525981425409034078100189
Offset: 0

Views

Author

Andrew Howroyd, Jan 11 2020

Keywords

Comments

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

Examples

			The a(2) = 28 matrices include 6 with 2 rows, 10 with 3 rows, 8 with 4 rows, 3 with 5 rows and 1 with 6 rows. The 16 with 2 or 3 rows are:
   [3 2]  [3 1]  [3 0]  [2 3]  [2 1]  [2 0]  [3 1]  [3 0]
   [0 1]  [0 2]  [0 3]  [1 0]  [1 2]  [1 3]  [0 1]  [0 2]
                                             [0 1]  [0 1]
.
   [2 2]  [2 1]  [2 1]  [2 0]  [2 0]  [2 0]  [1 3]  [1 2]
   [1 0]  [1 1]  [1 0]  [1 2]  [1 1]  [1 0]  [1 0]  [1 1]
   [0 1]  [0 1]  [0 2]  [0 1]  [0 2]  [0 3]  [1 0]  [1 0]

Links

Crossrefs

Row n=3 of A331161.
Cf. A322487.

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A322487(k).

A331197 Number of nonnegative integer matrices with 2 distinct columns and any number of nonzero rows with each column sum being n and rows in nonincreasing lexicographic order.

Original entry on oeis.org

0, 1, 7, 28, 104, 332, 1032, 2983, 8384, 22622, 59479, 151902, 379616, 927521, 2224100, 5236410, 12130549, 27669296, 62229605, 138095206, 302672402, 655627183, 1404598865, 2977830134, 6251059210, 12999297747, 26791987616, 54750232180, 110977385294, 223204454700, 445590973235
Offset: 0

Views

Author

Andrew Howroyd, Jan 11 2020

Keywords

Comments

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

Examples

			The a(2) = 7 matrices are:
   [2 1]   [2 0]   [1 2]   [1 1]   [2 0]   [1 0]   [1 0]
   [0 1]   [0 2]   [1 0]   [1 0]   [0 1]   [1 0]   [1 0]
                           [0 1]   [0 1]   [0 2]   [0 1]
                                                   [0 1]
See the example in A331197 for the a(3) = 28 case.

Links

Crossrefs

Column k=2 of A331161.
Cf. A000041, A002774, A331197.

Formula

a(n) = A002774(n) - A000041(n).

A330158 Number of nonnegative integer matrices with total sum n, distinct columns with equal sums and any number of nonzero rows in nonincreasing lexicographic order.

Original entry on oeis.org

1, 3, 4, 13, 8, 83, 16, 530, 630, 5620, 57, 119889, 102, 1901749, 1498322, 50091143, 298, 1649302673, 491, 54919254373, 16723808377, 2269075359300, 1256, 110133775311277, 26852941986, 5707320920415984, 622570195638208, 339574840900645411, 4566, 22572315004012650868
Offset: 1

Views

Author

Andrew Howroyd, Jan 11 2020

Keywords

Comments

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

Examples

			The a(4) = 13 matrices are:
  [1 0 0 0]  [1 0]  [1]  [1 1]  [2 0]  [1 0]  [2]
  [0 1 0 0]  [1 0]  [1]  [1 0]  [0 1]  [1 0]  [1]
  [0 0 1 0]  [0 1]  [1]  [0 1]  [0 1]  [0 2]  [1]
  [0 0 0 1]  [0 1]  [1]
.
  [2 1]  [2 0]  [1 2]  [3]  [2]  [4]
  [0 1]  [0 2]  [1 0]  [1]  [2]

Crossrefs

Cf. A331161.

Formula

a(n) = Sum_{d|n} A331161(n/d, d).
Showing 1-8 of 8 results.

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

Content is available under The OEIS End-User License Agreement.