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 1-10 of 10 results.

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).

A331571 Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of distinct 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, 0, 1, 1, 4, 3, 0, 1, 1, 8, 23, 0, 0, 1, 1, 16, 290, 184, 0, 0, 1, 1, 32, 4298, 17488, 840, 0, 0, 1, 1, 64, 79143, 2780752, 771305, 0, 0, 0, 1, 1, 128, 1702923, 689187720, 1496866413, 21770070, 0, 0, 0, 1, 1, 256, 42299820, 236477490418, 5261551562405, 585897733896, 328149360, 0, 0, 0, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 20 2020

Keywords

Comments

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

Examples

			Array begins:
===============================================================
n\k | 0 1 2   3         4               5                 6
----+----------------------------------------------------------
  0 | 1 1 1   1         1               1                 1 ...
  1 | 1 1 2   4         8              16                32 ...
  2 | 1 0 3  23       290            4298             79143 ...
  3 | 1 0 0 184     17488         2780752         689187720 ...
  4 | 1 0 0 840    771305      1496866413     5261551562405 ...
  5 | 1 0 0   0  21770070    585897733896 30607728081550686 ...
  6 | 1 0 0   0 328149360 161088785679360 ...
  ...
The A(2,2) = 3 matrices are:
   [1 1]  [1 0]  [1 0]
   [1 0]  [1 1]  [0 1]
   [0 1]  [0 1]  [1 1]

Links

Crossrefs

Rows n=0..4 are A000012, A011782, A060090, A060491, A331652.
Cf. A330942, A331567, A331569, A331570, A331572, A331653.

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 - 1, 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)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, 1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }

Formula

A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A331567(n, j)/k!.
A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331569(n, j).
A(n, k) = 0 for k > 0, n > 2^(k-1).
A331653(n) = Sum_{d|n} A(n/d, d).

A331569 Array read by antidiagonals: 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 columns in decreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 3, 0, 1, 0, 1, 17, 0, 0, 1, 0, 1, 230, 184, 0, 0, 1, 0, 1, 3264, 16936, 840, 0, 0, 1, 0, 1, 60338, 2711904, 768785, 0, 0, 0, 1, 0, 1, 1287062, 675457000, 1493786233, 21770070, 0, 0, 0, 1, 0, 1, 31900620, 232383728378, 5254074934990, 585810653616, 328149360, 0, 0, 0, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 20 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 k-block n-uniform T_0 set systems without isolated vertices.

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 0 3  17       230            3264             60338 ...
  3 | 1 0 0 184     16936         2711904         675457000 ...
  4 | 1 0 0 840    768785      1493786233     5254074934990 ...
  5 | 1 0 0   0  21770070    585810653616 30604798810581906 ...
  6 | 1 0 0   0 328149360 161087473081920 ...
  ...
The A(2,2) = 3 matrices are:
   [1 1]  [1 0]  [1 0]
   [1 0]  [1 1]  [0 1]
   [0 1]  [0 1]  [1 1]

Links

Crossrefs

Rows n=1..4 are A000012, A331649, A094631, A331650.
Cf. A330942, A331039, A331567, A331570, A331571, A331572, A331651.

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)/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)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, k<=1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }

Formula

A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A331567(n, j)/k!.
A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A331571(n, j).
A331651(n) = Sum_{d|n} A(n/d, d).

A331570 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 columns in decreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 6, 3, 1, 0, 1, 46, 42, 3, 1, 0, 1, 544, 1900, 268, 5, 1, 0, 1, 7983, 184550, 73028, 1239, 11, 1, 0, 1, 144970, 29724388, 57835569, 2448599, 7278, 13, 1, 0, 1, 3097825, 7137090958, 99940181999, 16550232235, 75497242, 40828, 19, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 21 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.

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  1    6       46           544               7983 ...
  3 | 1  3   42     1900        184550           29724388 ...
  4 | 1  3  268    73028      57835569        99940181999 ...
  5 | 1  5 1239  2448599   16550232235    311353753947045 ...
  6 | 1 11 7278 75497242 4388476386528 896320470282357104 ...
  ...
The A(2,2) = 6 matrices are:
   [1 1]  [1 0]  [1 0]  [2 1]  [2 0]  [1 0]
   [1 0]  [1 1]  [0 1]  [0 1]  [0 2]  [1 2]
   [0 1]  [0 1]  [1 1]

Links

Crossrefs

Rows 1..3 are A000012, A331704, A331705.
Columns k=0..3 are A000012, A032020, A331706, A331707.
Cf. A331315, A331568, A331569, A331571, A331572, A331708.

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)/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)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, k<=1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }

Formula

A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A331568(n, j)/k!.
A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A331572(n, j).
A331708(n) = Sum_{d|n} A(n/d, d).

A331568 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k columns and any number of distinct nonzero rows with column sums n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 13, 13, 3, 1, 1, 75, 313, 87, 3, 1, 1, 541, 14797, 11655, 539, 5, 1, 1, 4683, 1095601, 4498191, 439779, 2483, 11, 1, 1, 47293, 119621653, 3611504823, 1390686419, 14699033, 14567, 13, 1, 1, 545835, 17943752233, 5192498314767, 12006713338683, 397293740555, 453027131, 81669, 19, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 21 2020

Keywords

Examples

			Array begins:
================================================================
n\k | 0  1     2         3               4                 5
----+-----------------------------------------------------------
  0 | 1  1     1         1               1                 1 ...
  1 | 1  1     3        13              75               541 ...
  2 | 1  1    13       313           14797           1095601 ...
  3 | 1  3    87     11655         4498191        3611504823 ...
  4 | 1  3   539    439779      1390686419    12006713338683 ...
  5 | 1  5  2483  14699033    397293740555 37366422896708825 ...
  6 | 1 11 14567 453027131 105326151279287 ...
  ...
The A(2,2) = 13 matrices are:
   [1 1]  [1 1]  [1 0]  [1 0]  [0 1]  [0 1]
   [1 0]  [0 1]  [1 1]  [0 1]  [1 1]  [1 0]
   [0 1]  [1 0]  [0 1]  [1 1]  [1 0]  [1 1]
.
   [2 1]  [2 0]  [1 2]  [1 0]  [0 2]  [0 1]  [2 2]
   [0 1]  [0 2]  [1 0]  [1 2]  [2 0]  [2 1]

Links

Crossrefs

Rows n=0..3 are A000012, A000670, A331644, A331645.
Columns k=0..3 are A000012, A032020, A331646, A331647.
Cf. A219585, A331315, A331567, A331570, A331572, A331648.

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]]++); EulerT(v)[n]^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)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, 1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }

Formula

A331648(n) = Sum_{d|n} A(n/d, d).

A331709 Number of nonnegative integer matrices with n columns and any number of distinct nonzero rows with column sums 2 and columns in nonincreasing lexicographic order.

Original entry on oeis.org

1, 1, 7, 59, 701, 10460, 190816, 4098997, 101523139, 2847014941, 89188733362, 3086888531896, 116982554539226, 4817701229837597, 214245144969388823, 10231975601963484807, 522307300100522413863, 28379690860876378241538, 1635356759307997113784404
Offset: 0

Views

Author

Andrew Howroyd, Jan 25 2020

Keywords

Comments

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

Examples

			The a(2) = 7 matrices are:
   [1 1]  [1 0]  [1 0]  [2 1]  [2 0]  [1 0]  [2 2]
   [1 0]  [1 1]  [0 1]  [0 1]  [0 2]  [1 2]
   [0 1]  [0 1]  [1 1]

Links

Crossrefs

Row n=2 of A331572.
Cf. A331644, A331710.

Formula

a(n) = (1/n!)*Sum_{k=0..n} abs(Stirling1(n, k)) * A331644(k).

A331710 Number of nonnegative integer matrices with n columns and any number of distinct nonzero rows with column sums 3 and columns in nonincreasing lexicographic order.

Original entry on oeis.org

1, 3, 45, 1987, 190379, 30474159, 7287577611, 2436916655479, 1085776582252197, 621663581843731535, 444746638465623906738, 388773810523972862494769, 407727415097448880517583006, 505268334502886263575349570013, 730406898110019766652845079212010
Offset: 0

Views

Author

Andrew Howroyd, Jan 25 2020

Keywords

Comments

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

Examples

			The a(1) = 3 matrices are:
  [2]  [1]  [3]
  [1]  [2]

Links

Crossrefs

Row n=3 of A331572.
Cf. A331645, A331709.

Formula

a(n) = (1/n!)*Sum_{k=0..n} abs(Stirling1(n, k)) * A331645(k).

A331711 Number of nonnegative integer matrices with 2 columns and any number of distinct nonzero rows with column sums n and columns in nonincreasing lexicographic order.

Original entry on oeis.org

1, 2, 7, 45, 271, 1244, 7289, 40841, 201103, 1044720, 5172055, 24532739, 116470697, 546142112, 2505755203, 11318525367, 50046273319, 219637249886, 944072864849, 4029243438335, 16977344151163, 70370874105726, 289702060533067, 1177283903981765, 4740700176816041
Offset: 0

Views

Author

Andrew Howroyd, Jan 25 2020

Keywords

Comments

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

Examples

			The a(2) = 2 matrices are:
   [1 1]  [1 0]  [1 0]  [2 1]  [2 0]  [1 0]  [2 2]
   [1 0]  [1 1]  [0 1]  [0 1]  [0 2]  [1 2]
   [0 1]  [0 1]  [1 1]

Crossrefs

Column k=2 of A331572.
Cf. A032020, A331646, A331712.

Formula

a(n) = (A331646(n) + A032020(n)) / 2.

A331712 Number of nonnegative integer matrices with 3 columns and any number of distinct nonzero rows with column sums n and columns in nonincreasing lexicographic order.

Original entry on oeis.org

1, 4, 59, 1987, 73567, 2451082, 75511809, 2339355873, 69675164655, 1979830442230, 54476566524395, 1451352437084341, 37417988223257669, 936855055418767566, 22833066207602691971, 542259452455606556087, 12569324678636155802527, 284784345516796080566156, 6313245421004111720679497
Offset: 0

Views

Author

Andrew Howroyd, Jan 25 2020

Keywords

Comments

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

Examples

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

Crossrefs

Column k=3 of A331572.
Cf. A331711.

A331713 Number of nonnegative integer matrices with pairwise distinct nonzero rows, the total sum of elements equal n, the same sum of elements in each column, and the columns coming in nonincreasing lexicographic order.

Original entry on oeis.org

1, 3, 7, 18, 21, 147, 77, 1119, 2270, 12273, 1089, 464200, 4229, 4148223, 32941976, 159812905, 66153, 10211281004, 263321, 205923976532, 2439257062679, 3086915165084, 4199061, 1526043270802300, 311419986358517, 4817701809545446, 621665561741298232, 1898993198795136209
Offset: 1

Views

Author

Andrew Howroyd, Jan 25 2020

Keywords

Comments

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

Examples

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

Crossrefs

Cf. A331572.

Formula

a(n) = Sum_{d|n} A331572(n/d, d).
Showing 1-10 of 10 results.

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