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 1-9 of 9 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).

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).

A331572 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 and columns in nonincreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 7, 3, 1, 1, 8, 59, 45, 3, 1, 1, 16, 701, 1987, 271, 5, 1, 1, 32, 10460, 190379, 73567, 1244, 11, 1, 1, 64, 190816, 30474159, 58055460, 2451082, 7289, 13, 1, 1, 128, 4098997, 7287577611, 100171963518, 16557581754, 75511809, 40841, 19, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 21 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
----+-----------------------------------------------------
  0 | 1  1    1        1             1               1 ...
  1 | 1  1    2        4             8              16 ...
  2 | 1  1    7       59           701           10460 ...
  3 | 1  3   45     1987        190379        30474159 ...
  4 | 1  3  271    73567      58055460    100171963518 ...
  5 | 1  5 1244  2451082   16557581754 311419969572540 ...
  6 | 1 11 7289 75511809 4388702900099 ...
  ...
The A(2,2) = 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

Rows n=0..3 are A000012, A011782, A331709, A331710.
Columns k=0..3 are A000012, A032020, A331711, A331712.
Cf. A331315, A331568, A331569, A331570, A331571, A331713.

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 - 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))*A331568(n, j)/k!.
A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331570(n, j).
A331713(n) = Sum_{d|n} A(n/d, d).

A331567 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.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 0, 1, 1, 13, 6, 0, 1, 1, 75, 120, 0, 0, 1, 1, 541, 6174, 1104, 0, 0, 1, 1, 4683, 449520, 413088, 5040, 0, 0, 1, 1, 47293, 49686726, 329520720, 18481080, 0, 0, 0, 1, 1, 545835, 7455901320, 491236986720, 179438982360, 522481680, 0, 0, 0, 1
Offset: 0

Views

Author

Andrew Howroyd, Jan 20 2020

Keywords

Examples

			Array begins:
===============================================================
n\k | 0 1 2    3          4              5                6
----+----------------------------------------------------------
  0 | 1 1 1    1          1              1                1 ...
  1 | 1 1 3   13         75            541             4683 ...
  2 | 1 0 6  120       6174         449520         49686726 ...
  3 | 1 0 0 1104     413088      329520720     491236986720 ...
  4 | 1 0 0 5040   18481080   179438982360 3785623968170400 ...
  5 | 1 0 0    0  522481680 70302503250720 ...
  6 | 1 0 0    0 7875584640 ...
  ...
The A(2,2) = 6 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]

Links

Crossrefs

Rows n=1..3 are A000670, A331640, A331641.
Column k=5 is A331642.
Cf. A188445, A330942, A331568, A331569, A331571, A331643.

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]]++); 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, 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) = 0 for k > 0, n > 2^(k-1).
A(2^(k-1), k) = (2^k-1)! for k > 0.
A331643(n) = Sum_{d|n} A(n/d, d).

A331644 Number of nonnegative integer matrices with n columns and any number of distinct nonzero rows with column sums 2.

Original entry on oeis.org

1, 1, 13, 313, 14797, 1095601, 119621653, 17943752233, 3550217905597, 894951415465441, 280109128908963493, 106563994919500072153, 48431841090878234155597, 25916389944261925021739281, 16128299505539619187866596533, 11549663940282035980807700520073
Offset: 0

Views

Author

Andrew Howroyd, Jan 23 2020

Keywords

Examples

			The a(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

Row n=2 of A331568.
Cf. A331645.

A331645 Number of nonnegative integer matrices with n columns and any number of distinct nonzero rows with column sums 3.

Original entry on oeis.org

1, 3, 87, 11655, 4498191, 3611504823, 5192498314767, 12172382140406295, 43436006002867938351, 224018914608150321575223, 1603777845822934662938327247, 15430102163147097295950086754135, 194280602846166109497542618768659311, 3131120528310366432743101620841328437623
Offset: 0

Views

Author

Andrew Howroyd, Jan 23 2020

Keywords

Examples

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

Links

Crossrefs

Row n=3 of A331568.
Cf. A331644.

A331646 Number of nonnegative integer matrices with 2 columns and any number of distinct nonzero rows with column sums n.

Original entry on oeis.org

1, 3, 13, 87, 539, 2483, 14567, 81669, 402187, 2089413, 10344053, 49065413, 232941293, 1092284091, 5011510213, 22637050383, 100092546203, 439274499155, 1888145728847, 8058486875493, 33954688300771, 140741748208701, 579404121062837, 2354567807958773, 9481400353625789
Offset: 0

Views

Author

Andrew Howroyd, Jan 23 2020

Keywords

Examples

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

Links

Crossrefs

Column k=2 of A331568.
Cf. A331647, A331706, A331711.

Programs

  • PARI
    a(n)={subst(serlaplace(polcoef(polcoef(prod(i=0, n, prod(j=0, n, 1 + x^i*y^j*z + O(x*x^n) + O(y*y^n))), n, x), n, y)/(1 + z)), z, 1)}

A331647 Number of nonnegative integer matrices with 3 columns and any number of distinct nonzero rows with column sums n.

Original entry on oeis.org

1, 13, 313, 11655, 439779, 14699033, 453027131, 14035890205, 418049781331, 11878976385087, 326859368114097, 8708114475309677, 224507928640721933, 5621130329235752857, 136998397230581620801, 3253556714665728184671, 75415948071516657175683, 1708706073099458659898237
Offset: 0

Views

Author

Andrew Howroyd, Jan 23 2020

Keywords

Examples

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

Crossrefs

Column k=3 of A331568.
Cf. A331646.

A331648 Number of nonnegative integer matrices with distinct nonzero rows, total sum n and columns with equal sums.

Original entry on oeis.org

1, 4, 16, 91, 546, 5094, 47306, 561190, 7098943, 103345704, 1622632638, 28216141886, 526858348514, 10659286804538, 230286817182060, 5319206290975993, 130370767029136518, 3386434807625742926, 92801587319328412310, 2677967917777310591454, 81124837170900250181072
Offset: 1

Views

Author

Andrew Howroyd, Jan 24 2020

Keywords

Examples

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

Crossrefs

Cf. A331568.

Formula

a(n) = Sum_{d|n} A331568(d/n, d).
Showing 1-9 of 9 results.

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