A331278 -id:A331278 - cp's OEIS frontend

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.

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

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 n-uniform k-block multisets of multisets.

			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]

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

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.

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)))}

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).

A173219 G.f.: A(x) = Sum_{n>=0} (1 + x)^(n(n+1)/2) / 2^(n+1).

Original entry on oeis.org

1, 2, 12, 124, 1800, 33648, 769336, 20796960, 648841680, 22945907520, 907036108432, 39631833652320, 1896696894062880, 98669609894805600, 5543804125505195040, 334563594743197602272, 21583554094995765302592

Offset: 0

Paul D. Hanna, Mar 05 2010

nonn

a(n) is the number of nonnegative integer matrices with n distinct columns and any number of nonzero rows with 2 ones in every column and columns in decreasing lexicographic order. - Andrew Howroyd, Jan 15 2020

Andrew Howroyd, Table of n, a(n) for n = 0..100

Row n=2 of A331278.

Cf. A121251, A173217, A173218.

Table[Sum[StirlingS1[n, j] * Sum[Binomial[j, s]*HurwitzLerchPhi[1/2, -j - s, 0], {s, 0, j}] / 2^(j+1), {j, 0, n}] / n!, {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2019 *)

{a(n)=local(A=sum(m=0,n^2+100,(1+x +O(x^(n+2)))^(m*(m+1)/2)/2^(m+1)));round(polcoeff(A,n))}

a(n) = A265937(n)/2. - Vaclav Kotesovec, Oct 08 2019
a(n) ~ 2^n * n^n / (2^(log(2)/4) * log(2)^(2*n+1) * exp(n)). - Vaclav Kotesovec, Oct 08 2019
a(n) = 2*A121251(n) for n > 0. - Andrew Howroyd, Jan 15 2020

A331277 Array read by antidiagonals: 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 columns in decreasing lexicographic order.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 6, 1, 1, 0, 1, 62, 31, 1, 1, 0, 1, 900, 2649, 160, 1, 1, 0, 1, 16824, 441061, 116360, 841, 1, 1, 0, 1, 384668, 121105865, 231173330, 5364701, 4494, 1, 1, 0, 1, 10398480, 49615422851, 974787170226, 131147294251, 256452714, 24319, 1, 1

Offset: 0

Andrew Howroyd, Jan 13 2020

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 labeled n-uniform hypergraphs with k edges and no isolated vertices. When n=2 these objects are graphs.

			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    6        62            900        16824      384668 ...
  3 | 1 1   31      2649         441061    121105865 49615422851 ...
  4 | 1 1  160    116360      231173330 974787170226 ...
  5 | 1 1  841   5364701   131147294251 ...
  6 | 1 1 4494 256452714 78649359753286 ...
  ...
The A(2,2) = 6 matrices are:
   [1 0]  [1 0]  [1 0]  [1 1]  [1 0]  [1 0]
   [1 0]  [0 1]  [0 1]  [1 0]  [1 1]  [0 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, A121251, A136245.
Columns k=0..3 are A000012, A000012, A047665, A137219.
The version with nonnegative integer entries is A331278.
The version with not necessarily distinct columns is A330942.
Cf. A262809 (unrestricted version), A331315, A331639.

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

A(n, k) = Sum_{j=0..n*k} binomial(binomial(j,n),k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).
A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A262809(n, j)/k!.
A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A330942(n, j).
A331639(n) = Sum_{d|n} A(n/d, d).

A331396 Number of nonnegative integer matrices with 2 distinct columns and any number of nonzero rows with column sums n and columns in decreasing lexicographic order.

Original entry on oeis.org

1, 12, 124, 1280, 13456, 143808, 1556416, 17006592, 187207936, 2072947712, 23063919616, 257634271232, 2887544049664, 32456082440192, 365710391885824, 4129672996585472, 46721752249729024, 529486122704437248, 6009576477811277824, 68299997524116111360

Offset: 1

Andrew Howroyd, Jan 15 2020

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 1..200

Column k=2 of A331278.

Cf. A011782, A047665, A052141, A331397.

seq(n)={Vec(1/(4*sqrt(1 - 12*x + 4*x^2 + O(x*x^n))) - 1/(4*(1-2*x)))}

a(n) = (A052141(n) - A011782(n))/2.
G.f.: 1/(4*sqrt(1 - 12*x + 4*x^2)) - 1/(4*(1-2*x)).
a(n) = A011782(n) * A047665(n).

A331395 Number of nonnegative integer matrices with total sum n, nonzero rows and distinct columns each with the same sum with columns in decreasing lexicographic order.

Original entry on oeis.org

1, 3, 5, 21, 17, 281, 65, 3209, 10853, 47617, 1025, 3610317, 4097, 22361569, 570275061, 2515267681, 65537, 229801424781, 262145, 10805763650257, 114290395025669, 39654899669089, 4194305, 150858428803209493, 140094551951590929, 98672497472409697

Offset: 1

Andrew Howroyd, Jan 15 2020

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A331278.

a(n) = Sum_{d|n} A331278(n/d, d).

a(p) = 2^(p-1) + 1 for prime p.

cp's OEIS Frontend

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.

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A173219 G.f.: A(x) = Sum_{n>=0} (1 + x)^(n(n+1)/2) / 2^(n+1).

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A331277 Array read by antidiagonals: 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 columns in decreasing lexicographic order.

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A331396 Number of nonnegative integer matrices with 2 distinct columns and any number of nonzero rows with column sums n and columns in decreasing lexicographic order.

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A331395 Number of nonnegative integer matrices with total sum n, nonzero rows and distinct columns each with the same sum with columns in decreasing lexicographic order.

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