A331644 -id:A331644 - cp's OEIS frontend

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.

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

Andrew Howroyd, Jan 21 2020

			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]

Andrew Howroyd, Table of n, a(n) for n = 0..209

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.

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]))); }

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

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

Andrew Howroyd, Jan 23 2020

nonn

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

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

Row n=3 of A331568.

Cf. A331644.

A331704 Number of nonnegative integer matrices with n distinct columns and any number of distinct nonzero rows with column sums 2 and columns in decreasing lexicographic order.

Original entry on oeis.org

1, 1, 6, 46, 544, 7983, 144970, 3097825, 76494540, 2139610590, 66898897827, 2311748912745, 87494097274959, 3599356204576335, 159917091369687135, 7631292367127171222, 389282192196378927707, 21138914821756778420757, 1217459545430430305769230

Offset: 0

Andrew Howroyd, Jan 25 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.

			The a(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]

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row n=2 of A331570.

Cf. A331644, A331705.

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

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

Andrew Howroyd, Jan 25 2020

nonn

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

			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]

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row n=2 of A331572.

Cf. A331644, A331710.

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

cp's OEIS Frontend

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.

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

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A331704 Number of nonnegative integer matrices with n distinct columns and any number of distinct nonzero rows with column sums 2 and columns in decreasing lexicographic order.

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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.

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