A331278 Array read by antidiagonals: A(n,k) is the number of nonnegative integer matrices with k distinct columns and any number of nonzero rows with column sums n and columns in decreasing lexicographic order.

1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 12, 4, 1, 0, 1, 124, 124, 8, 1, 0, 1, 1800, 10596, 1280, 16, 1, 0, 1, 33648, 1764244, 930880, 13456, 32, 1, 0, 1, 769336, 484423460, 1849386640, 85835216, 143808, 64, 1, 0, 1, 20796960, 198461691404, 7798297361808, 2098356708016, 8206486848, 1556416, 128, 1

Offset

0,9

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 n-uniform k-block sets of multisets.

Links

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

Formula

A(n,k) = Sum_{j=0..n*k} binomial(binomial(j+n-1,n),k) * (Sum_{i=j..n*k} (-1)^(i-j)*binomial(i,j)).

A(n, k) = Sum_{j=0..k} Stirling1(k, j)*A316674(n, j)/k!.

A(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k-1, k-j)*A331315(n, j).

A(n, k) = A011782(n) * A331277(n, k) for k > 0.

A331395(n) = Sum_{d|n} A(n/d, d).

Example

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  2     12        124             1800               33648 ...
  3 | 1  4    124      10596          1764244           484423460 ...
  4 | 1  8   1280     930880       1849386640       7798297361808 ...
  5 | 1 16  13456   85835216    2098356708016  140094551934813712 ...
  6 | 1 32 143808 8206486848 2516779512105152 ...
  ...
The A(2,2) matrices are:
  [1 0]  [1 0]  [1 0]  [2 0]  [1 1]  [1 0]
  [1 0]  [0 1]  [0 1]  [0 1]  [1 0]  [1 1]
  [0 1]  [1 0]  [0 1]  [0 1]  [0 1]  [0 1]
  [0 1]  [0 1]  [1 0]
.
  [1 0]  [1 0]  [1 0]  [2 1]  [2 0]  [1 0]
  [1 0]  [0 2]  [0 1]  [0 1]  [0 2]  [1 2]
  [0 2]  [1 0]  [1 1]

Prog

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

Crossrefs

Rows n=1..2 are A000012, A173219.

Columns k=0..2 are A000012, A011782, A331396.

The version with binary entries is A331277.

The version with not necessarily distinct columns is A331315.

Cf. A316674 (unrestricted version), A330942, A331395.

Sequence in context: A326476 A247864 A370398 * A266632 A062154 A359619

Adjacent sequences: A331275 A331276 A331277 * A331279 A331280 A331281

Keyword

nonn,tabl

Author

Andrew Howroyd, Jan 13 2020

Status

approved