A331461 Array read by antidiagonals: A(n,k) is the number of nonequivalent binary matrices with k columns and any number of nonzero rows with n ones in every column up to permutation of rows and columns.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 5, 8, 4, 1, 1, 1, 7, 23, 16, 5, 1, 1, 1, 11, 66, 93, 30, 6, 1, 1, 1, 15, 212, 652, 332, 50, 7, 1, 1, 1, 22, 686, 6369, 6414, 1062, 80, 8, 1, 1, 1, 30, 2389, 79568, 226041, 56712, 3117, 120, 9, 1, 1, 1, 42, 8682, 1256425, 12848128, 7295812, 441881, 8399, 175, 10, 1, 1

Offset

0,8

Comments

A(n,k) is the number of non-isomorphic set multipartitions (multiset of sets) with k parts each part has size n.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..152

Formula

A306018(n) = Sum_{d|n} A(n/d, d).

Example

Array begins:
===========================================================
n\k | 0 1 2   3    4       5          6              7
----+-----------------------------------------------------
  0 | 1 1 1   1    1       1          1              1 ...
  1 | 1 1 2   3    5       7         11             15 ...
  2 | 1 1 3   8   23      66        212            686 ...
  3 | 1 1 4  16   93     652       6369          79568 ...
  4 | 1 1 5  30  332    6414     226041       12848128 ...
  5 | 1 1 6  50 1062   56712    7295812     1817321457 ...
  6 | 1 1 7  80 3117  441881  195486906   200065951078 ...
  7 | 1 1 8 120 8399 3006771 4298181107 17131523059493 ...
  ...
The A(2,3) = 8 matrices are:
  [1 0 0]  [1 1 0]  [1 1 1]  [1 1 0]  [1 1 0]  [1 1 1]  [1 1 0]  [1 1 1]
  [1 0 0]  [1 0 0]  [1 0 0]  [1 1 0]  [1 0 1]  [1 1 0]  [1 0 1]  [1 1 1]
  [0 1 0]  [0 1 0]  [0 1 0]  [0 0 1]  [0 1 0]  [0 0 1]  [0 1 1]
  [0 1 0]  [0 0 1]  [0 0 1]  [0 0 1]  [0 0 1]
  [0 0 1]  [0 0 1]
  [0 0 1]

Prog

(PARI) \\ See A304942 for Blocks
T(n, k)={Blocks(k, n*k, n)}
{ for(n=0, 7, for(k=0, 6, print1(T(n, k), ", ")); print) }

Crossrefs

Rows n=0..6 are A000012, A000041, A050535, A050913, A058783, A058784, A058785.

Columns k=0..4 are A000012, A000012, A000027(n+1), A002624, A331720.

Cf. A188392, A262809, A304942, A306018, A330942, A331485, A331508, A331509, A331510.

Sequence in context: A130580 A220708 A110541 * A238016 A185812 A152798

Adjacent sequences: A331458 A331459 A331460 * A331462 A331463 A331464

Keyword

nonn,tabl

Author

Andrew Howroyd, Jan 18 2020

Status

approved