A331638 Number of binary matrices with nonzero rows, a total of n ones and each column with the same number of ones and columns in nonincreasing lexicographic order.

1, 3, 5, 16, 17, 140, 65, 1395, 2969, 22176, 1025, 1050766, 4097, 13010328, 128268897, 637598438, 65537, 64864962683, 262145, 1676258452736, 28683380484257, 24908619669860, 4194305, 30567710172480050, 8756434134071649, 62128557507554504, 21271147396968151093

Offset: 1

Author

Andrew Howroyd, Jan 23 2020

Keywords

• nonn

Comments

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

From Gus Wiseman, Apr 03 2025: (Start)

Also the number of multiset partitions such that (1) the blocks together cover an initial interval of positive integers, (2) the blocks are sets of a common size, and (3) the block-sizes sum to n. For example, the a(1) = 1 through a(4) = 16 multiset partitions are:

{{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}}

{{1},{1}} {{1},{1},{1}} {{1,2},{1,2}}

{{1},{2}} {{1},{1},{2}} {{1,2},{1,3}}

{{1},{2},{2}} {{1,2},{2,3}}

{{1},{2},{3}} {{1,2},{3,4}}

{{1,3},{2,3}}

{{1,3},{2,4}}

{{1,4},{2,3}}

{{1},{1},{1},{1}}

{{1},{1},{1},{2}}

{{1},{1},{2},{2}}

{{1},{1},{2},{3}}

{{1},{2},{2},{2}}

{{1},{2},{2},{3}}

{{1},{2},{3},{3}}

{{1},{2},{3},{4}}

(End)

Links

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

Crossrefs

Cf. A330942, A331639.

For constant instead of strict blocks we have A034729.

Without equal sizes we have A116540 (normal set multipartitions).

Without strict blocks we have A317583.

For distinct instead of equal sizes we have A382428, non-strict blocks A326517.

For equal sums instead of sizes we have A382429, non-strict blocks A326518.

Normal multiset partitions: A255903, A255906, A317532, A382203, A382204, A382216.

Cf. A000110, A000670, A007716, A034691, A035310, A050320, A050326, A116539, A306319, A381718.

Formula

a(n) = Sum_{d|n} A330942(n/d, d).

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