cp's OEIS Frontend

Also the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations, with no equal rows (or alternatively, with no equal columns).

Also the number of non-isomorphic multiset partitions of weight n with no equivalent vertices. In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second.

Also the number of nonnegative integer square symmetric matrices with sum of elements equal to n, under row and column permutations.

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity.

A cover of a set is a T_0-cover if for every two distinct points of the set there exists a member (block) of the cover containing one but not the other point.

From Gus Wiseman, Aug 13 2019: (Start)

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges). For example, the a(2) = 4 covers are:

{{1},{2}}

{{1},{1,2}}

{{2},{1,2}}

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

(End)

From Altug Alkan, Dec 22 2015: (Start)

a(p^k) == 1 (mod p) and a(n + p) == a(n + 1) (mod p) for all primes p.

a(0+19) == a(0+1) (mod 19) or a(19^1) == 1 (mod 19), that is, a(19) mod 19 = 1.

a(2+17) == a(2+1) (mod 17). So a(19) == 19 (mod 17), that is, a(19) mod 17 = 2.

a(6+13) == a(6+1) (mod 13). So a(19) == 6129859 (mod 13), that is, a(19) mod 13 = 8.

a(8+11) == a(8+1) (mod 11). So a(19) == 44511042511 (mod 11), that is, a(19) mod 11 = 1.

a(12+7) == a(12+1) (mod 7). So a(19) == 171850728381587059351 (mod 7), that is, a(19) mod 7 = 1.

a(14+5) == a(14+1) (mod 5). So a(19) == 77567171020440688353049939 (mod 5), that is, a(19) mod 5 = 4.

a(16+3) == a(16+1) (mod 3). So a(19) == 122152541250295322862941281269151 (mod 3), that is, a(19) mod 3 = 1.

a(17+2) == a(17+1) (mod 2). So a(19) mod 2 = 1.

In conclusion, a(19) is a number of the form 2*3*5*7*11*13*17*19*n - 1615151, that is, 9699690*n - 1615151.

Additionally, for n > 0, note that the last digit of a(n) has the simple periodic pattern: 1,3,9,9,1,3,9,9,1,3,9,9,...

(End)

Number of rank n sublattices of the Boolean algebra B_n. - Kevin Long, Nov 20 2018

a(n) is the number of n X n idempotent Boolean relation matrices (A121337) that have rank n. - Geoffrey Critzer, Aug 16 2023

a(19) == 163279579 (mod 232792560). - Didier Garcia, Feb 06 2025

Also number of fixed effects ANOVA models with n factors, which may be both crossed and nested.

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. For an integer partition the T_0 condition means the dual of the multiset partition obtained by factoring each part into prime numbers is strict (no repeated blocks).

Also the number of integer partitions of n with no equivalent primes. In an integer partition, two primes are equivalent if each part has in its prime factorization the same multiplicity of both primes. For example, in (6,5) the primes {2,3} are equivalent. See A316978 for more examples.

The dual of a set-system has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).