A326966 BII-numbers of set-systems whose dual is a weak antichain.

0, 1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 15, 16, 18, 25, 27, 30, 31, 32, 33, 42, 43, 45, 47, 51, 52, 53, 54, 55, 59, 60, 61, 62, 63, 64, 75, 76, 79, 82, 91, 94, 95, 97, 107, 109, 111, 115, 116, 117, 118, 119, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 135

Offset

1,3

Comments

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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

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.

Links

Table of n, a(n) for n=1..62.

Example

The sequence of all set-systems whose dual is a weak antichain together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  16: {{1,3}}
  18: {{2},{1,3}}
  25: {{1},{3},{1,3}}
  27: {{1},{2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  32: {{2,3}}
  33: {{1},{2,3}}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Select[Range[0, 100], stableQ[dual[bpe/@bpe[#]], SubsetQ]&]

Crossrefs

Set-systems whose dual is a weak antichain are counted by A326968, with covering case A326970, unlabeled version A326971, and unlabeled covering version A326973.

BII-numbers of set-systems whose dual is strict (T_0) are A326947.

BII-numbers of set-systems whose dual is a (strict) antichain (T_1) are A326979.

Cf. A059523, A319639, A326961, A326965, A326969, A326974, A326975, A326978.

Sequence in context: A039106 A047549 A039074 * A326784 A047337 A039049

Adjacent sequences: A326963 A326964 A326965 * A326967 A326968 A326969

Keyword

nonn

Author

Gus Wiseman, Aug 13 2019

Status

approved