

A326966


BIInumbers of setsystems whose dual is a weak antichain.


13



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
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OFFSET

1,3


COMMENTS

A setsystem is a finite set of finite nonempty sets. The dual of a setsystem 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 setsystem with BIInumber 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 BIInumber. 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 BIInumber of {{2},{1,3}} is 18. Elements of a setsystem are sometimes called edges.


LINKS



EXAMPLE

The sequence of all setsystems whose dual is a weak antichain together with their BIInumbers 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

Setsystems whose dual is a weak antichain are counted by A326968, with covering case A326970, unlabeled version A326971, and unlabeled covering version A326973.
BIInumbers of setsystems whose dual is strict (T_0) are A326947.
BIInumbers of setsystems whose dual is a (strict) antichain (T_1) are A326979.


KEYWORD

nonn


AUTHOR



STATUS

approved



