A326979 - OEIS

A326979

BII-numbers of T_1 set-systems.

0, 1, 2, 3, 7, 8, 9, 10, 11, 15, 25, 27, 30, 31, 42, 43, 45, 47, 51, 52, 53, 54, 55, 59, 60, 61, 62, 63, 75, 79, 91, 94, 95, 107, 109, 111, 115, 116, 117, 118, 119, 123, 124, 125, 126, 127, 128, 129, 130, 131, 135, 136, 137, 138, 139, 143, 153, 155, 158, 159

(list; graph; refs; listen; history; text; internal format)

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}}. The T_1 condition means that the dual is a (strict) antichain, meaning that none of its edges is a 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..60.

EXAMPLE

The sequence of all T_1 set-systems together with their BII-numbers begins:

0: {}

1: {{1}}

2: {{2}}

3: {{1},{2}}

7: {{1},{2},{1,2}}

8: {{3}}

9: {{1},{3}}

10: {{2},{3}}

11: {{1},{2},{3}}

15: {{1},{2},{1,2},{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}}

42: {{2},{3},{2,3}}

43: {{1},{2},{3},{2,3}}

45: {{1},{1,2},{3},{2,3}}

47: {{1},{2},{1,2},{3},{2,3}}

51: {{1},{2},{1,3},{2,3}}

52: {{1,2},{1,3},{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], UnsameQ@@dual[bpe/@bpe[#]]&&stableQ[dual[bpe/@bpe[#]], SubsetQ]&]

CROSSREFS

BII-numbers of T_0 set-systems are A326947.

T_1 set-systems are counted by A326965, A326961 (covering), A326972 (unlabeled), and A326974 (unlabeled covering).

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

Cf. A059201, A059523, A319639, A326970, A326973, A326976, A326977.

Sequence in context: A299497 A047534 A353651 * A326874 A118374 A344624

Adjacent sequences: A326976 A326977 A326978 * A326980 A326981 A326982

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 13 2019

STATUS

approved