John Tyler Rascoe, <a href="/A326704/b326704_1.txt">Table of n, a(n) for n = 1..7580</a>

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John Tyler Rascoe, <a href="/A326704/b326704_1.txt">Table of n, a(n) for n = 1..7580</a>

John Tyler Rascoe, <a href="/A326704/a326704.py.txt">Python Program</a>.

(Python) # see linked program

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A binary index of n is any position of a 1 in its reversed binary digitsexpansion. 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 digits expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, it follows that the BII-number of {{2},{1,3}} is 18.

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Elements of a set-system are sometimes called edges. In an antichain of sets, no edge is a subset or superset of any other edge.

allocated for Gus WisemanBII-numbers of antichains of nonempty sets.

0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 16, 18, 20, 32, 33, 36, 48, 52, 64, 128, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288, 292, 304, 308, 320, 512, 513, 516, 520, 521, 524

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A binary index of n is any position of a 1 in its reversed binary digits. 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 digits (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, it follows that the BII-number of {{2},{1,3}} is 18.

In an antichain of sets, no edge is a subset or superset of any other.

The sequence of all antichains of nonempty sets together with their BII-numbers begins:

0: {}

1: {{1}}

2: {{2}}

3: {{1},{2}}

4: {{1,2}}

8: {{3}}

9: {{1},{3}}

10: {{2},{3}}

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

12: {{1,2},{3}}

16: {{1,3}}

18: {{2},{1,3}}

20: {{1,2},{1,3}}

32: {{2,3}}

33: {{1},{2,3}}

36: {{1,2},{2,3}}

48: {{1,3},{2,3}}

52: {{1,2},{1,3},{2,3}}

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];

Select[Range[100], stableQ[bpe/@bpe[#], SubsetQ]&]

Antichains of sets are counted by A000372.

Antichains of nonempty sets are counted by A014466.

MM-numbers of antichains of multisets are A316476.

BII-numbers of chains of nonempty sets are A326703.

Cf. A000120, A029931, A035327, A048793, A070939, A291166, A302521, A326031, A326675, A326701, A326702.

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Gus Wiseman, Jul 21 2019

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