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A330110
BII-numbers of lexicographically normalized set-systems.
19
0, 1, 3, 4, 5, 7, 11, 13, 15, 20, 21, 23, 31, 33, 37, 45, 52, 53, 55, 63, 64, 65, 67, 68, 69, 71, 75, 77, 79, 84, 85, 87, 95, 97, 101, 109, 116, 117, 119, 127, 139, 141, 143, 149, 151, 159, 165, 173, 181, 183, 191, 193, 195, 197, 199, 203, 205, 207, 213, 215
OFFSET
1,3
COMMENTS
First differs from A330099 in having 13 and lacking 19.
First differs from A330123 in having 141 and lacking 180, with corresponding set-systems 141: {{1},{3},{4},{1,2}} and 180: {{4},{1,2},{1,3},{2,3}}.
We define the lexicographic normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the lexicographically least of these representatives.
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 set-system 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.
For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
Brute-force: 2067: {{1},{2},{1,3},{3,4}}
Lexicographic: 165: {{1},{4},{1,2},{2,3}}
VDD: 525: {{1},{3},{1,2},{2,4}}
MM: 270: {{2},{3},{1,2},{1,4}}
BII: 150: {{2},{4},{1,2},{1,3}}
EXAMPLE
The sequence of all nonempty lexicographically normalized set-systems together with their BII-numbers begins:
1: {1} 52: {12}{13}{23}
3: {1}{2} 53: {1}{12}{13}{23}
4: {12} 55: {1}{2}{12}{13}{23}
5: {1}{12} 63: {1}{2}{3}{12}{13}{23}
7: {1}{2}{12} 64: {123}
11: {1}{2}{3} 65: {1}{123}
13: {1}{3}{12} 67: {1}{2}{123}
15: {1}{2}{3}{12} 68: {12}{123}
20: {12}{13} 69: {1}{12}{123}
21: {1}{12}{13} 71: {1}{2}{12}{123}
23: {1}{2}{12}{13} 75: {1}{2}{3}{123}
31: {1}{2}{3}{12}{13} 77: {1}{3}{12}{123}
33: {1}{23} 79: {1}{2}{3}{12}{123}
37: {1}{12}{23} 84: {12}{13}{123}
45: {1}{3}{12}{23} 85: {1}{12}{13}{123}
CROSSREFS
A subset of A326754.
Unlabeled covering set-systems counted by vertices are A055621.
Unlabeled set-systems counted by weight are A283877.
BII-weight is A326031.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).
Sequence in context: A082378 A046642 A377317 * A330123 A354711 A140826
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 05 2019
STATUS
approved