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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 (finite set of finite nonempty sets of positive integers) 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.
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 (finite set of finite nonempty sets of positive integers) 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.
allocated for Gus WisemanBII-numbers of VDD-normalized set-systems.
0, 1, 3, 4, 5, 7, 11, 15, 19, 20, 21, 23, 31, 33, 37, 51, 52, 53, 55, 63, 64, 65, 67, 68, 69, 71, 75, 79, 83, 84, 85, 87, 95, 97, 101, 115, 116, 117, 119, 127, 139, 143, 159, 179, 180, 181, 183, 191, 203, 207, 211, 212, 213, 215, 223, 225, 229, 243, 244, 245, 247
0,3
First differs from A330099 in lacking 545 and having 179, with corresponding set-systems 545: {{1},{2,3},{2,4}} and 179: {{1},{2},{4},{1,3},{2,3}}.
A set-system is a finite set of finite nonempty sets of positive integers.
We define the VDD (vertex-degrees decreasing) normalization of a set-system 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, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering of sets is first by length and then lexicographically.
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}}
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 (finite set of finite nonempty sets of positive integers) 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.
The sequence of all nonempty VDD-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}
15: {1}{2}{3}{12} 67: {1}{2}{123}
19: {1}{2}{13} 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} 79: {1}{2}{3}{12}{123}
33: {1}{23} 83: {1}{2}{13}{123}
37: {1}{12}{23} 84: {12}{13}{123}
51: {1}{2}{13}{23} 85: {1}{12}{13}{123}
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]], sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]], i}, {i, Length[Union@@m]}]], First[Sort[sysnorm[m, 1]]]];
sysnorm[m_, aft_]:=If[Length[Union@@m]<=aft, {m}, With[{mx=Table[Count[m, i, {2}], {i, Select[Union@@m, #>=aft&]}]}, Union@@(sysnorm[#, aft+1]&/@Union[Table[Map[Sort, m/.{par+aft-1->aft, aft->par+aft-1}, {0, 1}], {par, First/@Position[mx, Max[mx]]}]])]];
Select[Range[0, 100], Sort[bpe/@bpe[#]]==sysnorm[bpe/@bpe[#]]&]
Equals the image/fixed points of the idempotent sequence A330102.
A subset of A326754.
Non-isomorphic multiset partitions are A007716.
Unlabeled spanning set-systems by vertices are A055621.
Unlabeled set-systems by weight are A283877.
Cf. A000120, A000612, A048793, A070939, A300913, A319559, A321405, A326031, A330061, A330101.
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).
allocated
nonn,eigen
Gus Wiseman, Dec 04 2019
approved
editing
allocated for Gus Wiseman
allocated
approved