

A330099


BIInumbers of bruteforce normalized setsystems.


19



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, 191, 203, 207, 223, 255, 267, 271, 275, 276, 277, 279, 287, 307, 308, 309, 311, 319, 331
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OFFSET

1,3


COMMENTS

First differs from A330100 in having 545 and lacking 179, with corresponding setsystems 545: {{1},{2,3},{2,4}} and 179: {{1},{2},{4},{1,3},{2,3}}.
A setsystem is a finite set of finite nonempty sets of positive integers.
We define the bruteforce normalization of a setsystem 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 least representative, where the ordering of sets is first by length and then lexicographically.
For example, 156 is the BIInumber of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BIInumbers:
Bruteforce: 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 setsystem with BIInumber n to be obtained by taking the binary indices of each binary index of n. Every setsystem 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.
There are A055621(n) entries m such that A326702(m) = n, where A326702(k) is the number of covered vertices in the setsystem with BIInumber k.
There are A283877(n) entries m such that A326031(m) = n, where A326031(k) is the weight of the setsystem with BIInumber k.


LINKS

Table of n, a(n) for n=1..61.


EXAMPLE

The sequence of all nonempty bruteforce normalized setsystems together with their BIInumbers 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}


MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]], brute[m/.Rule@@@Table[{(Union@@m)[[i]], i}, {i, Length[Union@@m]}]], First[Sort[brute[m, 1]]]];
brute[m_, 1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i, p[[i]]}, {i, Length[p]}])], {p, Permutations[Union@@m]}];
Select[Range[0, 100], Sort[bpe/@bpe[#]]==brute[bpe/@bpe[#]]&]


CROSSREFS

Equals the image/fixed points of the idempotent sequence A330101.
Nonisomorphic multiset partitions are A007716.
Unlabeled spanning setsystems by span are A055621.
Unlabeled spanning setsystems by weight are A283877.
Cf. A000612, A300913, A321405, A330061, A330102, A330105.
Other fixed points:
 Bruteforce: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (setsystems).
 Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (setsystems).
 VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (setsystems).
 MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (setsystems).
 BII: A330109 (setsystems).
Sequence in context: A076497 A137950 A330100 * A046413 A285224 A322991
Adjacent sequences: A330096 A330097 A330098 * A330100 A330101 A330102


KEYWORD

nonn,eigen


AUTHOR

Gus Wiseman, Dec 02 2019


STATUS

approved



