

A330100


BIInumbers of VDDnormalized 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, 179, 180, 181, 183, 191, 203, 207, 211, 212, 213, 215, 223, 225, 229, 243, 244, 245, 247
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OFFSET

0,3


COMMENTS

First differs from A330099 in lacking 545 and having 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 VDD (vertexdegrees decreasing) 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, then selecting only the representatives whose vertexdegrees are weakly decreasing, and finally taking the least of these representatives, where the ordering of sets is first by length and then lexicographically.
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 (finite set of finite nonempty sets of positive integers) 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.
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}}


LINKS

Table of n, a(n) for n=0..60.


EXAMPLE

The sequence of all nonempty VDDnormalized 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];
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+aft1>aft, aft>par+aft1}, {0, 1}], {par, First/@Position[mx, Max[mx]]}]])]];
Select[Range[0, 100], Sort[bpe/@bpe[#]]==sysnorm[bpe/@bpe[#]]&]


CROSSREFS

Equals the image/fixed points of the idempotent sequence A330102.
A subset of A326754.
Nonisomorphic multiset partitions are A007716.
Unlabeled spanning setsystems counted by vertices are A055621.
Unlabeled setsystems counted by weight are A283877.
Cf. A000120, A000612, A048793, A070939, A300913, A319559, A321405, A326031, A330061, A330101.
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: A095880 A076497 A137950 * A330099 A046413 A285224
Adjacent sequences: A330097 A330098 A330099 * A330101 A330102 A330103


KEYWORD

nonn,eigen


AUTHOR

Gus Wiseman, Dec 04 2019


STATUS

approved



