A326872 BII-numbers of connectedness systems.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 24, 25, 26, 27, 32, 33, 34, 35, 40, 41, 42, 43, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset

1,3

Comments

We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping 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 finite set of finite nonempty sets 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 enumeration of these set-systems by number of covered vertices is given by A326870.

Links

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Example

The sequence of all connectedness systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  18: {{2},{1,3}}
  19: {{1},{2},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  26: {{2},{3},{1,3}}
  27: {{1},{2},{3},{1,3}}
  32: {{2,3}}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
connsysQ[eds_]:=SubsetQ[eds, Union@@@Select[Tuples[eds, 2], Intersection@@#!={}&]];
Select[Range[0, 100], connsysQ[bpe/@bpe[#]]&]

Prog

(Python)
from itertools import count, islice, combinations
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen():
    for n in count(0):
        E, f = [bin_i(k) for k in bin_i(n)], 0
        for i in combinations(E, 2):
            if list(set(i[0])|set(i[1])) not in E and len(set(i[0])&set(i[1])) > 0:
                f += 1
                break
        if f < 1:
            yield n
A326872_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Mar 07 2025

Crossrefs

Connectedness systems are counted by A326866, with unlabeled version A326867.

The case without singletons is A326873.

The connected case is A326879.

Set-systems closed under union are counted by A102896, with BII numbers A326875.

Cf. A029931, A048793, A072446, A326031, A326749, A326753, A326870, A326876, A326879.

Sequence in context: A057605 A247759 A377912 * A396694 A110548 A249815

Adjacent sequences: A326869 A326870 A326871 * A326873 A326874 A326875

Keyword

nonn,base

Author

Gus Wiseman, Jul 29 2019

Status

approved