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

Author

Gus Wiseman, Jul 29 2019

Keywords

• nonn
• base

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.

Examples

			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}}
		

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.

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.

Programs

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[#]]&]

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