A326947 BII-numbers of T_0 set-systems.

0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 69, 70, 71, 73, 74, 75, 77, 78

Offset: 1

Author

Gus Wiseman, Aug 08 2019

Keywords

• nonn
• base

Comments

The dual of a set-system has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated 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.

Examples

			The sequence of all T_0 set-systems together with their BII numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{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}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
		

Links

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

Crossrefs

T_0 set-systems are counted by A326940, with unlabeled version A326946.

Cf. A059201, A316978, A319559, A319564, A326939, A326941, A326949.

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
TZQ[sys_]:=UnsameQ@@dual[sys];
Select[Range[0,100],TZQ[bpe/@bpe[#]]&]

Python

from itertools import count, chain, islice
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):
        a,b,s = [bin_i(k) for k in bin_i(n)],[],set()
        for i in {i for i in chain.from_iterable(a)}:
            b.append([])
            for j in range(len(a)):
                if i in a[j]:
                    b[-1].append(j)
            s.add(tuple(b[-1]))
        if len(s) == len(b):
            yield n
A326947_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Jul 25 2024