Integer sequences related to Boolean functions/clans

These sequences are related to clans, i.e. equivalence classes created by negation and permutation of arguments.

sequences by arity edit

arity n		0	1	2	3	4	5
number of functions with arity ≤ n	A001146	2	4	16	256	65 536	4 294 967 296
number of functions with arity = n	A000371	2	2	10	218	64 594	4 294 642 034
number of clans with arity ≤ n	A000616	2	3	6	22	402	1 228 158
number of clans with arity = n	A000618	2	1	3	16	380	1 227 756
number of blightless clans with arity ≤ n		2	0	3	14	357
number of blightless clans with arity = n		2	0	1	11	343

triangles edit

clans by weight edit

Triangle A039754 shows the number of clans with arity ≤ n and weight k.
k n	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	A000616	row 3
0	1	1																2
1	1	1	1															3
2	1	1	2	1	1													6
3	1	1	3	3	6	3	3	1	1									22
4	1	1	4	6	19	27	50	56	74	56	50	27	19	6	4	1	1	402

This triangle shows the number of clans with arity = n and weight k.
k n	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	A000618	row 3
0	1	1																2
1	0	1	0															1
2	0	1	1	1	0													3
3	0	1	2	3	4	3	2	1	0									16
4	0	1	3	6	16	27	47	56	68	56	47	27	16	6	3	1	0	380

blightless clans by weight edit

See Studies of Euler diagrams/blightless.

This triangle shows the number of clans with arity = n and weight k.
k n	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16		row 3
0	1	1																2
1	0	0	0															0
2	0	0	0	1	0													1
3	0	0	0	1	4	3	2	1	0									11
4	0	0	0	0	5	19	41	54	68	56	47	27	16	6	3	1	0	343

blightless clans by number of bundles edit

       0    1    2    3    4      blightless (row sums)      blighted    all (A000618)
        
0      2                               2                        0         2
1      0    0                          0                        1         1
2      0    0    1                     1                        2         3
3      0    6    2    3               11                        5        16
4      0  292   36   10    5         343                       37       380 

          298                        357                       45       402

Python fragment edit

calculation of blightless clans by weight

This code is part of a Python library called discrete helpers, which is not yet published.

from discretehelpers.boolf import Boolf
from discretehelpers.binv import Binv


n = 4

bec_indices_by_weight = []
for weight in range(2 ** n + 1):
    bec_indices_by_weight.append(set())
    
for i in range(2 ** 2 ** n):
    binv = Binv(intval=i, length=2 ** n)
    boolf = Boolf(binv)
    if boolf.sarity == n and boolf.is_blightless:
        weight = binv.weight
        bec_index = boolf.bec_index_from_fingerprint()
        bec_indices_by_weight[weight].add(bec_index)
        
[len(_) for _ in bec_indices_by_weight]  # [0, 0, 0, 0, 5, 19, 41, 54, 68, 56, 47, 27, 16, 6, 3, 1, 0]