4-ary Boolean functions; clans in rational order
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In this table the Boolean functions are sorted as fractions in colex order.
This order is independent of arity (unlike that of integers in A227723).
The number of BECs by actual arity is found in A000618. (Compare
A000616 for acual or smaller arity.)
n 0, 1, 2, 3, 4 arity 2^n 1, 2, 4, 8, 16 period length A000618(n) 2, 1, 3, 16, 380 number of BECs of arity = n A000616(n) 2, 3, 6, 22, 402 number of BECs of arity ≤ n
The same information in a Python dict: https://pastebin.com/q2d4aWVE
Map from the 65536 functions to the BEC index numbers: https://pastebin.com/3fBShT7M
The 168 monotonic functions belonging to 30 BECs: https://pastebin.com/JUeKsasG
I accidentally reflected the period in the table below. E.g. 1/3 in binary is .0101... - not .1010. I will fix it later.
c | a | weight | so na |
no mo |
rational number value | f/s |
s/b ary |
f/b ary |
s/b | f/b | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
num over den | period | num | den | dec | 0 | 1 | 2 | 3 | 4 | 0 | 1 | 2 | 3 | 4 | |||||||||||||||||
0 | 1 | 0 | 0 / 1 | 0 | 0 | ![]() |
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0000000000000000 | ![]() |
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0 | 1 | 0.000000 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
1 | 0 | 0 | 1 / 1 | 16 | s | 0 | ![]() |
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1111111111111111 | ![]() |
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1 | 1 | 1.000000 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 1 / 2 | 8 | s | 0 | ![]() |
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1010101010101010 | ![]() |
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1 | 3 | 0.333333 | 2 | 1 | 2 | 1 | 2 | 3 | 4 | 2 | 4 | 6 | 8 | |||
3 | 2 | 1 / 2 | 8 | s | 4 | ![]() |
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0110011001100110 | ![]() |
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2 | 5 | 0.400000 | 2 | 1 | 2 | 1 | 3 | 6 | 2 | 6 | 12 | |||||
4 | 5 | 2 | 1 / 4 | 4 | s | 0 | ![]() |
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1000100010001000 | ![]() |
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1 | 15 | 0.066667 | 4 | 1 | 4 | 1 | 3 | 6 | 4 | 12 | 24 | ||||
5 | 4 | 2 | 3 / 4 | 12 | 0 | ![]() |
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1110111011101110 | ![]() |
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7 | 15 | 0.466667 | 4 | 1 | 4 | 1 | 3 | 6 | 4 | 12 | 24 | |||||
6 | 3 | 1 / 2 | 8 | 2 | ![]() |
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0111100001111000 | ![]() |
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2 | 17 | 0.117647 | 8 | 3 | 24 | 3 | 12 | 24 | 96 | ||||||||
7 | 3 | 1 / 2 | 8 | s | 4 | ![]() |
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1001011010010110 | ![]() |
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7 | 17 | 0.411765 | 2 | 1 | 2 | 1 | 4 | 2 | 8 | |||||||
8 | 13 | 3 | 1 / 4 | 4 | s | 2 | ![]() |
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0010100000101000 | ![]() |
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4 | 51 | 0.078431 | 4 | 3 | 12 | 3 | 12 | 12 | 48 | ||||||
9 | 12 | 3 | 3 / 8 | 6 | 2 | ![]() |
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1001100010011000 | ![]() |
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5 | 51 | 0.098039 | 8 | 3 | 24 | 3 | 12 | 24 | 96 | |||||||
10 | 11 | 3 | 3 / 8 | 6 | 0 | ![]() |
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1100010011000100 | ![]() |
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7 | 51 | 0.137255 | 8 | 3 | 24 | 3 | 12 | 24 | 96 | |||||||
11 | 10 | 3 | 5 / 8 | 10 | 0 | ![]() |
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1110110011101100 | ![]() |
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11 | 51 | 0.215686 | 8 | 3 | 24 | 3 | 12 | 24 | 96 | |||||||
12 | 9 | 3 | 5 / 8 | 10 | 2 | ![]() |
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0111011001110110 | ![]() |
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22 | 51 | 0.431373 | 8 | 3 | 24 | 3 | 12 | 24 | 96 | |||||||
13 | 8 | 3 | 3 / 4 | 12 | 2 | ![]() |
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1011111010111110 | ![]() |
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25 | 51 | 0.490196 | 4 | 3 | 12 | 3 | 12 | 12 | 48 | |||||||
14 | 16 | 3 | 1 / 4 | 4 | s | 2 | ![]() |
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0001100000011000 | ![]() |
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8 | 85 | 0.094118 | 4 | 1 | 4 | 1 | 4 | 4 | 16 | ||||||
15 | 3 | 1 / 2 | 8 | 2 | ![]() |
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1101100011011000 | ![]() |
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9 | 85 | 0.105882 | 8 | 3 | 24 | 3 | 12 | 24 | 96 | ||||||||
16 | 14 | 3 | 3 / 4 | 12 | 2 | ![]() |
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0111111001111110 | ![]() |
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42 | 85 | 0.494118 | 4 | 1 | 4 | 1 | 4 | 4 | 16 | |||||||
17 | 21 | 3 | 1 / 8 | 2 | s | 0 | ![]() |
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1000000010000000 | ![]() |
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1 | 255 | 0.003922 | 8 | 1 | 8 | 1 | 4 | 8 | 32 | ||||||
18 | 20 | 3 | 3 / 8 | 6 | 2 | ![]() |
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0110100001101000 | ![]() |
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22 | 255 | 0.086275 | 8 | 1 | 8 | 1 | 4 | 8 | 32 | |||||||
19 | 3 | 1 / 2 | 8 | 0 | ![]() |
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1110100011101000 | ![]() |
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23 | 255 | 0.090196 | 8 | 1 | 8 | 1 | 4 | 8 | 32 | ||||||||
20 | 18 | 3 | 5 / 8 | 10 | 2 | ![]() |
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1101011011010110 | ![]() |
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107 | 255 | 0.419608 | 8 | 1 | 8 | 1 | 4 | 8 | 32 | |||||||
21 | 17 | 3 | 7 / 8 | 14 | 0 | ![]() |
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1111111011111110 | ![]() |
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127 | 255 | 0.498039 | 8 | 1 | 8 | 1 | 4 | 8 | 32 | |||||||
22 | 4 | 1 / 2 | 8 | 1 | ![]() |
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0111111110000000 | ![]() |
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2 | 257 | 0.007782 | 16 | 4 | 64 | 4 | 64 | ||||||||||
23 | 4 | 1 / 2 | 8 | 3 | ![]() |
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1001111101100000 | ![]() |
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7 | 257 | 0.027237 | 8 | 12 | 96 | 12 | 96 | ||||||||||
24 | 4 | 1 / 2 | 8 | 3 | ![]() |
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0001111111100000 | ![]() |
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8 | 257 | 0.031128 | 16 | 12 | 192 | 12 | 192 | ||||||||||
25 | 4 | 1 / 2 | 8 | 4 | ![]() |
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1001011101101000 | ![]() |
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23 | 257 | 0.089494 | 16 | 4 | 64 | 4 | 64 | ||||||||||
26 | 4 | 1 / 2 | 8 | 4 | ![]() |
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0001011111101000 | ![]() |
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24 | 257 | 0.093385 | 8 | 4 | 32 | 4 | 32 | ||||||||||
27 | 4 | 1 / 2 | 8 | 2 | ![]() |
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1110011100011000 | ![]() |
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25 | 257 | 0.097276 | 8 | 4 | 32 | 4 | 32 | ||||||||||
28 | 4 | 1 / 2 | 8 | 3 | ![]() |
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0110011110011000 | ![]() |
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26 | 257 | 0.101167 | 16 | 12 | 192 | 12 | 192 | ||||||||||
29 | 4 | 1 / 2 | 8 | 4 | ![]() |
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0010011111011000 | ![]() |
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28 | 257 | 0.108949 | 8 | 12 | 96 | 12 | 96 | ||||||||||
30 | 4 | 1 / 2 | 8 | 4 | ![]() |
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1000011101111000 | ![]() |
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31 | 257 | 0.120623 | 8 | 6 | 48 | 6 | 48 | ||||||||||
31 | 4 | 1 / 2 | 8 | s | 5 | ![]() |
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0110100110010110 | ![]() |
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106 | 257 | 0.412451 | 2 | 1 | 2 | 1 | 2 | |||||||||
32 | 95 | 4 | 1 / 4 | 4 | 1 | ![]() |
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0010101010000000 | ![]() |
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4 | 771 | 0.005188 | 16 | 12 | 192 | 12 | 192 | |||||||||
33 | 94 | 4 | 5 / 16 | 5 | 2 | ![]() |
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1001010110000000 | ![]() |
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5 | 771 | 0.006485 | 16 | 12 | 192 | 12 | 192 | |||||||||
34 | 109 | 4 | 5 / 16 | 5 | 2 | ![]() |
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1100101001000000 | ![]() |
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7 | 771 | 0.009079 | 16 | 12 | 192 | 12 | 192 | |||||||||
35 | 96 | 4 | 7 / 16 | 7 | 1 | ![]() |
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1110010111000000 | ![]() |
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11 | 771 | 0.014267 | 16 | 24 | 384 | 24 | 384 | |||||||||
36 | 44 | 4 | 7 / 16 | 7 | 1 | ![]() |
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1111001001100000 | ![]() |
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19 | 771 | 0.024643 | 16 | 12 | 192 | 12 | 192 | |||||||||
37 | 97 | 4 | 5 / 16 | 5 | 3 | ![]() |
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0010010101100000 | ![]() |
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20 | 771 | 0.025940 | 16 | 12 | 192 | 12 | 192 | |||||||||
38 | 67 | 4 | 7 / 16 | 7 | 2 | ![]() |
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0111001011100000 | ![]() |
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22 | 771 | 0.028534 | 16 | 12 | 192 | 12 | 192 | |||||||||
39 | 98 | 4 | 7 / 16 | 7 | 2 | ![]() |
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1100010111100000 | ![]() |
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23 | 771 | 0.029831 | 16 | 12 | 192 | 12 | 192 | |||||||||
40 | 78 | 4 | 5 / 16 | 5 | 1 | ![]() |
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1011001000010000 | ![]() |
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25 | 771 | 0.032425 | 16 | 12 | 192 | 12 | 192 | |||||||||
41 | 82 | 4 | 5 / 16 | 5 | 2 | ![]() |
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0011001010010000 | ![]() |
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28 | 771 | 0.036316 | 16 | 24 | 384 | 24 | 384 | |||||||||
42 | 91 | 4 | 3 / 8 | 6 | 1 | ![]() |
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1101001001010000 | ![]() |
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31 | 771 | 0.040208 | 16 | 24 | 384 | 24 | 384 | |||||||||
43 | 93 | 4 | 3 / 8 | 6 | 1 | ![]() |
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0101001011010000 | ![]() |
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34 | 771 | 0.044099 | 16 | 12 | 192 | 12 | 192 | |||||||||
44 | 36 | 4 | 9 / 16 | 9 | 1 | ![]() |
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1111100111010000 | ![]() |
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35 | 771 | 0.045396 | 16 | 12 | 192 | 12 | 192 | |||||||||
45 | 90 | 4 | 5 / 16 | 5 | 1 | ![]() |
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1001001000110000 | ![]() |
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37 | 771 | 0.047990 | 16 | 6 | 96 | 6 | 96 | |||||||||
46 | 57 | 4 | 7 / 16 | 7 | 1 | ![]() |
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0111100100110000 | ![]() |
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38 | 771 | 0.049287 | 16 | 12 | 192 | 12 | 192 | |||||||||
47 | 65 | 4 | 1 / 2 | 8 | 1 | ![]() |
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1011100110110000 | ![]() |
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41 | 771 | 0.053178 | 16 | 12 | 192 | 12 | 192 | |||||||||
48 | 72 | 4 | 7 / 16 | 7 | 2 | ![]() |
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0011100101110000 | ![]() |
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44 | 771 | 0.057069 | 16 | 24 | 384 | 24 | 384 | |||||||||
49 | 50 | 4 | 7 / 16 | 7 | 2 | ![]() |
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0110001011110000 | ![]() |
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46 | 771 | 0.059663 | 16 | 12 | 192 | 12 | 192 | |||||||||
50 | 49 | 4 | 9 / 16 | 9 | 2 | ![]() |
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1101100111110000 | ![]() |
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47 | 771 | 0.060960 | 16 | 12 | 192 | 12 | 192 | |||||||||
51 | 110 | 4 | 5 / 16 | 5 | 2 | ![]() |
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1100001001001000 | ![]() |
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55 | 771 | 0.071336 | 16 | 24 | 384 | 24 | 384 | |||||||||
52 | 111 | 4 | 5 / 16 | 5 | 2 | ![]() |
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0100001011001000 | ![]() |
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58 | 771 | 0.075227 | 16 | 24 | 384 | 24 | 384 | |||||||||
53 | 99 | 4 | 1 / 2 | 8 | 1 | ![]() |
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1110100111001000 | ![]() |
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59 | 771 | 0.076524 | 16 | 12 | 192 | 12 | 192 | |||||||||
54 | 107 | 4 | 1 / 4 | 4 | s | 2 | ![]() |
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1000001000101000 | ![]() |
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61 | 771 | 0.079118 | 4 | 4 | 16 | 4 | 16 | ||||||||
55 | 100 | 4 | 3 / 8 | 6 | 3 | ![]() |
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0110100100101000 | ![]() |
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62 | 771 | 0.080415 | 8 | 12 | 96 | 12 | 96 | |||||||||
56 | 101 | 4 | 7 / 16 | 7 | 1 | ![]() |
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1010100110101000 | ![]() |
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65 | 771 | 0.084306 | 16 | 12 | 192 | 12 | 192 | |||||||||
57 | 46 | 4 | 9 / 16 | 9 | 1 | ![]() |
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1111110001101000 | ![]() |
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67 | 771 | 0.086900 | 16 | 12 | 192 | 12 | 192 | |||||||||
58 | 102 | 4 | 3 / 8 | 6 | 3 | ![]() |
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0010100101101000 | ![]() |
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68 | 771 | 0.088197 | 8 | 12 | 96 | 12 | 96 | |||||||||
59 | 69 | 4 | 9 / 16 | 9 | 1 | ![]() |
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0111110011101000 | ![]() |
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70 | 771 | 0.090791 | 16 | 6 | 96 | 6 | 96 | |||||||||
60 | 103 | 4 | 1 / 2 | 8 | 1 | ![]() |
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1100100111101000 | ![]() |
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71 | 771 | 0.092088 | 16 | 12 | 192 | 12 | 192 | |||||||||
61 | 79 | 4 | 7 / 16 | 7 | 3 | ![]() |
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1011110000011000 | ![]() |
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73 | 771 | 0.094682 | 16 | 24 | 384 | 24 | 384 | |||||||||
62 | 104 | 4 | 5 / 16 | 5 | 3 | ![]() |
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0100100100011000 | ![]() |
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74 | 771 | 0.095979 | 16 | 12 | 192 | 12 | 192 | |||||||||
63 | 87 | 4 | 7 / 16 | 7 | 3 | ![]() |
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0011110010011000 | ![]() |
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76 | 771 | 0.098573 | 16 | 24 | 384 | 24 | 384 | |||||||||
64 | 105 | 4 | 3 / 8 | 6 | 2 | ![]() |
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1000100110011000 | ![]() |
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77 | 771 | 0.099870 | 8 | 6 | 48 | 6 | 48 | |||||||||
65 | 47 | 4 | 1 / 2 | 8 | 1 | ![]() |
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1101110001011000 | ![]() |
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79 | 771 | 0.102464 | 16 | 12 | 192 | 12 | 192 | |||||||||
66 | 71 | 4 | 1 / 2 | 8 | 2 | ![]() |
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0101110011011000 | ![]() |
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82 | 771 | 0.106355 | 16 | 12 | 192 | 12 | 192 | |||||||||
67 | 38 | 4 | 9 / 16 | 9 | 2 | ![]() |
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1111000111011000 | ![]() |
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83 | 771 | 0.107652 | 16 | 12 | 192 | 12 | 192 | |||||||||
68 | 89 | 4 | 7 / 16 | 7 | 3 | ![]() |
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1001110000111000 | ![]() |
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85 | 771 | 0.110246 | 16 | 12 | 192 | 12 | 192 | |||||||||
69 | 59 | 4 | 7 / 16 | 7 | 1 | ![]() |
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0111000100111000 | ![]() |
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86 | 771 | 0.111543 | 16 | 6 | 96 | 6 | 96 | |||||||||
70 | 81 | 4 | 7 / 16 | 7 | 4 | ![]() |
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0001110010111000 | ![]() |
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88 | 771 | 0.114137 | 16 | 12 | 192 | 12 | 192 | |||||||||
71 | 66 | 4 | 1 / 2 | 8 | 2 | ![]() |
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1011000110111000 | ![]() |
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89 | 771 | 0.115435 | 16 | 12 | 192 | 12 | 192 | |||||||||
72 | 48 | 4 | 9 / 16 | 9 | 2 | ![]() |
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1110110001111000 | ![]() |
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91 | 771 | 0.118029 | 16 | 24 | 384 | 24 | 384 | |||||||||
73 | 86 | 4 | 3 / 8 | 6 | 1 | ![]() |
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0100110011000100 | ![]() |
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106 | 771 | 0.137484 | 8 | 12 | 96 | 12 | 96 | |||||||||
74 | 106 | 4 | 7 / 16 | 7 | 2 | ![]() |
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1110000111000100 | ![]() |
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107 | 771 | 0.138781 | 16 | 12 | 192 | 12 | 192 | |||||||||
75 | 80 | 4 | 3 / 8 | 6 | 2 | ![]() |
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1011010000010100 | ![]() |
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121 | 771 | 0.156939 | 16 | 24 | 384 | 24 | 384 | |||||||||
76 | 88 | 4 | 3 / 8 | 6 | 3 | ![]() |
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0011010010010100 | ![]() |
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124 | 771 | 0.160830 | 16 | 24 | 384 | 24 | 384 | |||||||||
77 | 92 | 4 | 7 / 16 | 7 | 0 | ![]() |
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1101010001010100 | ![]() |
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127 | 771 | 0.164721 | 16 | 12 | 192 | 12 | 192 | |||||||||
78 | 40 | 4 | 11 / 16 | 11 | 1 | ![]() |
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1111111011010100 | ![]() |
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131 | 771 | 0.169909 | 16 | 12 | 192 | 12 | 192 | |||||||||
79 | 61 | 4 | 9 / 16 | 9 | 3 | ![]() |
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0111111000110100 | ![]() |
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134 | 771 | 0.173800 | 16 | 24 | 384 | 24 | 384 | |||||||||
80 | 75 | 4 | 5 / 8 | 10 | 2 | ![]() |
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1011111010110100 | ![]() |
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137 | 771 | 0.177691 | 16 | 24 | 384 | 24 | 384 | |||||||||
81 | 70 | 4 | 9 / 16 | 9 | 4 | ![]() |
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0011111001110100 | ![]() |
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140 | 771 | 0.181582 | 16 | 12 | 192 | 12 | 192 | |||||||||
82 | 41 | 4 | 11 / 16 | 11 | 2 | ![]() |
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1101111011110100 | ![]() |
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143 | 771 | 0.185473 | 16 | 24 | 384 | 24 | 384 | |||||||||
83 | 4 | 1 / 2 | 8 | 2 | ![]() |
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0110111000101100 | ![]() |
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158 | 771 | 0.204929 | 16 | 24 | 384 | 24 | 384 | ||||||||||
84 | 108 | 4 | 9 / 16 | 9 | 1 | ![]() |
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![]() |
1010111010101100 | ![]() |
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161 | 771 | 0.208820 | 16 | 24 | 384 | 24 | 384 | |||||||||
85 | 4 | 1 / 2 | 8 | 3 | ![]() |
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![]() |
0010111001101100 | ![]() |
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164 | 771 | 0.212711 | 16 | 24 | 384 | 24 | 384 | ||||||||||
86 | 73 | 4 | 5 / 8 | 10 | 1 | ![]() |
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![]() |
1100111011101100 | ![]() |
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167 | 771 | 0.216602 | 8 | 12 | 96 | 12 | 96 | |||||||||
87 | 63 | 4 | 9 / 16 | 9 | 3 | ![]() |
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0111011000111100 | ![]() |
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182 | 771 | 0.236057 | 16 | 24 | 384 | 24 | 384 | |||||||||
88 | 76 | 4 | 5 / 8 | 10 | 3 | ![]() |
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