Gender of Boolean functions

Studies of Boolean functions

gender ratio for arities 0...5

A Boolean function shall be called male, iff its root is sharp (i. e. iff its compressed truth table has odd weight).
(Equivalently, it is female, iff after removing all repetitions, the weight of the truth table is still even.)

For positive arities, there are more males than females. The imbalance peaks for arity 2. For higher arities, the ratio is almost balanced.
The ratio is balanced for the infinite set of all Boolean functions. Both sets are countable, so there is a trivial bijection. But is there a meaningful bijection?

The number triangles in the following boxes show the numbers of Boolean functions by arity/adicity and valency.
Row indices on the left (🌊) are the arity, on the right (💧) the adicity. Entries on the left are the sums of columns on the right.
These triangles are based on Pascals triangle, with columns multiplied by consecutive entries of a sequence, which becomes the diagonal.

diagonals and row sums

name OEIS 0 1 2 3 4 5 ♀ diag. Violet ~ 2·A007537 1 0 2 90 31826 2147158386 🌊 Primula A246537 1 1 3 97 32199 2147318437 💧 Petunia 1 0 2 94 32102 2147286238 ♂ diag. SmallGrass ~A058891 1 2 8 128 32768 2147483648 🌊 SmallCrocus A246418 1 3 13 159 33337 2147648859 💧 SmallHibiscus 1 2 10 146 33178 2147615522 all diag. Lotus A000371 2 2 10 218 64594 4294642034 🌊 Grass A001146 2 4 16 256 65536 4294967296 💧 Moss ~A111403 2 2 12 240 65280 4294901760 ♂ − ♀ diag. Orchid A005530 0 2 6 38 942 325262 🌊 Mallow 2·A059301 0 2 10 62 1138 330422 💧 Mullein 0 2 8 52 1076 329284

♀

Pascal's triangle with columns multiplied by Violet. 🌊    triangle Robinia    row sums Primula (A246537) v a 0 1 2 3 4 5 sums 0 1 · 1 1 1 1 1 · 1 1 1 · 0 0 1 2 1 · 1 1 2 · 0 0 1 · 2 2 3 3 1 · 1 1 3 · 0 0 3 · 2 6 1 · 90 90 97 4 1 · 1 1 4 · 0 0 6 · 2 12 4 · 90 360 1 · 31826 31826 32199 5 1 · 1 1 5 · 0 0 10 · 2 20 10 · 90 900 5 · 31826 159130 1 · 2147158386 2147158386 2147318437 Offset Pascal's triangle with columns multiplied by Violet. 💧    triangle Acacia    row sums Petunia v a 0 1 2 3 4 5 sums 0   1 1 1   0 1 · 0 0 0 2   0 1 · 0 0 1 · 2 2 2 3   0 1 · 0 0 2 · 2 4 1 · 90 90 94 4   0 1 · 0 0 3 · 2 6 3 · 90 270 1 · 31826 31826 32102 5   0 1 · 0 0 4 · 2 8 6 · 90 540 4 · 31826 127304 1 · 2147158386 2147158386 2147286238

♂

Pascal's triangle with columns multiplied by SmallGrass. 🌊    triangle SmallCedar    row sums SmallCrocus (A246418) v a 0 1 2 3 4 5 sums 0 1 · 1 1 1 1 1 · 1 1 1 · 2 2 3 2 1 · 1 1 2 · 2 4 1 · 8 8 13 3 1 · 1 1 3 · 2 6 3 · 8 24 1 · 128 128 159 4 1 · 1 1 4 · 2 8 6 · 8 48 4 · 128 512 1 · 32768 32768 33337 5 1 · 1 1 5 · 2 10 10 · 8 80 10 · 128 1280 5 · 32768 163840 1 · 2147483648 2147483648 2147648859 Offset Pascal's triangle with columns multiplied by SmallGrass. 💧    triangle SmallCypress    row sums SmallHibiscus v a 0 1 2 3 4 5 sums 0   1 1 1   0 1 · 2 2 2 2   0 1 · 2 2 1 · 8 8 10 3   0 1 · 2 2 2 · 8 16 1 · 128 128 146 4   0 1 · 2 2 3 · 8 24 3 · 128 384 1 · 32768 32768 33178 5   0 1 · 2 2 4 · 8 32 6 · 128 768 4 · 32768 131072 1 · 2147483648 2147483648 2147615522 illustration These images show the blue patterns also seen in the two 8×256 matrices.   (Only a few of the 128 on the right side of the matrix are shown.) They are ordered by adicity (vertical) and valency (horizontal). Zhegalkin indices truth tables

all

Pascal's triangle with columns multiplied by Lotus. 🌊    triangle Willow    row sums Grass (A001146) v a 0 1 2 3 4 5 sums 0 1 · 2 2 2 1 1 · 2 2 1 · 2 2 4 2 1 · 2 2 2 · 2 4 1 · 10 10 16 3 1 · 2 2 3 · 2 6 3 · 10 30 1 · 218 218 256 4 1 · 2 2 4 · 2 8 6 · 10 60 4 · 218 872 1 · 64594 64594 65536 5 1 · 2 2 5 · 2 10 10 · 10 100 10 · 218 2180 5 · 64594 322970 1 · 4294642034 4294642034 4294967296 Offset Pascal's triangle with columns multiplied by Lotus. 💧    triangle Poplar    row sums Moss (~A111403) v a 0 1 2 3 4 5 sums 0   2 2 1   0 1 · 2 2 2 2   0 1 · 2 2 1 · 10 10 12 3   0 1 · 2 2 2 · 10 20 1 · 218 218 240 4   0 1 · 2 2 3 · 10 30 3 · 218 654 1 · 64594 64594 65280 5   0 1 · 2 2 4 · 10 40 6 · 218 1308 4 · 64594 258376 1 · 4294642034 4294642034 4294901760

♂ − ♀

Pascal's triangle with columns multiplied by Orchid. 🌊    triangle Chestnut    row sums Mallow (2·A059301) v a 0 1 2 3 4 5 sums 0 1 · 0 0 0 1 1 · 0 0 1 · 2 2 2 2 1 · 0 0 2 · 2 4 1 · 6 6 10 3 1 · 0 0 3 · 2 6 3 · 6 18 1 · 38 38 62 4 1 · 0 0 4 · 2 8 6 · 6 36 4 · 38 152 1 · 942 942 1138 5 1 · 0 0 5 · 2 10 10 · 6 60 10 · 38 380 5 · 942 4710 1 · 325262 325262 330422 Offset Pascal's triangle with columns multiplied by Orchid. 💧    triangle Chinkapin    row sums Mullein v a 0 1 2 3 4 5 sums 0   0 0 1   0 1 · 2 2 2 2   0 1 · 2 2 1 · 6 6 8 3   0 1 · 2 2 2 · 6 12 1 · 38 38 52 4   0 1 · 2 2 3 · 6 18 3 · 38 114 1 · 942 942 1076 5   0 1 · 2 2 4 · 6 24 6 · 38 228 4 · 942 3768 1 · 325262 325262 329284

Pascal's triangle
🌊    Pascal's triangle k n 0 1 2 3 4 5 sums 0 1 1 1 1 1 2 2 1 2 1 4 3 1 3 3 1 8 4 1 4 6 4 1 16 5 1 5 10 10 5 1 32 💧    offset Pascal's triangle k n 0 1 2 3 4 5 sums 0 1 1 1 0 1 1 2 0 1 1 2 3 0 1 2 1 4 4 0 1 3 3 1 8 5 0 1 4 6 4 1 16

Lotus (A000371, the number of dense BF) is surprisingly the sum of a signed version of Cedar.

SignedCedar and row sums Lotus

triangle SignedCedar    row sums Lotus (A000371)   🌊 k n 0 1 2 3 4 5 sums 0 1 · 2 2 2 1 1 · 2 −2 1 · 4 4 2 2 1 · 2 2 2 · 4 −8 1 · 16 16 10 3 1 · 2 −2 3 · 4 12 3 · 16 −48 1 · 256 256 218 4 1 · 2 2 4 · 4 −16 6 · 16 96 4 · 256 −1024 1 · 65536 65536 64594 5 1 · 2 −2 5 · 4 20 10 · 16 −160 10 · 256 2560 5 · 65536 −327680 1 · 4294967296 4294967296 4294642034 Lily   💧 0 2 1 0 2 8 3 208 4 64376 5 4294577440 small triangle SmallSignedCedar    row sums SmallLotus (A003465)   🌊 k n 0 1 2 3 4 5 sums 0 1 · 1 1 1 1 1 · 1 −1 1 · 2 2 1 2 1 · 1 1 2 · 2 −4 1 · 8 8 5 3 1 · 1 −1 3 · 2 6 3 · 8 −24 1 · 128 128 109 4 1 · 1 1 4 · 2 −8 6 · 8 48 4 · 128 −512 1 · 32768 32768 32297 5 1 · 1 −1 5 · 2 10 10 · 8 −80 10 · 128 1280 5 · 32768 −163840 1 · 2147483648 2147483648 2147321017 SmallLily   💧 0 1 1 0 2 4 3 104 4 32188 5 2147288720

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truth tables   (Zhegalkin matrix)
These matrices show the 97 females and 159 males among the first 256 Boolean functions. The matrix above shows the Zhegalkin indices. The Zhegalkin matrix below shows the truth tables. The male truth tables on the left side of the matrix contain repetitions, i.e. their pattern can be described by a shorter truth table.

atomvals   (bitwise OR)
The upper matrix shows the Zhegalkin indices. The one below shows atomvals of the corresponding Boolean functions. This is sequence  A327041, the bitwise ORs of the binary exponents seen above.

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sequences of Zhegalkin indices

The counting sequences are: Primula, Petunia, SmallCrocus, SmallHibiscus, Grass, Moss a ♀ ♂ all 🌊 💧 Ж 🌊 💧 Ж 🌊 💧 0 1 1 0, 1 1 1, 2 2 1 1 0 3 2 2, 3, 4 2 2 3 2 6, 7, 13 10 4, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16 12 3 97 94 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 159 146 16, 17, 32, 33, 34, 35, 48, 49, 50, 51, 64, 65, 68, 69, 80, 81, 84, 85, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256 240 4 32199 32102 258, 259, 260, 261   ...   32764, 32765, 32766, 32767 33337 33178 256, 257, 512, 513   ...   21840, 21841, 21844, 21845, 32768, 32769, 32770, 32771   ...   65532, 65533, 65534, 65535 65536 65280 females = [0, 6, 7, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127] males = [1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 32, 33, 34, 35, 48, 49, 50, 51, 64, 65, 68, 69, 80, 81, 84, 85, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255] 4-ary females on Pastebin The male Zhegalkin indices are not to be confused with A359527.