Gender of Boolean functions

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
	💧	PrimulaDrop		1	0	2	94	32102	2147286238
♂	diag.	HalfGrass	~A058891	1	2	8	128	32768	2147483648
	🌊	HalfCrocus	A246418	1	3	13	159	33337	2147648859
	💧	HalfCrocusDrop		1	2	10	146	33178	2147615522
all	diag.	Lotus	A000371	2	2	10	218	64594	4294642034
	🌊	Grass	A001146	2	4	16	256	65536	4294967296
	💧	GrassDrop	~A111403	2	2	12	240	65280	4294901760
♂ − ♀	diag.	NonLotus	A005530	0	2	6	38	942	325262
	🌊	Hyacinth	2·A059301	0	2	10	62	1138	330422
	💧	HyacinthDrop		0	2	8	52	1076	329284

♀

Pascal 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

PascalDrop with columns multiplied by Violet

💧 triangle RobiniaDrop row sums PrimulaDrop
v a	0	1	2	3	4	5	sums
0	1 · 1 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

illustration

These images show the pink patterns also seen in the two 8×256 matrices.
They are ordered by adicity (vertical) and valency (horizontal).

♂

Pascal with columns multiplied by HalfGrass

🌊 triangle HalfCedar row sums HalfCrocus (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

PascalDrop with columns multiplied by HalfGrass

💧 triangle HalfCedarDrop row sums HalfCrocusDrop
v a	0	1	2	3	4	5	sums
0	1 · 1 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).

all

Pascal 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

PascalDrop with columns multiplied by Lotus

💧 triangle WillowDrop row sums GrassDrop (~A111403)
v a	0	1	2	3	4	5	sums
0	1 · 2 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 with columns multiplied by NonLotus

🌊 triangle Hickory row sums Hyacinth (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

PascalDrop with columns multiplied by NonLotus

💧 triangle HickoryDrop row sums HyacinthDrop
v a	0	1	2	3	4	5	sums
0	1 · 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
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

💧 PascalDrop
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

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.

sequences of Zhegalkin indices

The counting sequences are: Primula, Petunia, SmallCrocus, SmallHibiscus, Grass, Moss

a	♀			♂			all
a	🌊	💧	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.

oddacity and gender

The XOR of all members of a family is either the tautology or the contradiction. Where it is the tautology, the BF shall be called oddacious.
There are no evenacious males. So every evenacious BF is female, and every male BF is oddacious.

sequences
	0	1	2	3	4
female	1	1	3	97	32199
evenacious female	1	1	1	57	30537
oddacious female	0	0	2	40	1662
oddacious male	1	3	13	159	33337
oddacious	1	3	15	199	34999
all	2	4	16	256	65536

See triangles by weight, e.g. male, oddacious.

The following images show 3-ary Boolean functions. See also: Boolf prop/3-ary/oddacity & gender
The pattern of Zhegalkin indices for evenacious BF is similar to a Walsh matrix (with columns 3 and 7 exchanged). But that is specific to arity 3. So is the fact, that all non-trivial evenacious BF are balanced.

blunt (128)

female (97)

evenacious (57)

balanced (70)
These are the BF whose truth table has weight 4. Almost all evenacious BF are balanced. Only the contradiction is not. (This is not the case for other arities. See triangle by weight.)

oddacious female (40)

sharp (128)

male (159)

oddacious (199)