Gentle sets of Boolean functions

Studies of Boolean functions

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ANF   matrix   twins

noble   gentle

smallest Zhegalkin index

This faction is gentle, because it contains the twins of its elements. The integers describe the truth tables. Compare this file, which is also shown below.

It is possible two extend the twin relationship to sets of Boolean functions. Two sets $A,B$ are twins, iff $a\in A\implies \operatorname {twin} (a)\in B$.
Sets that are their own twins shall be called gentle. (A set of nobles is gentle, but a gentle set does not need to contain nobles.)

Transposing the matrix positions of Zhegalkin indices often leads to gentle sets.
The upper row in the following table shows some sets of Zhegalkin indices. (The tt links lead to the corresponding truth tables.)
The lower row shows the transposed positions. Fields marked with the same color form gentle sets.

super-clan     (tt)	tribes     (tt)	quaestor 10     (tt)
transposed super-clan	transposed tribes	patron 72

The following table shows the eight Boolean functions marked pink in the transposed super-clan.
Twins are shown below each other, and the corresponding patrons are shown in the third row.

9	7	11	13
119	153	221	187
126	158	214	182

gentle factions

The table below shows representative Zhegalkin indices of gentle factions in their respective quadrants.
Representatives of noble factions are bold. Odd gentle factions (blue and green) seem to exist only for even arities.

arity	total size	size		evil	odious
1	2	1	even	0	2
2	6	2	even	0, 6	8, 14
		1	odd	3	11
3	12	6	even	0, 10, 30, 40, 46, 126	128, 138, 158, 168, 174, 254
4	224	76	even	0, 60, 136, 238, 510, 680, 718, 854, 874, 918, 938, 940, 990, 1632, 1678, 1690, 1696, 1768, 1950, 1954, 2176, 2216, 2278, 2454, 2474, 2752, 2770, 2812, 2992, 3030, 3070, 3808, 3826, 4086, 5760, 6014, 6018, 6078, 6280, 6298, 6304, 6324, 6558, 6648, 6830, 6856, 6876, 7134, 7138, 7158, 7912, 8190, 10432, 10492, 10654, 10732, 10750, 10920, 10926, 10952, 10958, 11464, 11470, 11482, 11508, 11766, 12286, 15572, 15592, 15830, 26752, 26812, 27606, 27626, 28384, 32766	32768, 32828, 32904, 33006, 33278, 33448, 33486, 33622, 33642, 33686, 33706, 33708, 33758, 34400, 34446, 34458, 34464, 34536, 34718, 34722, 34944, 34984, 35046, 35222, 35242, 35520, 35538, 35580, 35760, 35798, 35838, 36576, 36594, 36854, 38528, 38782, 38786, 38846, 39048, 39066, 39072, 39092, 39326, 39416, 39598, 39624, 39644, 39902, 39906, 39926, 40680, 40958, 43200, 43260, 43422, 43500, 43518, 43688, 43694, 43720, 43726, 44232, 44238, 44250, 44276, 44534, 45054, 48340, 48360, 48598, 59520, 59580, 60374, 60394, 61152, 65534
		36	odd	15, 175, 857, 907, 921, 1005, 1647, 1683, 1711, 1725, 2227, 2457, 2469, 2767, 2803, 3045, 4089, 6027, 6279, 6331, 6573, 6907, 7121, 7149, 7911, 10447, 10669, 10699, 10991, 11477, 11517, 11769, 15579, 26767, 27609, 28399	32783, 32943, 33625, 33675, 33689, 33773, 34415, 34451, 34479, 34493, 34995, 35225, 35237, 35535, 35571, 35813, 36857, 38795, 39047, 39099, 39341, 39675, 39889, 39917, 40679, 43215, 43437, 43467, 43759, 44245, 44285, 44537, 48347, 59535, 60377, 61167

3-ary gentle great factions

There are 20 great factions with arity 3. The 6 gentle ones among them are shown below.
The even factions in them (with red and yellow borders) are gentle, while the odd ones (with blue and green borders) are twins.
4 of the 12 gentle factions are not noble.

noble   (faction size 1)

noble   (faction size 3)
consul weight 1	consul weight 2

not noble   (faction size 6)