Families of Boolean functions

Studies of Boolean functions

Boolean functions belong to the same family, when they can be transformed into each other by negating arguments.
The size of a family is always a power of two. The maximal size is 2adicity, i.e. the period length of the truth table.
(While the size of faction and clan increases with the chosen arity, the size of a family is fixed.)

The simplest property of a family is its weight, i.e. the number of true entries in each truth table.
An important property is the parity of the weight. Those with odd weight shall be called sharp, those with even weight blunt.
(It is the same as the parity of the quaestor weight.)

A family belongs to a clan (where the arguments are not just negated, but also permuted).
Together with its complement, it forms a super-family. Adding a half-complement forms an ultra-family.

## family matrix

[edit | edit source]

A family matrix is a bitwise XOR table with integers replaced by binary values. Each row (or column) is the truth table of a BF in the family.

bitwise XOR table with labels on the left showing how the arguments are negated | red family matrix of a 4-ary function, containing each function twice | eight members of a family as Venn diagrams and rows of a family matrix |

two family matrices of 3-ary seals, one with 4 functions (left), the other with 2 (right)

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## sequences

[edit | edit source]

arity to count

|  | arity  n {\displaystyle n}

|

0 | 1 | 2 | 3 | 4 | 5
| Saffron | balanced families  | 0 | 1 | 3 | 14 | 870 | 18796230
( 2 2 n − 1 ) ( 2 n − 1 ) 2 n {\displaystyle {\frac {(2^{2^{n-1}})(2^{n}-1)}{2^{n}}}} | NonDubSage | closed families  | 0 | 1 | 3 | 14 | 240 | 63488
2 2 n 2 n = 2 ( 2 n − n ) {\displaystyle {\frac {2^{2^{n}}}{2^{n}}}~=~2^{(2^{n}-n)}} | DubSage | unclosed families
(BF with a particular consul) | 2 | 2 | 4 | 32 | 4096 | 134217728
2 n − n − 1 {\displaystyle 2^{n}-n-1} |  | Eulerian numbers  | 0 | 0 | 1 | 4 | 11 | 26
2 2 n 2 n + 1 = 2 ( 2 n − n − 1 ) {\displaystyle {\frac {2^{2^{n}}}{2^{n{\color {Red}+1}}}}~=~2^{(2^{n}-n-1)}} | Sage | sharp families
(BF with a particular prefect) | 1 | 1 | 2 | 16 | 2048 | 67108864
2 2 n + ( 2 2 n − 1 + 1 ) ( 2 n − 1 ) 2 n + 1 {\displaystyle {\frac {2^{2^{n}}+(2^{2^{n-1}{\color {Red}+1}})(2^{n}-1)}{2^{n{\color {Red}+1}}}}} | NonSage | blunt families
super-families | 1 | 2 | 5 | 30 | 2288 | 67172352
2 2 n + ( 2 2 n − 1 ) ( 2 n − 1 ) 2 n {\displaystyle {\frac {2^{2^{n}}+(2^{2^{n-1}})(2^{n}-1)}{2^{n}}}} | Thyme | families  | 2 | 3 | 7 | 46 | 4336 | 134281216
| TornSage | sharp super-families  |  | 1 | 1 | 8 | 1024 | 33554432
| Clove | blunt super-families  |  | 1 | 4 | 22 | 1264 | 33617920
| Aloe | families with maximal size 2 n {\displaystyle 2^{n}} (diagonal of Olive) | 2 | 1 | 2 | 23 | 3904 | 134156284
|  | blunt families with maximal size 2 n {\displaystyle 2^{n}} | 1 | 0 | 0 | 7 | 1856 | 67047420
formulas in Python

def f_closed(n):
a = 2 ** (2 ** (n - 1))
b = 2 ** n - 1
c = 2 ** n
result = a * b / c
assert result % 1 == 0  # is integer
return int(result)

def f_unclosed(n):
a = 2 ** n - n
result = 2 ** a
assert result % 1 == 0  # is integer
return int(result)

def f_sharp(n):
a = 2 ** n - n - 1
result = 2 ** a
assert result % 1 == 0  # is integer
return int(result)

def f_blunt(n):
a = 2 ** (2 ** n)
b = 2 ** (2 ** (n - 1) + 1)
c = 2 ** n - 1
d = 2 ** (n + 1)
result = (a + b * c) / d
assert result % 1 == 0  # is integer
return int(result)

def f_total(n):
a = 2 ** (2 ** n)
b = 2 ** (2 ** (n - 1))
c = 2 ** n - 1
d = 2 ** n
result = (a + b * c) / d
assert result % 1 == 0  # is integer
return int(result)

closed = [0, 1, 3, 14, 240, 63488, 4227858432]
unclosed = [2, 2, 4, 32, 4096, 134217728, 288230376151711744]
sharp = [1, 1, 2, 16, 2048, 67108864, 144115188075855872]
blunt = [1, 2, 5, 30, 2288, 67172352, 144115192303714304]
total = [2, 3, 7, 46, 4336, 134281216, 288230380379570176]

assert [f_closed(n) for n in range(7)] == closed
assert [f_unclosed(n) for n in range(7)] == unclosed
assert [f_sharp(n) for n in range(7)] == sharp
assert [f_blunt(n) for n in range(7)] == blunt
assert [f_total(n) for n in range(7)] == total

for n in range(7):

assert closed[n] == blunt[n] - sharp[n]
assert unclosed[n] == 2 * sharp[n]

assert sharp[n] == blunt[n] - closed[n]
assert blunt[n] == sharp[n] + closed[n] == (total[n] + closed[n]) / 2

assert total[n] == sharp[n] + blunt[n] == closed[n] + unclosed[n]

family representatives

The representative is the smallest Zhegalkin index.
The length of the sequence for arity n is A000231(n), the number of n-ary families.

| n |  | sequence

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|

0  | 2  | 0, 1,
1  | 1  | 2,
2  | 4  | 4, 6, 8, 9,
3  | 39  | 16, 18, 20, 22, 24, 32, 33, 36, 40, 41, 44, 64, 65, 66, 72, 73, 74, 96, 97, 98, 104, 106, 107, 128, 129, 130, 131, 132, 133, 134, 135, 144, 145, 146, 147, 148, 149, 150, 151,
4  | 4290  | 256, 258, 260, 262, 264, 272, 274, 276, 278, 280, 288, 292, 296, 300, 320, 322, 328, 330, 352, 354, 360, 362, 384, 386, 388, 390, 400, 402, 404, 406, 512, 513, 516, 520, 521, 524, 528, 532, 536, 540, 544, 545, 548, 552, 553, 556, 560, 564, 568, 572, 576, 577, 584, 585, 608, 609, 616, 617, 640, 641, 644, 645, 656, 657, 660, 661, 704, 705, 708, 709, 720, 721, 724, 725, 1024, 1025, 1026, 1032, 1033, 1034, 1040, 1042, 1048, 1050, 1056, 1057, 1064, 1065, 1088, 1089, 1090, 1096, 1097, 1098, 1104, 1106, 1112, 1114, 1120, 1121, 1128, 1129, 1152, 1153, 1154, 1155, 1168, 1169, 1170, 1171, 1184, 1185, 1186, 1187, 1200, 1201, 1202, 1203, 1536, 1537, 1538, 1544, 1546, 1547, 1552, 1554, 1560, 1562, 1568, 1569, 1576, 1577, 1600, 1601, 1608, 1609, 1632, 1633, 1634, 1640, 1642, 1643, 1648, 1650, 1656, 1658, 1664, 1665, 1666, 1667, 1680, 1681, 1682, 1683, 1696, 1697, 1698, 1699, 1712, 1713, 1714, 1715, 2048, 2049, 2050, 2051, 2052, 2053, 2054, 2055, 2064, 2066, 2068, 2070, 2080, 2081, 2084, 2085, 2096, 2097, 2100, 2101, 2112, 2113, 2114, 2115, 2128, 2129, 2130, 2131, 2144, 2145, 2146, 2147, 2160, 2161, 2162, 2163, 2176, 2177, 2178, 2179, 2180, 2181, 2182, 2183, 2192, 2194, 2196, 2198, 2208, 2209, 2212, 2213, 2224, 2225, 2228, 2229, 2240, 2241, 2242, 2243, 2256, 2257, 2258, 2259, 2272, 2273, 2274, 2275, 2288, 2289, 2290, 2291, 2304, 2305, 2306, 2307, 2308, 2309, 2310, 2311, 2320, 2322, 2324, 2326, 2336, 2337, 2340, 2341, 2352, 2353, 2356, 2357, 2368, 2369, 2370, 2371, 2384, 2385, 2386, 2387, 2400, 2401, 2402, 2403, 2416, 2417, 2418, 2419, 2432, 2434, 2436, 2438, 2448, 2449, 2450, 2451, 2452, 2453, 2454, 2455, 2464, 2465, 2468, 2469, 2480, 2481, 2484, 2485, 2496, 2497, 2498, 2499, 2512, 2513, 2514, 2515, 2528, 2529, 2530, 2531, 2544, 2545, 2546, 2547, 4096, 4097, 4098, 4100, 4102, 4104, 4105, 4128, 4129, 4130, 4132, 4134, 4136, 4137, 4160, 4161, 4162, 4164, 4166, 4168, 4169, 4192, 4193, 4194, 4196, 4198, 4200, 4201, 4224, 4225, 4226, 4227, 4228, 4229, 4230, 4231, 4232, 4233, 4234, 4235, 4236, 4237, 4238, 4239, 4608, 4609, 4610, 4612, 4614, 4616, 4617, 4640, 4642, 4643, 4644, 4646, 4648, 4649, 4672, 4673, 4680, 4681, 4682, 4684, 4686, 4704, 4705, 4712, 4714, 4715, 4716, 4718, 4736, 4737, 4738, 4739, 4740, 4741, 4742, 4743, 4744, 4745, 4746, 4747, 4748, 4749, 4750, 4751, 5120, 5121, 5122, 5124, 5126, 5128, 5129, 5152, 5153, 5160, 5161, 5162, 5164, 5166, 5184, 5186, 5188, 5189, 5190, 5192, 5193, 5216, 5217, 5224, 5226, 5228, 5229, 5230, 5248, 5249, 5250, 5251, 5252, 5253, 5254, 5255, 5256, 5257, 5258, 5259, 5260, 5261, 5262, 5263, 5632, 5633, 5634, 5636, 5638, 5640, 5641, 5664, 5665, 5672, 5674, 5675, 5676, 5678, 5696, 5697, 5704, 5706, 5708, 5709, 5710, 5728, 5730, 5732, 5734, 5735, 5736, 5737, 5760, 5761, 5762, 5763, 5764, 5765, 5766, 5767, 5768, 5769, 5770, 5771, 5772, 5773, 5774, 5775, 6144, 6145, 6146, 6147, 6148, 6149, 6150, 6151, 6152, 6153, 6154, 6155, 6156, 6157, 6158, 6159, 6176, 6177, 6178, 6179, 6180, 6181, 6182, 6183, 6184, 6185, 6186, 6187, 6188, 6189, 6190, 6191, 6208, 6209, 6210, 6211, 6212, 6213, 6214, 6215, 6216, 6217, 6218, 6219, 6220, 6221, 6222, 6223, 6240, 6241, 6242, 6243, 6244, 6245, 6246, 6247, 6248, 6249, 6250, 6251, 6252, 6253, 6254, 6255, 6272, 6273, 6274, 6275, 6276, 6277, 6278, 6279, 6280, 6281, 6282, 6283, 6284, 6285, 6286, 6287, 6304, 6305, 6306, 6307, 6308, 6309, 6310, 6311, 6312, 6313, 6314, 6315, 6316, 6317, 6318, 6319, 6336, 6337, 6338, 6339, 6340, 6341, 6342, 6343, 6344, 6345, 6346, 6347, 6348, 6349, 6350, 6351, 6368, 6369, 6370, 6371, 6372, 6373, 6374, 6375, 6376, 6377, 6378, 6379, 6380, 6381, 6382, 6383, 8192, 8193, 8194, 8195, 8196, 8198, 8200, 8201, 8204, 8205, 8208, 8209, 8210, 8211, 8212, 8214, 8216, 8217, 8220, 8221, 8256, 8257, 8258, 8259, 8260, 8261, 8262, 8263, 8264, 8265, 8266, 8267, 8268, 8269, 8270, 8271, 8320, 8321, 8322, 8323, 8324, 8326, 8328, 8329, 8332, 8333, 8336, 8337, 8338, 8339, 8340, 8342, 8344, 8345, 8348, 8349, 8384, 8385, 8386, 8387, 8388, 8389, 8390, 8391, 8392, 8393, 8394, 8395, 8396, 8397, 8398, 8399, 8448, 8449, 8450, 8451, 8452, 8454, 8456, 8457, 8460, 8461, 8464, 8465, 8466, 8467, 8468, 8470, 8472, 8473, 8476, 8477, 8512, 8513, 8514, 8515, 8516, 8517, 8518, 8519, 8520, 8521, 8522, 8523, 8524, 8525, 8526, 8527, 8576, 8578, 8580, 8581, 8582, 8583, 8584, 8585, 8588, 8589, 8592, 8594, 8596, 8597, 8598, 8599, 8600, 8601, 8604, 8605, 8640, 8641, 8642, 8643, 8644, 8645, 8646, 8647, 8648, 8649, 8650, 8651, 8652, 8653, 8654, 8655, 9216, 9217, 9218, 9219, 9220, 9221, 9222, 9223, 9224, 9225, 9226, 9227, 9228, 9229, 9230, 9231, 9232, 9233, 9234, 9235, 9236, 9237, 9238, 9239, 9240, 9241, 9242, 9243, 9244, 9245, 9246, 9247, 9280, 9281, 9282, 9283, 9284, 9285, 9286, 9287, 9288, 9289, 9290, 9291, 9292, 9293, 9294, 9295, 9296, 9297, 9298, 9299, 9300, 9301, 9302, 9303, 9304, 9305, 9306, 9307, 9308, 9309, 9310, 9311, 9344, 9345, 9346, 9347, 9348, 9349, 9350, 9351, 9352, 9353, 9354, 9355, 9356, 9357, 9358, 9359, 9360, 9361, 9362, 9363, 9364, 9365, 9366, 9367, 9368, 9369, 9370, 9371, 9372, 9373, 9374, 9375, 9408, 9409, 9410, 9411, 9412, 9413, 9414, 9415, 9416, 9417, 9418, 9419, 9420, 9421, 9422, 9423, 9424, 9425, 9426, 9427, 9428, 9429, 9430, 9431, 9432, 9433, 9434, 9435, 9436, 9437, 9438, 9439, 10240, 10241, 10242, 10243, 10244, 10246, 10248, 10249, 10252, 10253, 10256, 10258, 10260, 10261, 10262, 10263, 10264, 10265, 10268, 10269, 10304, 10305, 10306, 10307, 10308, 10309, 10310, 10311, 10312, 10313, 10314, 10315, 10316, 10317, 10318, 10319, 10368, 10369, 10372, 10373, 10376, 10377, 10378, 10379, 10380, 10382, 10384, 10385, 10388, 10389, 10392, 10394, 10396, 10397, 10398, 10399, 10432, 10433, 10434, 10435, 10436, 10437, 10438, 10439, 10440, 10441, 10442, 10443, 10444, 10445, 10446, 10447, 10496, 10497, 10498, 10499, 10500, 10502, 10504, 10505, 10508, 10509, 10512, 10514, 10516, 10517, 10518, 10519, 10520, 10521, 10524, 10525, 10560, 10561, 10562, 10563, 10564, 10565, 10566, 10567, 10568, 10569, 10570, 10571, 10572, 10573, 10574, 10575, 10624, 10625, 10628, 10629, 10632, 10634, 10636, 10637, 10638, 10639, 10640, 10641, 10644, 10645, 10648, 10649, 10650, 10651, 10652, 10654, 10688, 10689, 10690, 10691, 10692, 10693, 10694, 10695, 10696, 10697, 10698, 10699, 10700, 10701, 10702, 10703, 11264, 11265, 11266, 11267, 11268, 11269, 11270, 11271, 11272, 11273, 11274, 11275, 11276, 11277, 11278, 11279, 11280, 11281, 11282, 11283, 11284, 11285, 11286, 11287, 11288, 11289, 11290, 11291, 11292, 11293, 11294, 11295, 11328, 11329, 11330, 11331, 11332, 11333, 11334, 11335, 11336, 11337, 11338, 11339, 11340, 11341, 11342, 11343, 11344, 11345, 11346, 11347, 11348, 11349, 11350, 11351, 11352, 11353, 11354, 11355, 11356, 11357, 11358, 11359, 11392, 11393, 11394, 11395, 11396, 11397, 11398, 11399, 11400, 11401, 11402, 11403, 11404, 11405, 11406, 11407, 11408, 11409, 11410, 11411, 11412, 11413, 11414, 11415, 11416, 11417, 11418, 11419, 11420, 11421, 11422, 11423, 11456, 11457, 11458, 11459, 11460, 11461, 11462, 11463, 11464, 11465, 11466, 11467, 11468, 11469, 11470, 11471, 11472, 11473, 11474, 11475, 11476, 11477, 11478, 11479, 11480, 11481, 11482, 11483, 11484, 11485, 11486, 11487, 16384, 16385, 16386, 16388, 16389, 16390, 16392, 16393, 16394, 16395, 16400, 16401, 16402, 16404, 16405, 16406, 16408, 16409, 16410, 16411, 16416, 16417, 16418, 16419, 16420, 16421, 16422, 16423, 16424, 16425, 16426, 16427, 16428, 16429, 16430, 16431, 16512, 16513, 16514, 16516, 16517, 16518, 16520, 16521, 16522, 16523, 16528, 16529, 16530, 16532, 16533, 16534, 16536, 16537, 16538, 16539, 16544, 16545, 16546, 16547, 16548, 16549, 16550, 16551, 16552, 16553, 16554, 16555, 16556, 16557, 16558, 16559, 16640, 16641, 16642, 16644, 16645, 16646, 16648, 16649, 16650, 16651, 16656, 16657, 16658, 16660, 16661, 16662, 16664, 16665, 16666, 16667, 16672, 16673, 16674, 16675, 16676, 16677, 16678, 16679, 16680, 16681, 16682, 16683, 16684, 16685, 16686, 16687, 16768, 16770, 16771, 16772, 16774, 16775, 16776, 16777, 16778, 16779, 16784, 16786, 16787, 16788, 16790, 16791, 16792, 16793, 16794, 16795, 16800, 16801, 16802, 16803, 16804, 16805, 16806, 16807, 16808, 16809, 16810, 16811, 16812, 16813, 16814, 16815, 16896, 16897, 16898, 16899, 16900, 16901, 16902, 16903, 16904, 16905, 16906, 16907, 16908, 16909, 16910, 16911, 16912, 16913, 16914, 16915, 16916, 16917, 16918, 16919, 16920, 16921, 16922, 16923, 16924, 16925, 16926, 16927, 16928, 16929, 16930, 16931, 16932, 16933, 16934, 16935, 16936, 16937, 16938, 16939, 16940, 16941, 16942, 16943, 16944, 16945, 16946, 16947, 16948, 16949, 16950, 16951, 16952, 16953, 16954, 16955, 16956, 16957, 16958, 16959, 17024, 17025, 17026, 17027, 17028, 17029, 17030, 17031, 17032, 17033, 17034, 17035, 17036, 17037, 17038, 17039, 17040, 17041, 17042, 17043, 17044, 17045, 17046, 17047, 17048, 17049, 17050, 17051, 17052, 17053, 17054, 17055, 17056, 17057, 17058, 17059, 17060, 17061, 17062, 17063, 17064, 17065, 17066, 17067, 17068, 17069, 17070, 17071, 17072, 17073, 17074, 17075, 17076, 17077, 17078, 17079, 17080, 17081, 17082, 17083, 17084, 17085, 17086, 17087, 18432, 18433, 18434, 18436, 18437, 18438, 18440, 18441, 18442, 18443, 18448, 18450, 18451, 18452, 18454, 18455, 18456, 18457, 18458, 18459, 18464, 18465, 18466, 18467, 18468, 18469, 18470, 18471, 18472, 18473, 18474, 18475, 18476, 18477, 18478, 18479, 18560, 18561, 18562, 18563, 18568, 18569, 18570, 18572, 18573, 18574, 18576, 18577, 18578, 18579, 18584, 18586, 18587, 18588, 18590, 18591, 18592, 18593, 18594, 18595, 18596, 18597, 18598, 18599, 18600, 18601, 18602, 18603, 18604, 18605, 18606, 18607, 18688, 18689, 18690, 18692, 18693, 18694, 18696, 18697, 18698, 18699, 18704, 18706, 18707, 18708, 18710, 18711, 18712, 18713, 18714, 18715, 18720, 18721, 18722, 18723, 18724, 18725, 18726, 18727, 18728, 18729, 18730, 18731, 18732, 18733, 18734, 18735, 18816, 18817, 18818, 18819, 18824, 18826, 18827, 18828, 18830, 18831, 18832, 18833, 18834, 18835, 18840, 18841, 18842, 18844, 18845, 18846, 18848, 18849, 18850, 18851, 18852, 18853, 18854, 18855, 18856, 18857, 18858, 18859, 18860, 18861, 18862, 18863, 18944, 18945, 18946, 18947, 18948, 18949, 18950, 18951, 18952, 18953, 18954, 18955, 18956, 18957, 18958, 18959, 18960, 18961, 18962, 18963, 18964, 18965, 18966, 18967, 18968, 18969, 18970, 18971, 18972, 18973, 18974, 18975, 18976, 18977, 18978, 18979, 18980, 18981, 18982, 18983, 18984, 18985, 18986, 18987, 18988, 18989, 18990, 18991, 18992, 18993, 18994, 18995, 18996, 18997, 18998, 18999, 19000, 19001, 19002, 19003, 19004, 19005, 19006, 19007, 19072, 19073, 19074, 19075, 19076, 19077, 19078, 19079, 19080, 19081, 19082, 19083, 19084, 19085, 19086, 19087, 19088, 19089, 19090, 19091, 19092, 19093, 19094, 19095, 19096, 19097, 19098, 19099, 19100, 19101, 19102, 19103, 19104, 19105, 19106, 19107, 19108, 19109, 19110, 19111, 19112, 19113, 19114, 19115, 19116, 19117, 19118, 19119, 19120, 19121, 19122, 19123, 19124, 19125, 19126, 19127, 19128, 19129, 19130, 19131, 19132, 19133, 19134, 19135, 24576, 24577, 24578, 24580, 24582, 24583, 24584, 24585, 24586, 24587, 24592, 24593, 24594, 24596, 24598, 24599, 24600, 24601, 24602, 24603, 24608, 24609, 24610, 24611, 24612, 24613, 24614, 24615, 24616, 24617, 24618, 24619, 24620, 24621, 24622, 24623, 24704, 24705, 24706, 24707, 24708, 24709, 24710, 24711, 24712, 24713, 24714, 24715, 24716, 24717, 24718, 24719, 24736, 24737, 24738, 24740, 24742, 24743, 24744, 24745, 24746, 24747, 24752, 24753, 24754, 24756, 24758, 24759, 24760, 24761, 24762, 24763, 24832, 24833, 24834, 24836, 24838, 24839, 24840, 24841, 24842, 24843, 24848, 24849, 24850, 24852, 24854, 24855, 24856, 24857, 24858, 24859, 24864, 24865, 24866, 24867, 24868, 24869, 24870, 24871, 24872, 24873, 24874, 24875, 24876, 24877, 24878, 24879, 24960, 24961, 24962, 24963, 24964, 24965, 24966, 24967, 24968, 24969, 24970, 24971, 24972, 24973, 24974, 24975, 24992, 24994, 24995, 24996, 24997, 24998, 25000, 25001, 25002, 25003, 25008, 25010, 25011, 25012, 25013, 25014, 25016, 25017, 25018, 25019, 25088, 25089, 25090, 25091, 25092, 25093, 25094, 25095, 25096, 25097, 25098, 25099, 25100, 25101, 25102, 25103, 25104, 25105, 25106, 25107, 25108, 25109, 25110, 25111, 25112, 25113, 25114, 25115, 25116, 25117, 25118, 25119, 25120, 25121, 25122, 25123, 25124, 25125, 25126, 25127, 25128, 25129, 25130, 25131, 25132, 25133, 25134, 25135, 25136, 25137, 25138, 25139, 25140, 25141, 25142, 25143, 25144, 25145, 25146, 25147, 25148, 25149, 25150, 25151, 25216, 25217, 25218, 25219, 25220, 25221, 25222, 25223, 25224, 25225, 25226, 25227, 25228, 25229, 25230, 25231, 25232, 25233, 25234, 25235, 25236, 25237, 25238, 25239, 25240, 25241, 25242, 25243, 25244, 25245, 25246, 25247, 25248, 25249, 25250, 25251, 25252, 25253, 25254, 25255, 25256, 25257, 25258, 25259, 25260, 25261, 25262, 25263, 25264, 25265, 25266, 25267, 25268, 25269, 25270, 25271, 25272, 25273, 25274, 25275, 25276, 25277, 25278, 25279, 26624, 26625, 26626, 26627, 26628, 26629, 26630, 26631, 26632, 26633, 26634, 26635, 26636, 26637, 26638, 26639, 26640, 26641, 26642, 26643, 26644, 26645, 26646, 26647, 26648, 26649, 26650, 26651, 26652, 26653, 26654, 26655, 26656, 26657, 26658, 26659, 26660, 26661, 26662, 26663, 26664, 26665, 26666, 26667, 26668, 26669, 26670, 26671, 26672, 26673, 26674, 26675, 26676, 26677, 26678, 26679, 26680, 26681, 26682, 26683, 26684, 26685, 26686, 26687, 26752, 26753, 26754, 26755, 26756, 26757, 26758, 26759, 26760, 26761, 26762, 26763, 26764, 26765, 26766, 26767, 26768, 26769, 26770, 26771, 26772, 26773, 26774, 26775, 26776, 26777, 26778, 26779, 26780, 26781, 26782, 26783, 26784, 26785, 26786, 26787, 26788, 26789, 26790, 26791, 26792, 26793, 26794, 26795, 26796, 26797, 26798, 26799, 26800, 26801, 26802, 26803, 26804, 26805, 26806, 26807, 26808, 26809, 26810, 26811, 26812, 26813, 26814, 26815, 27136, 27137, 27138, 27140, 27142, 27143, 27144, 27145, 27146, 27147, 27152, 27154, 27155, 27156, 27157, 27158, 27160, 27161, 27162, 27163, 27168, 27169, 27170, 27171, 27172, 27173, 27174, 27175, 27176, 27177, 27178, 27179, 27180, 27181, 27182, 27183, 27264, 27265, 27266, 27267, 27268, 27269, 27270, 27271, 27272, 27273, 27274, 27275, 27276, 27277, 27278, 27279, 27296, 27297, 27298, 27299, 27304, 27306, 27307, 27308, 27309, 27310, 27312, 27313, 27314, 27315, 27320, 27321, 27322, 27324, 27326, 27327, 27392, 27393, 27394, 27396, 27398, 27399, 27400, 27401, 27402, 27403, 27408, 27410, 27411, 27412, 27413, 27414, 27416, 27417, 27418, 27419, 27424, 27425, 27426, 27427, 27428, 27429, 27430, 27431, 27432, 27433, 27434, 27435, 27436, 27437, 27438, 27439, 27520, 27521, 27522, 27523, 27524, 27525, 27526, 27527, 27528, 27529, 27530, 27531, 27532, 27533, 27534, 27535, 27552, 27553, 27554, 27555, 27560, 27561, 27562, 27564, 27566, 27567, 27568, 27569, 27570, 27571, 27576, 27578, 27579, 27580, 27581, 27582, 32768, 32769, 32770, 32771, 32772, 32773, 32774, 32775, 32776, 32777, 32778, 32779, 32780, 32781, 32782, 32783, 32784, 32785, 32786, 32787, 32788, 32789, 32790, 32791, 32792, 32793, 32794, 32795, 32796, 32797, 32798, 32799, 32800, 32801, 32802, 32803, 32804, 32805, 32806, 32807, 32808, 32809, 32810, 32811, 32812, 32813, 32814, 32815, 32816, 32817, 32818, 32819, 32820, 32821, 32822, 32823, 32824, 32825, 32826, 32827, 32828, 32829, 32830, 32831, 32832, 32833, 32834, 32835, 32836, 32837, 32838, 32839, 32840, 32841, 32842, 32843, 32844, 32845, 32846, 32847, 32848, 32849, 32850, 32851, 32852, 32853, 32854, 32855, 32856, 32857, 32858, 32859, 32860, 32861, 32862, 32863, 32864, 32865, 32866, 32867, 32868, 32869, 32870, 32871, 32872, 32873, 32874, 32875, 32876, 32877, 32878, 32879, 32880, 32881, 32882, 32883, 32884, 32885, 32886, 32887, 32888, 32889, 32890, 32891, 32892, 32893, 32894, 32895, 33024, 33025, 33026, 33027, 33028, 33029, 33030, 33031, 33032, 33033, 33034, 33035, 33036, 33037, 33038, 33039, 33040, 33041, 33042, 33043, 33044, 33045, 33046, 33047, 33048, 33049, 33050, 33051, 33052, 33053, 33054, 33055, 33056, 33057, 33058, 33059, 33060, 33061, 33062, 33063, 33064, 33065, 33066, 33067, 33068, 33069, 33070, 33071, 33072, 33073, 33074, 33075, 33076, 33077, 33078, 33079, 33080, 33081, 33082, 33083, 33084, 33085, 33086, 33087, 33088, 33089, 33090, 33091, 33092, 33093, 33094, 33095, 33096, 33097, 33098, 33099, 33100, 33101, 33102, 33103, 33104, 33105, 33106, 33107, 33108, 33109, 33110, 33111, 33112, 33113, 33114, 33115, 33116, 33117, 33118, 33119, 33120, 33121, 33122, 33123, 33124, 33125, 33126, 33127, 33128, 33129, 33130, 33131, 33132, 33133, 33134, 33135, 33136, 33137, 33138, 33139, 33140, 33141, 33142, 33143, 33144, 33145, 33146, 33147, 33148, 33149, 33150, 33151, 33280, 33281, 33282, 33283, 33284, 33285, 33286, 33287, 33288, 33289, 33290, 33291, 33292, 33293, 33294, 33295, 33296, 33297, 33298, 33299, 33300, 33301, 33302, 33303, 33304, 33305, 33306, 33307, 33308, 33309, 33310, 33311, 33312, 33313, 33314, 33315, 33316, 33317, 33318, 33319, 33320, 33321, 33322, 33323, 33324, 33325, 33326, 33327, 33328, 33329, 33330, 33331, 33332, 33333, 33334, 33335, 33336, 33337, 33338, 33339, 33340, 33341, 33342, 33343, 33344, 33345, 33346, 33347, 33348, 33349, 33350, 33351, 33352, 33353, 33354, 33355, 33356, 33357, 33358, 33359, 33360, 33361, 33362, 33363, 33364, 33365, 33366, 33367, 33368, 33369, 33370, 33371, 33372, 33373, 33374, 33375, 33376, 33377, 33378, 33379, 33380, 33381, 33382, 33383, 33384, 33385, 33386, 33387, 33388, 33389, 33390, 33391, 33392, 33393, 33394, 33395, 33396, 33397, 33398, 33399, 33400, 33401, 33402, 33403, 33404, 33405, 33406, 33407, 33536, 33537, 33538, 33539, 33540, 33541, 33542, 33543, 33544, 33545, 33546, 33547, 33548, 33549, 33550, 33551, 33552, 33553, 33554, 33555, 33556, 33557, 33558, 33559, 33560, 33561, 33562, 33563, 33564, 33565, 33566, 33567, 33568, 33569, 33570, 33571, 33572, 33573, 33574, 33575, 33576, 33577, 33578, 33579, 33580, 33581, 33582, 33583, 33584, 33585, 33586, 33587, 33588, 33589, 33590, 33591, 33592, 33593, 33594, 33595, 33596, 33597, 33598, 33599, 33600, 33601, 33602, 33603, 33604, 33605, 33606, 33607, 33608, 33609, 33610, 33611, 33612, 33613, 33614, 33615, 33616, 33617, 33618, 33619, 33620, 33621, 33622, 33623, 33624, 33625, 33626, 33627, 33628, 33629, 33630, 33631, 33632, 33633, 33634, 33635, 33636, 33637, 33638, 33639, 33640, 33641, 33642, 33643, 33644, 33645, 33646, 33647, 33648, 33649, 33650, 33651, 33652, 33653, 33654, 33655, 33656, 33657, 33658, 33659, 33660, 33661, 33662, 33663, 33792, 33793, 33794, 33795, 33796, 33797, 33798, 33799, 33800, 33801, 33802, 33803, 33804, 33805, 33806, 33807, 33808, 33809, 33810, 33811, 33812, 33813, 33814, 33815, 33816, 33817, 33818, 33819, 33820, 33821, 33822, 33823, 33824, 33825, 33826, 33827, 33828, 33829, 33830, 33831, 33832, 33833, 33834, 33835, 33836, 33837, 33838, 33839, 33840, 33841, 33842, 33843, 33844, 33845, 33846, 33847, 33848, 33849, 33850, 33851, 33852, 33853, 33854, 33855, 33856, 33857, 33858, 33859, 33860, 33861, 33862, 33863, 33864, 33865, 33866, 33867, 33868, 33869, 33870, 33871, 33872, 33873, 33874, 33875, 33876, 33877, 33878, 33879, 33880, 33881, 33882, 33883, 33884, 33885, 33886, 33887, 33888, 33889, 33890, 33891, 33892, 33893, 33894, 33895, 33896, 33897, 33898, 33899, 33900, 33901, 33902, 33903, 33904, 33905, 33906, 33907, 33908, 33909, 33910, 33911, 33912, 33913, 33914, 33915, 33916, 33917, 33918, 33919, 34048, 34049, 34050, 34051, 34052, 34053, 34054, 34055, 34056, 34057, 34058, 34059, 34060, 34061, 34062, 34063, 34064, 34065, 34066, 34067, 34068, 34069, 34070, 34071, 34072, 34073, 34074, 34075, 34076, 34077, 34078, 34079, 34080, 34081, 34082, 34083, 34084, 34085, 34086, 34087, 34088, 34089, 34090, 34091, 34092, 34093, 34094, 34095, 34096, 34097, 34098, 34099, 34100, 34101, 34102, 34103, 34104, 34105, 34106, 34107, 34108, 34109, 34110, 34111, 34112, 34113, 34114, 34115, 34116, 34117, 34118, 34119, 34120, 34121, 34122, 34123, 34124, 34125, 34126, 34127, 34128, 34129, 34130, 34131, 34132, 34133, 34134, 34135, 34136, 34137, 34138, 34139, 34140, 34141, 34142, 34143, 34144, 34145, 34146, 34147, 34148, 34149, 34150, 34151, 34152, 34153, 34154, 34155, 34156, 34157, 34158, 34159, 34160, 34161, 34162, 34163, 34164, 34165, 34166, 34167, 34168, 34169, 34170, 34171, 34172, 34173, 34174, 34175, 34304, 34305, 34306, 34307, 34308, 34309, 34310, 34311, 34312, 34313, 34314, 34315, 34316, 34317, 34318, 34319, 34320, 34321, 34322, 34323, 34324, 34325, 34326, 34327, 34328, 34329, 34330, 34331, 34332, 34333, 34334, 34335, 34336, 34337, 34338, 34339, 34340, 34341, 34342, 34343, 34344, 34345, 34346, 34347, 34348, 34349, 34350, 34351, 34352, 34353, 34354, 34355, 34356, 34357, 34358, 34359, 34360, 34361, 34362, 34363, 34364, 34365, 34366, 34367, 34368, 34369, 34370, 34371, 34372, 34373, 34374, 34375, 34376, 34377, 34378, 34379, 34380, 34381, 34382, 34383, 34384, 34385, 34386, 34387, 34388, 34389, 34390, 34391, 34392, 34393, 34394, 34395, 34396, 34397, 34398, 34399, 34400, 34401, 34402, 34403, 34404, 34405, 34406, 34407, 34408, 34409, 34410, 34411, 34412, 34413, 34414, 34415, 34416, 34417, 34418, 34419, 34420, 34421, 34422, 34423, 34424, 34425, 34426, 34427, 34428, 34429, 34430, 34431, 34560, 34561, 34562, 34563, 34564, 34565, 34566, 34567, 34568, 34569, 34570, 34571, 34572, 34573, 34574, 34575, 34576, 34577, 34578, 34579, 34580, 34581, 34582, 34583, 34584, 34585, 34586, 34587, 34588, 34589, 34590, 34591, 34592, 34593, 34594, 34595, 34596, 34597, 34598, 34599, 34600, 34601, 34602, 34603, 34604, 34605, 34606, 34607, 34608, 34609, 34610, 34611, 34612, 34613, 34614, 34615, 34616, 34617, 34618, 34619, 34620, 34621, 34622, 34623, 34624, 34625, 34626, 34627, 34628, 34629, 34630, 34631, 34632, 34633, 34634, 34635, 34636, 34637, 34638, 34639, 34640, 34641, 34642, 34643, 34644, 34645, 34646, 34647, 34648, 34649, 34650, 34651, 34652, 34653, 34654, 34655, 34656, 34657, 34658, 34659, 34660, 34661, 34662, 34663, 34664, 34665, 34666, 34667, 34668, 34669, 34670, 34671, 34672, 34673, 34674, 34675, 34676, 34677, 34678, 34679, 34680, 34681, 34682, 34683, 34684, 34685, 34686, 34687, 36864, 36865, 36866, 36867, 36868, 36869, 36870, 36871, 36872, 36873, 36874, 36875, 36876, 36877, 36878, 36879, 36880, 36881, 36882, 36883, 36884, 36885, 36886, 36887, 36888, 36889, 36890, 36891, 36892, 36893, 36894, 36895, 36896, 36897, 36898, 36899, 36900, 36901, 36902, 36903, 36904, 36905, 36906, 36907, 36908, 36909, 36910, 36911, 36912, 36913, 36914, 36915, 36916, 36917, 36918, 36919, 36920, 36921, 36922, 36923, 36924, 36925, 36926, 36927, 36928, 36929, 36930, 36931, 36932, 36933, 36934, 36935, 36936, 36937, 36938, 36939, 36940, 36941, 36942, 36943, 36944, 36945, 36946, 36947, 36948, 36949, 36950, 36951, 36952, 36953, 36954, 36955, 36956, 36957, 36958, 36959, 36960, 36961, 36962, 36963, 36964, 36965, 36966, 36967, 36968, 36969, 36970, 36971, 36972, 36973, 36974, 36975, 36976, 36977, 36978, 36979, 36980, 36981, 36982, 36983, 36984, 36985, 36986, 36987, 36988, 36989, 36990, 36991, 37120, 37121, 37122, 37123, 37124, 37125, 37126, 37127, 37128, 37129, 37130, 37131, 37132, 37133, 37134, 37135, 37136, 37137, 37138, 37139, 37140, 37141, 37142, 37143, 37144, 37145, 37146, 37147, 37148, 37149, 37150, 37151, 37152, 37153, 37154, 37155, 37156, 37157, 37158, 37159, 37160, 37161, 37162, 37163, 37164, 37165, 37166, 37167, 37168, 37169, 37170, 37171, 37172, 37173, 37174, 37175, 37176, 37177, 37178, 37179, 37180, 37181, 37182, 37183, 37184, 37185, 37186, 37187, 37188, 37189, 37190, 37191, 37192, 37193, 37194, 37195, 37196, 37197, 37198, 37199, 37200, 37201, 37202, 37203, 37204, 37205, 37206, 37207, 37208, 37209, 37210, 37211, 37212, 37213, 37214, 37215, 37216, 37217, 37218, 37219, 37220, 37221, 37222, 37223, 37224, 37225, 37226, 37227, 37228, 37229, 37230, 37231, 37232, 37233, 37234, 37235, 37236, 37237, 37238, 37239, 37240, 37241, 37242, 37243, 37244, 37245, 37246, 37247, 37376, 37377, 37378, 37379, 37380, 37381, 37382, 37383, 37384, 37385, 37386, 37387, 37388, 37389, 37390, 37391, 37392, 37393, 37394, 37395, 37396, 37397, 37398, 37399, 37400, 37401, 37402, 37403, 37404, 37405, 37406, 37407, 37408, 37409, 37410, 37411, 37412, 37413, 37414, 37415, 37416, 37417, 37418, 37419, 37420, 37421, 37422, 37423, 37424, 37425, 37426, 37427, 37428, 37429, 37430, 37431, 37432, 37433, 37434, 37435, 37436, 37437, 37438, 37439, 37440, 37441, 37442, 37443, 37444, 37445, 37446, 37447, 37448, 37449, 37450, 37451, 37452, 37453, 37454, 37455, 37456, 37457, 37458, 37459, 37460, 37461, 37462, 37463, 37464, 37465, 37466, 37467, 37468, 37469, 37470, 37471, 37472, 37473, 37474, 37475, 37476, 37477, 37478, 37479, 37480, 37481, 37482, 37483, 37484, 37485, 37486, 37487, 37488, 37489, 37490, 37491, 37492, 37493, 37494, 37495, 37496, 37497, 37498, 37499, 37500, 37501, 37502, 37503, 37632, 37633, 37634, 37635, 37636, 37637, 37638, 37639, 37640, 37641, 37642, 37643, 37644, 37645, 37646, 37647, 37648, 37649, 37650, 37651, 37652, 37653, 37654, 37655, 37656, 37657, 37658, 37659, 37660, 37661, 37662, 37663, 37664, 37665, 37666, 37667, 37668, 37669, 37670, 37671, 37672, 37673, 37674, 37675, 37676, 37677, 37678, 37679, 37680, 37681, 37682, 37683, 37684, 37685, 37686, 37687, 37688, 37689, 37690, 37691, 37692, 37693, 37694, 37695, 37696, 37697, 37698, 37699, 37700, 37701, 37702, 37703, 37704, 37705, 37706, 37707, 37708, 37709, 37710, 37711, 37712, 37713, 37714, 37715, 37716, 37717, 37718, 37719, 37720, 37721, 37722, 37723, 37724, 37725, 37726, 37727, 37728, 37729, 37730, 37731, 37732, 37733, 37734, 37735, 37736, 37737, 37738, 37739, 37740, 37741, 37742, 37743, 37744, 37745, 37746, 37747, 37748, 37749, 37750, 37751, 37752, 37753, 37754, 37755, 37756, 37757, 37758, 37759, 37888, 37889, 37890, 37891, 37892, 37893, 37894, 37895, 37896, 37897, 37898, 37899, 37900, 37901, 37902, 37903, 37904, 37905, 37906, 37907, 37908, 37909, 37910, 37911, 37912, 37913, 37914, 37915, 37916, 37917, 37918, 37919, 37920, 37921, 37922, 37923, 37924, 37925, 37926, 37927, 37928, 37929, 37930, 37931, 37932, 37933, 37934, 37935, 37936, 37937, 37938, 37939, 37940, 37941, 37942, 37943, 37944, 37945, 37946, 37947, 37948, 37949, 37950, 37951, 37952, 37953, 37954, 37955, 37956, 37957, 37958, 37959, 37960, 37961, 37962, 37963, 37964, 37965, 37966, 37967, 37968, 37969, 37970, 37971, 37972, 37973, 37974, 37975, 37976, 37977, 37978, 37979, 37980, 37981, 37982, 37983, 37984, 37985, 37986, 37987, 37988, 37989, 37990, 37991, 37992, 37993, 37994, 37995, 37996, 37997, 37998, 37999, 38000, 38001, 38002, 38003, 38004, 38005, 38006, 38007, 38008, 38009, 38010, 38011, 38012, 38013, 38014, 38015, 38144, 38145, 38146, 38147, 38148, 38149, 38150, 38151, 38152, 38153, 38154, 38155, 38156, 38157, 38158, 38159, 38160, 38161, 38162, 38163, 38164, 38165, 38166, 38167, 38168, 38169, 38170, 38171, 38172, 38173, 38174, 38175, 38176, 38177, 38178, 38179, 38180, 38181, 38182, 38183, 38184, 38185, 38186, 38187, 38188, 38189, 38190, 38191, 38192, 38193, 38194, 38195, 38196, 38197, 38198, 38199, 38200, 38201, 38202, 38203, 38204, 38205, 38206, 38207, 38208, 38209, 38210, 38211, 38212, 38213, 38214, 38215, 38216, 38217, 38218, 38219, 38220, 38221, 38222, 38223, 38224, 38225, 38226, 38227, 38228, 38229, 38230, 38231, 38232, 38233, 38234, 38235, 38236, 38237, 38238, 38239, 38240, 38241, 38242, 38243, 38244, 38245, 38246, 38247, 38248, 38249, 38250, 38251, 38252, 38253, 38254, 38255, 38256, 38257, 38258, 38259, 38260, 38261, 38262, 38263, 38264, 38265, 38266, 38267, 38268, 38269, 38270, 38271, 38400, 38401, 38402, 38403, 38404, 38405, 38406, 38407, 38408, 38409, 38410, 38411, 38412, 38413, 38414, 38415, 38416, 38417, 38418, 38419, 38420, 38421, 38422, 38423, 38424, 38425, 38426, 38427, 38428, 38429, 38430, 38431, 38432, 38433, 38434, 38435, 38436, 38437, 38438, 38439, 38440, 38441, 38442, 38443, 38444, 38445, 38446, 38447, 38448, 38449, 38450, 38451, 38452, 38453, 38454, 38455, 38456, 38457, 38458, 38459, 38460, 38461, 38462, 38463, 38464, 38465, 38466, 38467, 38468, 38469, 38470, 38471, 38472, 38473, 38474, 38475, 38476, 38477, 38478, 38479, 38480, 38481, 38482, 38483, 38484, 38485, 38486, 38487, 38488, 38489, 38490, 38491, 38492, 38493, 38494, 38495, 38496, 38497, 38498, 38499, 38500, 38501, 38502, 38503, 38504, 38505, 38506, 38507, 38508, 38509, 38510, 38511, 38512, 38513, 38514, 38515, 38516, 38517, 38518, 38519, 38520, 38521, 38522, 38523, 38524, 38525, 38526, 38527, 38656, 38657, 38658, 38659, 38660, 38661, 38662, 38663, 38664, 38665, 38666, 38667, 38668, 38669, 38670, 38671, 38672, 38673, 38674, 38675, 38676, 38677, 38678, 38679, 38680, 38681, 38682, 38683, 38684, 38685, 38686, 38687, 38688, 38689, 38690, 38691, 38692, 38693, 38694, 38695, 38696, 38697, 38698, 38699, 38700, 38701, 38702, 38703, 38704, 38705, 38706, 38707, 38708, 38709, 38710, 38711, 38712, 38713, 38714, 38715, 38716, 38717, 38718, 38719, 38720, 38721, 38722, 38723, 38724, 38725, 38726, 38727, 38728, 38729, 38730, 38731, 38732, 38733, 38734, 38735, 38736, 38737, 38738, 38739, 38740, 38741, 38742, 38743, 38744, 38745, 38746, 38747, 38748, 38749, 38750, 38751, 38752, 38753, 38754, 38755, 38756, 38757, 38758, 38759, 38760, 38761, 38762, 38763, 38764, 38765, 38766, 38767, 38768, 38769, 38770, 38771, 38772, 38773, 38774, 38775, 38776, 38777, 38778, 38779, 38780, 38781, 38782, 38783

A227725 T ( n , k ) {\displaystyle T(n,k)} is the number of  n {\displaystyle n} -ary families of size 2 k {\displaystyle 2^{k}} .
A227724 T ( n , k ) {\displaystyle T(n,k)} is the number of balanced n {\displaystyle n} -ary families of size 2 k {\displaystyle 2^{k}} .
A054724 T ( n , w ) {\displaystyle T(n,w)} is the number of  n {\displaystyle n} -ary families with weight w {\displaystyle w} . See here.

## blunt and sharp

[edit | edit source]

Sharp families always have the maximal size.
Each member of sharp family has a unique consul, while all members of a blunt family have the same consul.
All sharp families belong to the same tribe. The tribe of a blunt family is denoted by the binary weight of the consul.

It is easy to calculate a family representative of a sharp BF (a male to be precise), namely the one with consul 0.

There are many bijections between blunt and sharp BF.
It would be practical to have one, that assigns all members of a blunt family to members of the same sharp family.
So one sharp family would correspond to a set of blunt families. That would be nice to have, but may not exist.

An important fact (visible in the table of sequences above) is that the number of blunt families is equal to the number of super-families.
The numbers of sharp and closed (self-complementary) families always sum up to the number of blunt families (= the number of super-families).
Arity 1 is a special case, because the one sharp and the one closed family are the same. Apart from that, no sharp familiy is closed.

possible bijections

The quantities suggest two kinds of bijection for arities greater 1:

From the union of sharp and closed families to blunt families.
Closed families are themselves blunt. So they are assigned to themselves.
This leaves a bijection from sharp families to unclosed blunt families.

From the union of sharp and closed families to super-families.
Closed families are themselves complete super-families. So they are assigned to themselves.
This leaves a bijection from sharp families to unclosed super-families - which include sharp super-families.

E.g. the light sharp families could be assigned to the super-families containing them, and the heavy sharp families to the unclosed blunt super-families.
This seems silly. There is probably no meaningful bijection of this kind.

sharp to unclosed blunt

These images show the 32 unclosed families for arity 3 within their super-families.
On the left are the 16 (unclosed) sharp families. On the right are the 16 unclosed blunt families

| truth tables

255

3, 12, 48, 192
63, 207, 243, 252

5, 10, 80, 160
95, 175, 245, 250

6, 9, 96, 144
111, 159, 246, 249

17, 34, 68, 136
119, 187, 221, 238

18, 33, 72, 132
123, 183, 222, 237

20, 40, 65, 130
125, 190, 215, 235

24, 36, 66, 129
126, 189, 219, 231

Zhegalkin indices

The families on the left belong to two villages, which correspond to the rows of the 2×8 grid.
Those on the right belong to four villages, which correspond to the columns.

128

130

132

134

144

146

148

150

64, 68, 80, 85
65, 69, 81, 84

32, 34, 48, 51
33, 35, 49, 50

96, 102, 112, 119
97, 103, 113, 118

8, 10, 12, 15
9, 11, 13, 14

72, 76, 90, 95
73, 77, 91, 94

40, 42, 60, 63
41, 43, 61, 62

106, 108, 120, 127
107, 109, 121, 126

closed

These images show the 14 closed (blunt) families for arity 3.
On the left are the linear functions except contradiction and tautology. Their Walsh indices 1...7 go from top left to bottom right.
On the right are the families with size 8. Their consuls correspond to the Walsh indices on the left.

| truth tables

85, 170

51, 204

102, 153

15, 240

90, 165

60, 195

105, 150

Zhegalkin indices

2, 3

4, 5

6, 7

16, 17

18, 19

20, 21

22, 23

twin prefects of blunt families

blunt Boolean functions

The twin prefect (prefect of the twin) is the essentially the consul, but refined by a sign.
The eight images below are in the same 2×8 grid as the matrices shown above.
(Here truth tables and Zhegalkin indices are shown within the same image.)
The blunt families shown above (extroverted and introverted) are within these patterns.

## super-clans

[edit | edit source]

The following table shows the 46 3-ary families within the 14 super-clans. That means, each family is shown in its clan, and together with its complement.
In each matrix a family has a distinct color. Complements have the same base color (RGB or beige). Clans have the same shade (light or dark).
(The number of families in a super-clan is 1, 2, 3 or 6.)

tables of super-clans

| sharp families by quaestor weight

quaestor weight 1

quaestor weight 3

sharp (unclosed)

| f/c | weight  | quaestor
weight  | rep. | truth
tables  | Zhegalkin
indices  | reverse

|
|
|
|
|
|

1  | 1, 7  | 1  | 128

128

Ж 128  |  |  |
1  | 3, 5  | 3  | 150

Ж 150  |  |  |
3  | 3, 5  | 3  | 130

Ж 130  |  |  |
3  | 3, 5  | 1  | 134

230

Ж 134  |  |  |
blunt unclosed

| f/c | family
size  | weight  | consul
weight  | quaestor
weight  | rep. | truth
tables  | Zhegalkin
indices  | reverse

|
|
|
|
|
|
|
|

1  | 1  | 0, 8  | 0  | 0  | 0

Ж 0  |  |  |
1  | 4  | 2, 6  | 3  | 0  | 106

Ж 106  |  |  |
3  | 4  | 2, 6  | 2  | 2  | 40

Ж 40  |  |  |
3  | 4  | 2, 6  | 1  | 2  | 8

136

Ж 8  |  |  |
blunt closed

| f/c | family
size  | consul
weight  | quaestor
weight  | 🤥  | rep. | truth
tables  | Zhegalkin
indices  | reverse

|
|
|
|
|
|
|
|

1  | 2  | 0  | 4  | honest  | 22

150

Ж 22  |  |  |
3  | 2  | 0  | 4  | honest  | 2

170

Ж 2  |  |  |
3  | 2  | 0  | 0  | honest  | 6

102

Ж 6  |  |  |
1  | 8  | 3  | 4  | dishonest  | 104

232

Ж 104  |  |  |
3  | 8  | 1  | 2  | dishonest  | 24

120

Ж 24  |  |  |
3  | 8  | 2  | 2  | dishonest  | 44

228

Ж 44  |  |  |

## super-families

[edit | edit source]

The following table shows the 46 3-ary families within the 30 super-families. That means, each family is shown together with its complement.
An important property (that links the 3-ary to 2-ary families) is the village - seen as columns in the matrices of Zhegalkin indices.
Blunt families have a consul and a solidity. (Sharp families have four solid and four fluid members.)

tables of super-families

| sharp (unclosed)

| weight  | quaestor
weight  | reps. | truth
tables  | Zhegalkin
indices  | village

|
|
|
|
|

sc | sf | size  |  | reverse  | rep.
1, 7  | 1  | Ж 128  | 128

128

Ж 128  |  |  | 4  | {1, 3, 5, 15}  | T {8, 4, 2, 1}
Z {8, 12, 10, 15}  | 128

Ж 8
3, 5  | 3  | Ж 130  | 130

Ж 130  |  |  | 4  | {1, 3, 5, 15}  | T {8, 4, 2, 1}
Z {8, 12, 10, 15}  | 130

Ж 8
3, 5  | 3  | Ж 130  | 132

Ж 132  |  |  | 4  | {1, 3, 5, 15}  | T {8, 4, 2, 1}
Z {8, 12, 10, 15}  | 132

Ж 8
3, 5  | 1  | Ж 134  | 134

230

Ж 134  |  |  | 4  | {1, 3, 5, 15}  | T {8, 4, 2, 1}
Z {8, 12, 10, 15}  | 134

Ж 8
3, 5  | 3  | Ж 130  | 144

112

Ж 144  |  |  | 4  | {7, 9, 11, 13}  | T {14, 7, 11, 13}
Z {14, 9, 13, 11}  | 144

Ж 9
3, 5  | 1  | Ж 134  | 146

218

Ж 146  |  |  | 4  | {7, 9, 11, 13}  | T {14, 7, 11, 13}
Z {14, 9, 13, 11}  | 146

Ж 9
3, 5  | 1  | Ж 134  | 148

188

Ж 148  |  |  | 4  | {7, 9, 11, 13}  | T {14, 7, 11, 13}
Z {14, 9, 13, 11}  | 148

Ж 9
3, 5  | 3  | Ж 150  | 150

Ж 150  |  |  | 4  | {7, 9, 11, 13}  | T {14, 7, 11, 13}
Z {14, 9, 13, 11}  | 150

Ж 9
blunt unclosed

| size  | weight  | consul | 🧊  | quaestor
weight  | reps. | truth
tables  | Zhegalkin
indices  | village

|
|
|
|
|
|
|
|

|  | sc | sf | size  |  | reverse  | rep.
1  | 0, 8  | 0  | 0  | fluid  | 0  | Ж 0  | 0

Ж 0  |  |  | 1  | {0}  | T {0}
Z {0}  | 0

Ж 0
4  | 2, 6  | 1  | 4  | solid  | 2  | Ж 8  | 8

136

Ж 8  |  |  | 1  | {0}  | T {0}
Z {0}  | 8

Ж 0
4  | 2, 6  | 1  | 2  | solid  | 2  | Ж 8  | 32

160

Ж 32  |  |  | 2  | {4, 12}  | T {10, 5}
Z {2, 3}  | 32

Ж 2
4  | 2, 6  | 2  | 6  | fluid  | 2  | Ж 40  | 40

Ж 40  |  |  | 2  | {4, 12}  | T {10, 5}
Z {2, 3}  | 40

Ж 2
4  | 2, 6  | 1  | 1  | fluid  | 2  | Ж 8  | 64

192

Ж 64  |  |  | 2  | {2, 10}  | T {12, 3}
Z {4, 5}  | 64

Ж 4
4  | 2, 6  | 2  | 5  | solid  | 2  | Ж 40  | 72

Ж 72  |  |  | 2  | {2, 10}  | T {12, 3}
Z {4, 5}  | 72

Ж 4
4  | 2, 6  | 2  | 3  | solid  | 2  | Ж 40  | 96

Ж 96  |  |  | 2  | {6, 14}  | T {6, 9}
Z {6, 7}  | 96

Ж 6
4  | 2, 6  | 3  | 7  | fluid  | 0  | Ж 106  | 106

Ж 106  |  |  | 2  | {6, 14}  | T {6, 9}
Z {6, 7}  | 106

Ж 6
blunt closed

| size  | consul | 🧊  | 🤥  | quaestor
weight  | reps. | truth
tables  | Zhegalkin
indices  | village

|
|
|
|
|
|
|
|

|  | sc | sf | size  |  | reverse  | rep.
2  | 0  | 0  | fluid  | honest  | 4  | Ж 2  | 2

170

Ж 2  |  |  | 1  | {0}  | T {0}
Z {0}  | 2

Ж 0
2  | 0  | 0  | fluid  | honest  | 4  | Ж 2  | 4

204

Ж 4  |  |  | 1  | {0}  | T {0}
Z {0}  | 4

Ж 0
2  | 0  | 0  | fluid  | honest  | 0  | Ж 6  | 6

102

Ж 6  |  |  | 1  | {0}  | T {0}
Z {0}  | 6

Ж 0
2  | 0  | 0  | fluid  | honest  | 4  | Ж 2  | 16

240

Ж 16  |  |  | 1  | {8}  | T {15}
Z {1}  | 16

Ж 1
2  | 0  | 0  | fluid  | honest  | 0  | Ж 6  | 18

Ж 18  |  |  | 1  | {8}  | T {15}
Z {1}  | 18

Ж 1
2  | 0  | 0  | fluid  | honest  | 0  | Ж 6  | 20

Ж 20  |  |  | 1  | {8}  | T {15}
Z {1}  | 20

Ж 1
2  | 0  | 0  | fluid  | honest  | 4  | Ж 22  | 22

150

Ж 22  |  |  | 1  | {8}  | T {15}
Z {1}  | 22

Ж 1
8  | 1  | 4  | solid  | dishonest  | 2  | Ж 24  | 24

120

Ж 24  |  |  | 1  | {8}  | T {15}
Z {1}  | 24

Ж 1
8  | 1  | 2  | solid  | dishonest  | 2  | Ж 24  | 36

108

Ж 36  |  |  | 2  | {4, 12}  | T {10, 5}
Z {2, 3}  | 36

Ж 2
8  | 2  | 6  | fluid  | dishonest  | 2  | Ж 44  | 44

228

Ж 44  |  |  | 2  | {4, 12}  | T {10, 5}
Z {2, 3}  | 44

Ж 2
8  | 1  | 1  | fluid  | dishonest  | 2  | Ж 24  | 66

106

Ж 66  |  |  | 2  | {2, 10}  | T {12, 3}
Z {4, 5}  | 66

Ж 4
8  | 2  | 5  | solid  | dishonest  | 2  | Ж 44  | 74

226

Ж 74  |  |  | 2  | {2, 10}  | T {12, 3}
Z {4, 5}  | 74

Ж 4
8  | 2  | 3  | solid  | dishonest  | 2  | Ж 44  | 98

202

Ж 98  |  |  | 2  | {6, 14}  | T {6, 9}
Z {6, 7}  | 98

Ж 6
8  | 3  | 7  | fluid  | dishonest  | 4  | Ж 104  | 104

232

Ж 104  |  |  | 2  | {6, 14}  | T {6, 9}
Z {6, 7}  | 104

Ж 6

## misc.

[edit | edit source]

3-ary partitions: family (46), reverse family (46), super family (30), ultra family (18), family size (4), quaestor weight (5), tribe (5), is sharp (2)

* c:Category:3-ary truth tables in octeract matrix; ultra-families
* c:Category:3-ary Boolean functions in octeract matrix; super-clans
* c:Category:3-ary Boolean functions in octeract matrix; reverse super-clans

* c:Category:3-ary Boolean functions; small equivalence classes; cubes
* c:Category:Walsh spectra of 3-ary Boolean functions; families

* c:Category:Family to senior nobles to prefects (image set)

*[f/c]: families per clan
*[rep.]: representative
*[reps.]: representatives
*[sc]: clan
*[sf]: family
*[consul]: The column on the left is the consul weight.