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 2^adicity, 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 dull.
(It is the same as the parity of the quaestor weight.)

Sharp families always have the maximal size.
Each member of sharp family has a unique consul, while all members of a dull family have the same consul.
This makes it easy to calculate a family representative for sharp Boolean functions. (For males to be precise.)

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.
Clans belong to a tribe. Sharp BF form a tribe on their own. The tribe of a dull BF is the binary weight of the consul.

sequences

arity to integer
sequence		arity
sequence		0	1	2	3	4	5
A000231	families	2	3	7	46	4336	134281216
	super-families	1	2	5	30	1973	57805981
	balanced families (central values of A054724)		1	3	14	870	18796230
A001320	self-complementary families		1	3	14	240	63488
A051502	number of families with maximal size $2^{arity}$ (right column of A227725)	2	1	2	23	3904	134156284

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
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 $T(n,k)$ is the number of $n$ -ary families of size $2^{k}$ .
A227724 $T(n,k)$ is the number of balanced $n$ -ary families of size $2^{k}$ .
A054724 $T(n,k)$ is the number of $n$ -ary families with weight $k$ .

super-clans

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.)

table of super-clans

sharp families by quaestor weight
quaestor weight 1 quaestor weight 3

s.-c. size	family size	weight	tribe	quaestor weight	rep.
2	1	0, 8	dull 0	0	0 Ж 0
2	2	4	dull 0	4	150 Ж 22
6	2	4	dull 0	4	170 Ж 2
6	2	4	dull 0	0	102 Ж 6
8	4	2, 6	dull 3	0	66 Ж 106
8	8	4	dull 3	4	232 Ж 104
16	8	1, 7	sharp	1	128 Ж 128
16	8	3, 5	sharp	3	22 Ж 150
24	4	2, 6	dull 2	2	40 Ж 40
24	4	2, 6	dull 1	2	136 Ж 8
24	8	4	dull 1	2	120 Ж 24
24	8	4	dull 2	2	228 Ж 44
48	8	3, 5	sharp	3	42 Ж 130
48	8	3, 5	sharp	1	230 Ж 134