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Permutations with four variables: (mathematica code link1, simplify this code)


Permutation table C for x=4 binary variables has 2^x=2^4=16 permutations, ie 0000,0001,0010,0011,0100,0101,0110,0111,1000,1001,1010,1011,1100,1101,1110,1111.

Table C: binary table for four variables
n
 variable1
 variable2
 variable3
 variable4
1
 0
 0
 0
 0
2
 0
 0
 0
 1
3
 0
 0
 1
 0
4
 0
 0
 1
 1
5
 0
 1
 0
 0
6
 0
 1
 0
 1
7
 0
 1
 1
 0
8
 0
 1
 1
 1
9
 1
 0
 0
 0
10
 1
 0
 0
 1
11
 1
 0
 1
 0
12
 1
 0
 1
 1
13
 1
 1
 0
 0
14
 1
 1
 0
 1
15
 1
 1
 1
 0
16
 1
 1
 1
 1

Proportions:

For the range 1 to 2^x for x=4, for n:
0000 occurs 1/(2^x)=1/16 of the time
0001 occurs 1/16 of the time
0010 occurs 1/16 of the time
0011 occurs 1/16 of the time
0100 occurs 1/16 of the time
0101 occurs 1/16 of the time
0110 occurs 1/16 of the time
0111 occurs 1/16 of the time
1000 occurs 1/16 of the time
1001 occurs 1/16 of the time
1010 occurs 1/16 of the time
1011 occurs 1/16 of the time
1100 occurs 1/16 of the time
1101 occurs 1/16 of the time
1110 occurs 1/16 of the time
1111 occurs 1/16 of the time

These proportions sum to 1, corresponding to all natural numbers.

1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16=1. 

Sequences:

The n values where these permutations occur give the four sequences: (only first 4 values shown)

0000: n: {1, 17, 33, 49}
0001: n: {2, 18, 34, 50}
0010: n: {3, 19, 35, 51}
0011: n: {4, 20, 36, 52}
0100: n: {5, 21, 37, 53}
0101: n: {6, 22, 38, 54}
0110: n: {7, 23, 39, 55}
0111: n: {8, 24, 40, 56}
1000: n: {9, 25, 41, 57}
1001: n: {10, 26, 42, 58}
1010: n: {11, 27, 43, 59}
1011: n: {12, 28, 44, 60}
1100: n: {13, 29, 45, 61}
1101: n: {14, 30, 46, 62}
1110: n: {15, 31, 47, 63}
1111: n: {16, 32, 48, 64}

The difference sequences of the above:
0000: Differences[n]: {16,16,16}
0001: Differences[n]: {16,16,16}
0010: Differences[n]: {16,16,16}
0011: Differences[n]: {16,16,16}
0100: Differences[n]: {16,16,16}
0101: Differences[n]: {16,16,16}
0110: Differences[n]: {16,16,16}
0111: Differences[n]: {16,16,16}
1000: Differences[n]: {16,16,16}
1001: Differences[n]: {16,16,16}
1010: Differences[n]: {16,16,16}
1011: Differences[n]: {16,16,16}
1100: Differences[n]: {16,16,16}
1101: Differences[n]: {16,16,16}
1110: Differences[n]: {16,16,16}
1111: Differences[n]: {16,16,16}


Permutation table D for x=4 prime divisors has maximum 2^x=2^4=16 distinct permutations, ie
for four prime divisors (2, 3, 5, 7) the divisibility pattern 0000,0001,0010,0011,0100,0101,0110,0111,1000,1001,1010,1011,1100,1101,1110,1111 gives the 16 distinct permutations in order of first appearance in Table D: 0000,1000,0100,0010,1100,0001,1010,1001,0110,0101,1110,0011,1101,1011,0111,1111.
The 16 first appearances occur at n: 1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210.  These are the divisors of 210.
Ie. in Table D, for n=3, 3 is divisible by 3 and not 2 and not 5 and not 7, so the value of the four columns are 0,1,0,0 for row n=3.

Table D: Prime Factor table for 4 prime factors
n
 prime2
 prime3
 prime5
 prime7
1
 0
 0
 0
 0
2
 1
 0
 0
 0
3
 0
 1
 0
 0
4
 1
 0
 0
 0
5
 0
 0
 1
 0
6
 1
 1
 0
 0
7
 0
 0
 0
 1
8
 1
 0
 0
 0
9
 0
 1
 0
 0
10
 1
 0
 1
 0
11
 0
 0
 0
 0
12
 1
 1
 0
 0
13
 0
 0
 0
 0
14
 1
 0
 0
 1
15
 0
 1
 1
 0
16
 1
 0
 0
 0
17
 0
 0
 0
 0
18
 1
 1
 0
 0
19
 0
 0
 0
 0
20
 1
 0
 1
 0
21
 0
 1
 0
 1
22
 1
 0
 0
 0
23
 0
 0
 0
 0
24
 1
 1
 0
 0
25
 0
 0
 1
 0
26
 1
 0
 0
 0
27
 0
 1
 0
 0
28
 1
 0
 0
 1
29
 0
 0
 0
 0
30
 1
 1
 1
 0
31
 0
 0
 0
 0
32
 1
 0
 0
 0
33
 0
 1
 0
 0
34
 1
 0
 0
 0
35
 0
 0
 1
 1
36
 1
 1
 0
 0
37
 0
 0
 0
 0
38
 1
 0
 0
 0
39
 0
 1
 0
 0
40
 1
 0
 1
 0
41
 0
 0
 0
 0
42
 1
 1
 0
 1
43
 0
 0
 0
 0
44
 1
 0
 0
 0
45
 0
 1
 1
 0
46
 1
 0
 0
 0
47
 0
 0
 0
 0
48
 1
 1
 0
 0
49
 0
 0
 0
 1
50
 1
 0
 1
 0
51
 0
 1
 0
 0
52
 1
 0
 0
 0
53
 0
 0
 0
 0
54
 1
 1
 0
 0
55
 0
 0
 1
 0
56
 1
 0
 0
 1
57
 0
 1
 0
 0
58
 1
 0
 0
 0
59
 0
 0
 0
 0
60
 1
 1
 1
 0
61
 0
 0
 0
 0
62
 1
 0
 0
 0
63
 0
 1
 0
 1
64
 1
 0
 0
 0
65
 0
 0
 1
 0
66
 1
 1
 0
 0
67
 0
 0
 0
 0
68
 1
 0
 0
 0
69
 0
 1
 0
 0
70
 1
 0
 1
 1
71
 0
 0
 0
 0
72
 1
 1
 0
 0
73
 0
 0
 0
 0
74
 1
 0
 0
 0
75
 0
 1
 1
 0
76
 1
 0
 0
 0
77
 0
 0
 0
 1
78
 1
 1
 0
 0
79
 0
 0
 0
 0
80
 1
 0
 1
 0
81
 0
 1
 0
 0
82
 1
 0
 0
 0
83
 0
 0
 0
 0
84
 1
 1
 0
 1
85
 0
 0
 1
 0
86
 1
 0
 0
 0
87
 0
 1
 0
 0
88
 1
 0
 0
 0
89
 0
 0
 0
 0
90
 1
 1
 1
 0
91
 0
 0
 0
 1
92
 1
 0
 0
 0
93
 0
 1
 0
 0
94
 1
 0
 0
 0
95
 0
 0
 1
 0
96
 1
 1
 0
 0
97
 0
 0
 0
 0
98
 1
 0
 0
 1
99
 0
 1
 0
 0
100
 1
 0
 1
 0
101
 0
 0
 0
 0
102
 1
 1
 0
 0
103
 0
 0
 0
 0
104
 1
 0
 0
 0
105
 0
 1
 1
 1
106
 1
 0
 0
 0
107
 0
 0
 0
 0
108
 1
 1
 0
 0
109
 0
 0
 0
 0
110
 1
 0
 1
 0
111
 0
 1
 0
 0
112
 1
 0
 0
 1
113
 0
 0
 0
 0
114
 1
 1
 0
 0
115
 0
 0
 1
 0
116
 1
 0
 0
 0
117
 0
 1
 0
 0
118
 1
 0
 0
 0
119
 0
 0
 0
 1
120
 1
 1
 1
 0
121
 0
 0
 0
 0
122
 1
 0
 0
 0
123
 0
 1
 0
 0
124
 1
 0
 0
 0
125
 0
 0
 1
 0
126
 1
 1
 0
 1
127
 0
 0
 0
 0
128
 1
 0
 0
 0
129
 0
 1
 0
 0
130
 1
 0
 1
 0
131
 0
 0
 0
 0
132
 1
 1
 0
 0
133
 0
 0
 0
 1
134
 1
 0
 0
 0
135
 0
 1
 1
 0
136
 1
 0
 0
 0
137
 0
 0
 0
 0
138
 1
 1
 0
 0
139
 0
 0
 0
 0
140
 1
 0
 1
 1
141
 0
 1
 0
 0
142
 1
 0
 0
 0
143
 0
 0
 0
 0
144
 1
 1
 0
 0
145
 0
 0
 1
 0
146
 1
 0
 0
 0
147
 0
 1
 0
 1
148
 1
 0
 0
 0
149
 0
 0
 0
 0
150
 1
 1
 1
 0
151
 0
 0
 0
 0
152
 1
 0
 0
 0
153
 0
 1
 0
 0
154
 1
 0
 0
 1
155
 0
 0
 1
 0
156
 1
 1
 0
 0
157
 0
 0
 0
 0
158
 1
 0
 0
 0
159
 0
 1
 0
 0
160
 1
 0
 1
 0
161
 0
 0
 0
 1
162
 1
 1
 0
 0
163
 0
 0
 0
 0
164
 1
 0
 0
 0
165
 0
 1
 1
 0
166
 1
 0
 0
 0
167
 0
 0
 0
 0
168
 1
 1
 0
 1
169
 0
 0
 0
 0
170
 1
 0
 1
 0
171
 0
 1
 0
 0
172
 1
 0
 0
 0
173
 0
 0
 0
 0
174
 1
 1
 0
 0
175
 0
 0
 1
 1
176
 1
 0
 0
 0
177
 0
 1
 0
 0
178
 1
 0
 0
 0
179
 0
 0
 0
 0
180
 1
 1
 1
 0
181
 0
 0
 0
 0
182
 1
 0
 0
 1
183
 0
 1
 0
 0
184
 1
 0
 0
 0
185
 0
 0
 1
 0
186
 1
 1
 0
 0
187
 0
 0
 0
 0
188
 1
 0
 0
 0
189
 0
 1
 0
 1
190
 1
 0
 1
 0
191
 0
 0
 0
 0
192
 1
 1
 0
 0
193
 0
 0
 0
 0
194
 1
 0
 0
 0
195
 0
 1
 1
 0
196
 1
 0
 0
 1
197
 0
 0
 0
 0
198
 1
 1
 0
 0
199
 0
 0
 0
 0
200
 1
 0
 1
 0
201
 0
 1
 0
 0
202
 1
 0
 0
 0
203
 0
 0
 0
 1
204
 1
 1
 0
 0
205
 0
 0
 1
 0
206
 1
 0
 0
 0
207
 0
 1
 0
 0
208
 1
 0
 0
 0
209
 0
 0
 0
 0
210
 1
 1
 1
 1

Proportions:

Table E: Proportions of n values that each permutation occurs sorted descending by proportion.
n
 2
 3
 5
 7
 proportion
 reduced proportion
1
 0
 0
 0
 0
 48/210
 8/35
2
 1
 0
 0
 0
 48/210
 8/35
3
 0
 1
 0
 0
 24/210
 4/35
4
 1
 1
 0
 0
 24/210
 4/35
5
 0
 0
 1
 0
 12/210
 2/35
6
 1
 0
 1
 0
 12/210
 2/35
7
 0
 0
 0
 1
 8/210
 4/105
8
 1
 0
 0
 1
 8/210
 4/105
9
 0
 1
 1
 0
 6/210
 1/35
10
 1
 1
 1
 0
 6/210
 1/35
11
 0
 1
 0
 1
 4/210
 2/105
12
 1
 1
 0
 1
 4/210
 2/105
13
 0
 0
 1
 1
 2/210
 1/105
14
 1
 0
 1
 1
 2/210
 1/105
15
 0
 1
 1
 1
 1/210
 1/210
16
 1
 1
 1
 1
 1/210
 1/210

For the range 1 to 210 for n:

0000 occurs 48/210=8/35 of the time, so 8/35 of numbers don't have any of prime factors 2 or 3 or 5 or 7.
1000 occurs 48/210=8/35 of the time, so 8/35 of numbers have prime factor 2 and don't have any of prime factors 3 or 5 or 7.
0100 occurs 24/210=4/35 of the time, so 4/35 of numbers have prime factor 3 and don't have any of prime factors 2 or 5 or 7.
1100 occurs 24/210=4/35 of the time, so 4/35 of numbers have prime factor 2 and 3 and don't have either of prime factors 5 or 7.
0010 occurs 12/210=2/35 of the time, so 2/35 of numbers have prime factor 5 and don't have any of prime factors 2 or 3 or 7.
1010 occurs 12/210=2/35 of the time, so 2/35 of numbers have prime factor 2 and 5 and don't have either of prime factors 3 or 7.
0001 occurs 8/210=4/105 of the time, so 4/105 of numbers have prime factor 7 and don't have any of prime factors 2 or 3 or 5.
1001 occurs 8/210=4/105 of the time, so 4/105 of numbers have prime factor 2 and 7 and don't have either of prime factors 3 or 5.
0110 occurs 6/210=1/35 of the time, so 1/35 of numbers have prime factor 3 and 5 and don't have either of prime factors 2 or 7.
1110 occurs 6/210=1/35 of the time, so 1/35 of numbers have prime factor 2 and 3 and 5 and don't have prime factor 7.
0101 occurs 4/210=2/105 of the time, so 2/105 of numbers have prime factor 3 and 7 and don't have either of prime factors 2 or 5.
1101 occurs 4/210=2/105 of the time, so 2/105 of numbers have prime factor 2 and 3 and 7 and don't have prime factor 5.
0011 occurs 2/210=1/105 of the time, so 1/105 of numbers have prime factor 5 and 7 and don't have either of prime factors 2 or 3.
1011 occurs 2/210=1/105 of the time, so 1/105 of numbers have prime factor 2 and 5 and 7 and don't have prime factor 3.
0111 occurs 1/210 of the time, so 1/210 of numbers have prime factor 3 and 5 and 7 and don't have prime factor 2.
1111 occurs 1/210 of the time, so 1/210 of numbers have prime factor 2 and 3 and 5 and 7.

[additional proportions: (check these for larger primorials)]


Distinct numerators on the non-reduced proportions: 1, 2, 4, 6, 8, 12, 24, 48.

The 16 proportions sum to 1, corresponding to all natural numbers.

8/35+8/35+4/35+4/35+2/35+2/35+4/105+4/105+1/35+1/35+2/105+2/105+1/105+1/105+1/210+1/210=1

There are 8 distinct proportions.

{8/35, 4/35, 2/35, 4/105, 1/35, 2/105, 1/105, 1/210}

Distinct numerators of the reduced proportions: 1,2,4,8.
Distinct denominators of the reduced proportions: 35,105,210.

Each of the 8 proportions corresponds to 2 distinct prime factor permutations each, ie 8/35 corresponds to the {0,0,0,0} and {1,0,0,0} prime factor permutations.

Sequences:

The n values where these permutations occur give the 16 sequences: (only first 20 values shown)
Each sequence starts with a distinct divisor of 210.  The sequences are sorted in descending order by the corresponding proportions.

0000: n: {1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83}
1000: n: {2, 4, 8, 16, 22, 26, 32, 34, 38, 44, 46, 52, 58, 62, 64, 68, 74, 76, 82, 86}

0100: n: {3, 9, 27, 33, 39, 51, 57, 69, 81, 87, 93, 99, 111, 117, 123, 129, 141, 153, 159, 171}
1100: n: {6, 12, 18, 24, 36, 48, 54, 66, 72, 78, 96, 102, 108, 114, 132, 138, 144, 156, 162, 174}

0010: n: {5, 25, 55, 65, 85, 95, 115, 125, 145, 155, 185, 205, 215, 235, 265, 275, 295, 305, 325, 335}
1010: n: {10, 20, 40, 50, 80, 100, 110, 130, 160, 170, 190, 200, 220, 230, 250, 260, 290, 310, 320, 340}

0001: n: {7, 49, 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329, 343, 371, 413, 427, 469, 497, 511}
1001: n: {14, 28, 56, 98, 112, 154, 182, 196, 224, 238, 266, 308, 322, 364, 392, 406, 434, 448, 476, 518}

0110: n: {15, 45, 75, 135, 165, 195, 225, 255, 285, 345, 375, 405, 435, 465, 495, 555, 585, 615, 645, 675}
1110: n: {30, 60, 90, 120, 150, 180, 240, 270, 300, 330, 360, 390, 450, 480, 510, 540, 570, 600, 660, 690}

0101: n: {21, 63, 147, 189, 231, 273, 357, 399, 441, 483, 567, 609, 651, 693, 777, 819, 861, 903, 987, 1029}
1101: n: {42, 84, 126, 168, 252, 294, 336, 378, 462, 504, 546, 588, 672, 714, 756, 798, 882, 924, 966, 1008}

0011: n: {35, 175, 245, 385, 455, 595, 665, 805, 875, 1015, 1085, 1225, 1295, 1435, 1505, 1645, 1715, 1855, 1925, 2065}
1011: n: {70, 140, 280, 350, 490, 560, 700, 770, 910, 980, 1120, 1190, 1330, 1400, 1540, 1610, 1750, 1820, 1960, 2030}

0111: n: {105, 315, 525, 735, 945, 1155, 1365, 1575, 1785, 1995, 2205, 2415, 2625, 2835, 3045, 3255, 3465, 3675, 3885, 4095}
1111: n: {210, 420, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2310, 2520, 2730, 2940, 3150, 3360, 3570, 3780, 3990, 4200}

First three values in each of the 16 sequences:

{{1, 11, 13}, {2, 4, 8}, {3, 9, 27}, {6, 12, 18}, {5, 25, 55}, {10, 20, 40}, {15, 45, 75}, {30, 60, 90}, {7, 49, 77}, {14, 28, 56}, {21, 63, 147}, {42, 84, 126}, {35, 175, 245}, {70, 140, 280}, {105, 315, 525}, {210, 420, 630}}

Difference Sequences:

First differences sequences for the first 20 values:

0000: Differences1[n]: {10, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4}
1000: Differences1[n]: {2, 4, 8, 6, 4, 6, 2, 4, 6, 2, 6, 6, 4, 2, 4, 6, 2, 6, 4}
0100: Differences1[n]: {6, 18, 6, 6, 12, 6, 12, 12, 6, 6, 6, 12, 6, 6, 6, 12, 12, 6, 12}
1100: Differences1[n]: {6, 6, 6, 12, 12, 6, 12, 6, 6, 18, 6, 6, 6, 18, 6, 6, 12, 6, 12}
0010: Differences1[n]: {20, 30, 10, 20, 10, 20, 10, 20, 10, 30, 20, 10, 20, 30, 10, 20, 10, 20, 10}
1010: Differences1[n]: {10, 20, 10, 30, 20, 10, 20, 30, 10, 20, 10, 20, 10, 20, 10, 30, 20, 10, 20}
0001: Differences1[n]: {42, 28, 14, 28, 14, 28, 42, 14, 42, 28, 14, 28, 14, 28, 42, 14, 42, 28, 14}
1001: Differences1[n]: {14, 28, 42, 14, 42, 28, 14, 28, 14, 28, 42, 14, 42, 28, 14, 28, 14, 28, 42}
0110: Differences1[n]: {30, 30, 60, 30, 30, 30, 30, 30, 60, 30, 30, 30, 30, 30, 60, 30, 30, 30, 30}
1110: Differences1[n]: {30, 30, 30, 30, 30, 60, 30, 30, 30, 30, 30, 60, 30, 30, 30, 30, 30, 60, 30}
0101: Differences1[n]: {42, 84, 42, 42, 42, 84, 42, 42, 42, 84, 42, 42, 42, 84, 42, 42, 42, 84, 42}
1101: Differences1[n]: {42, 42, 42, 84, 42, 42, 42, 84, 42, 42, 42, 84, 42, 42, 42, 84, 42, 42, 42}
0011: Differences1[n]: {140, 70, 140, 70, 140, 70, 140, 70, 140, 70, 140, 70, 140, 70, 140, 70, 140, 70, 140}
1011: Differences1[n]: {70, 140, 70, 140, 70, 140, 70, 140, 70, 140, 70, 140, 70, 140, 70, 140, 70, 140, 70}
0111: Differences1[n]: {210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210}
1111: Differences1[n]: {210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210}

Sequence pattern lengths:

Ie. The repeating pattern length of 0000 and 1000 is 48 given by the numerator of the proportion 48/210.

Difference sequences offsets: (mathematica code for this, clean up this code)

Ie: 0000 is the same sequence as 1000, with a 48/2=24 offset. 

Shifting the difference sequence y by z amount x gives
0000->1000 offset by 48/2=24.
0100->1100 offset by x (double check these offsets are also the non reduced proportion numerator/2.
0010->1010 offset by x
0001->1001 offset by x
0110->1110 offset by x
0101->1101 offset by x
0011->1011 offset by x
0111->1111 offset by x

For any length of the sequences where the last value is greater or equal to 209, the below table gives the distinct values in the first differences sequences.
For example for sequence 0000: {1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209}
The distinct values in the first difference sequence are: {2,4,6,8,10} and the count of distinct values is 5.

Table F: Table E with differences sequences count column added
n
 2
 3
 5
 7
 sequence first three terms
 proportion
 reduced proportion
 first differences sequence distinct terms first differences sequence distinct terms counts
1
 0
 0
 0
 0
 1, 11, 13 48/210
 8/35
 2,4,6,8,10
 5
2
 1
 0
 0
 0
 2, 4, 8 48/210
 8/35
 2,4,6,8,10
 5
3
 0
 1
 0
 0
 3, 9, 27 24/210
 4/35
 6,12,18
 3
4
 1
 1
 0
 0
 6, 12, 18 24/210
 4/35
 6,12,18
 3
5
 0
 0
 1
 0
 5, 25, 55 12/210
 2/35
 10,20,30
 3
6
 1
 0
 1
 0
 10, 20, 40 12/210
 2/35
 10,20,30
 3
7
 0
 0
 0
 1
 7, 49, 77 8/210
 4/105
 14,28,42
 3
8
 1
 0
 0
 1
 14, 28, 56 8/210
 4/105
 14,28,42
 3
9
 0
 1
 1
 0
 15, 45, 75 6/210
 1/35
 30,60
 2
10
 1
 1
 1
 0
 30, 60, 90 6/210
 1/35
 30,60
 2
11
 0
 1
 0
 1
 21, 63, 147 4/210
 2/105
 42,84
 2
12
 1
 1
 0
 1
 42, 84, 126 4/210
 2/105
 42,84
 2
13
 0
 0
 1
 1
 35, 175, 245 2/210
 1/105
 70,140
 2
14
 1
 0
 1
 1
 70, 140, 280 2/210
 1/105
 70,140
 2
15
 0
 1
 1
 1
 105, 315, 525 1/210
 1/210
 210
 1
16
 1
 1
 1
 1
 210, 420, 630 1/210
 1/210
 210
 1

Values in column "first differences sequences values":
2,4,6,8,10,12,14,18,20,28,30,42,60,70,84,140,210. (double check)
(for differences sequence: {2, 2, 2, 2, 2, 2, 4, 2, 8, 2, 12, 18, 10, 14, 56, 70} 140-84=56 is the only value not in the original sequence, 56=48+8, 56/210=8/30, check more)
Total terms: 17. (double check)

Values in column "first differences distinct terms":
1,2,3,5.
Total terms: 4.
Counts of occurrences of each value {1,2,3,5} are {2,6,6,2}:
1->2, 2->6, 3->6, 5->2. (check these counts of occurrences more)

Pairwise difference sequences: (mathematica code link for pairwise differences)

Ie: take differences of two sequences with the same proportions:
a0000 = {1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83} (*48/210 proportion sequence*)
a1000 = {2, 4, 8, 16, 22, 26, 32, 34, 38, 44, 46, 52, 58, 62, 64, 68, 74, 76, 82, 86} (*48/210 proportion sequence*)
Abs[a0000 - a1000] (*outputs: 1, 7, 5, 1, 3, 3, 3, 3, 1, 3, 3, 5, 5, 3, 3, 1, 3, 3, 3, 3*)
Tally[Sort[Abs[a0000 - a1000]]] (*outputs: {1, 4}, {3, 12}, {5, 3}, {7, 1}*)
Sort[DeleteDuplicates[Abs[a0000 - a1000]]] (*outputs: 1, 3, 5, 7*)
CountDistinct[Abs[a0000 - a1000]] (*outputs: 4*)

Take pairwise difference sequence for range up to 210 for the full patterns etc.

show pairwise difference sequences for the first 20 values here:
0000-1000: Differences1[n]: {1, 7, 5, 1, 3, 3, 3, 3, 1, 3, 3, 5, 5, 3, 3, 1, 3, 3, 3, 3}
0100-1100: Differences1[n]: {3, 3, 9, 9, 3, 3, 3, 3, 9, 9, 3, 3, 3, 3, 9, 9, 3, 3, 3, 3}
0010-1010: Differences1[n]: {5, 5, 15, 15, 5, 5, 5, 5, 15, 15, 5, 5, 5, 5, 15, 15, 5, 5, 5, 5}
0001-1001: Differences1[n]: {7, 21, 21, 7, 7, 21, 21, 7, 7, 21, 21, 7, 7, 21, 21, 7, 7, 21, 21, 7}
0110-1110: Differences1[n]: {15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15}
0101-1101: Differences1[n]: {21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21}
0011-1011: Differences1[n]: {35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35, 35}
0111-1111: Differences1[n]: {105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105, 105}

Table G: Pairwise differences table
n
 pairwise differences
 tally
 distinct terms distinct terms counts
1
 0000-1000
 {1, 4}, {3, 12}, {5, 3}, {7, 1} 1, 3, 5, 7 4
2
 0100-1100
 {3, 14}, {9, 6}
 3, 9
 2
3
 0010-1010
 {5, 14}, {15, 6}
 5, 15
 2
4
 0001-1001
 {7, 10}, {21, 10}
 7, 21
 2
5
 0110-1110
 {15, 20}
 15
 1
6
 0101-1101
 {21, 20}
 21
 1
7
 0011-1011
 {35, 20}
 35
 1
8
 0111-1111
 {105, 20}
 105
 1



Add:

1. equivalent sequences using numerators and tuples etc
2. A309497 related transformed sequences, with example for constraint solver etc, also for sequences with equal proportions check for correspondence in their A309497 related transform sequences (ie similar to the first and half reversing for prime factor 2 added, since those two sequences also have equal proportions).
3. add info on sequence 1, 1, 2, 4, 8, 16, 26, 50, 88, 176, 352, 576, 824, 1248, 2040, 4080, 8160, 16320, 21728, 34480, 53520, 69648, 111808, 223616, 326464 (distinct proportions)
4. simplified fractions with equivalent values (ie for larger permutation patterns)
5. add two more pages, ie for 2,3,5,7,11 and 2,3,5,7,11,13 (don't add the full 2310 row and 30030 row tables, just the distinct 16 and 26 row tables)
6. add seperate H-tree visualization for page1, page2, page3, ... etc combined to show fractions etc for multiple permutation levels.
Related Mathematica code:

add code for tuples, and other code to generate sequences etc snippets

(link to related code from here)