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Orderings of compositions

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Binary representation ordering of the compositions

This ordering is based on binary run-length encoding of finite tuples in nonnegative integers.

 
n
n2
 		Ordered prime signature Numbers A-number Prime signature
0 0[1] { } {1}   { }
1 1 {1} {2, 3, 5, 7, 11, 13, 17, 19, ...} A000040 { }
2 10 {1,1} {6, 10, 15, ...} A?????? { }
3 11 {2} { , ...} A?????? { }
4 100 {1,2} { , ...} A?????? { }
5 101 {1,1,1} { , ...} A?????? { }
6 110 {2,1} { , ...} A?????? { }
7 111 {3} { , ...} A?????? { }
8 1000 {1,3} { , ...} A?????? { }
9 1001 {1,2,1} { , ...} A?????? { }
10 1010 {1,1,1,1} { , ...} A?????? { }
11 1011 {1,1,2} { , ...} A?????? { }
12 1100 {2,2} { , ...} A?????? { }
13 1101 {2,1,1} { , ...} A?????? { }
14 1110 {3,1} { , ...} A?????? { }
15 1111 {4} { , ...} A?????? { }
16 10000 {1,4} { , ...} A?????? { }
17 10001 {1,3,1} { , ...} A?????? { }
18 10010 {1,2,1,1} { , ...} A?????? { }
19 10011 {1,2,2} { , ...} A?????? { }
20 10100 {1,1,1,2} { , ...} A?????? { }
21 10101 {1,1,1,1,1} { , ...} A?????? { }
22 10110 {1,1,2,1} { , ...} A?????? { }
23 10111 {1,1,3} { , ...} A?????? { }
24 11000 {2,3} { , ...} A?????? { }
25 11001 {2,2,1} { , ...} A?????? { }
26 11010 {2,1,1,1} { , ...} A?????? { }
27 11011 {2,1,2} { , ...} A?????? { }
28 11100 {3,2} { , ...} A?????? { }
29 11101 {3,1,1} { , ...} A?????? { }
30 11110 {4,1} { , ...} A?????? { }
31 11111 {5} { , ...} A?????? { }
32 100000 {1,5} { , ...} A?????? { }
33 100001 {1,4,1} { , ...} A?????? { }
34 100010 {1,3,1,1} { , ...} A?????? { }
35 100011 {1,3,2} { , ...} A?????? { }
36 100100 {1,2,1,2} { , ...} A?????? { }
37 100101 {1,2,1,1,1} { , ...} A?????? { }
38 100110 {1,2,2,1} { , ...} A?????? { }
39 100111 {1,2,3} { , ...} A?????? { }
40 101000 {1,1,1,3} { , ...} A?????? { }
41 101001 {1,1,1,2,1} { , ...} A?????? { }
42 101010 {1,1,1,1,1,1} { , ...} A?????? { }
43 101011 {1,1,1,1,2} { , ...} A?????? { }
44 101100 {1,1,2,2} { , ...} A?????? { }
45 101101 {1,1,2,1,1} { , ...} A?????? { }
46 101110 {1,1,3,1} { , ...} A?????? { }
47 101111 {1,1,4} { , ...} A?????? { }
48 110000 {2,4} { , ...} A?????? { }
49 110001 {2,3,1} { , ...} A?????? { }
50 110010 {2,2,1,1} { , ...} A?????? { }
51 110011 {2,2,2} { , ...} A?????? { }
52 110100 {2,1,1,2} { , ...} A?????? { }
53 110101 {2,1,1,1,1} { , ...} A?????? { }
54 110110 {2,1,2,1} { , ...} A?????? { }
55 110111 {2,1,3} { , ...} A?????? { }
56 111000 {3,3} { , ...} A?????? { }
57 111001 {3,2,1} { , ...} A?????? { }
58 111010 {3,1,1,1} { , ...} A?????? { }
59 111011 {3,1,2} { , ...} A?????? { }
60 111100 {4,2} { , ...} A?????? { }
61 111101 {4,1,1} { , ...} A?????? { }
62 111110 {5,1} { , ...} A?????? { }
63 111111 {6} { , ...} A?????? { }
64 1000000 {1,6} { , ...} A?????? { }
65 1000001 {1,5,1} { , ...} A?????? { }
66 1000010 {1,4,1,1} { , ...} A?????? { }
67 1000011 {1,4,2} { , ...} A?????? { }
68 1000100 {1,3,1,2} { , ...} A?????? { }
69 1000101 {1,3,1,1,1} { , ...} A?????? { }
70 1000110 {1,3,2,1} { , ...} A?????? { }
71 1000111 {1,3,3} { , ...} A?????? { }
72 1001000 {1,2,1,3} { , ...} A?????? { }
73 1001001 {1,2,1,2,1} { , ...} A?????? { }
74 1001010 {1,2,1,1,1,1} { , ...} A?????? { }
75 1001011 {1,2,1,1,2} { , ...} A?????? { }
76 1001100 {1,2,2,2} { , ...} A?????? { }
77 1001101 {1,2,2,1,1} { , ...} A?????? { }
78 1001110 {1,2,3,1} { , ...} A?????? { }
79 1001111 {1,2,4} { , ...} A?????? { }
80 1010000 {1,1,1,4} { , ...} A?????? { }
81 1010001 {1,1,1,3,1} { , ...} A?????? { }
82 1010010 {1,1,1,2,1,1} { , ...} A?????? { }
83 1010011 {1,1,1,2,2} { , ...} A?????? { }
84 1010100 {1,1,1,1,1,2} { , ...} A?????? { }
85 1010101 {1,1,1,1,1,1,1} { , ...} A?????? { }
86 1010110 {1,1,1,1,2,1} { , ...} A?????? { }
87 1010111 {1,1,1,1,3} { , ...} A?????? { }
88 1011000 {1,1,2,3} { , ...} A?????? { }
89 1011001 {1,1,2,2,1} { , ...} A?????? { }
90 1011010 {1,1,2,1,1,1} { , ...} A?????? { }
91 1011011 {1,1,2,1,2} { , ...} A?????? { }
92 1011100 {1,1,3,2} { , ...} A?????? { }
93 1011101 {1,1,3,1,1} { , ...} A?????? { }
94 1011110 {1,1,4,1} { , ...} A?????? { }
95 1011111 {1,1,5} { , ...} A?????? { }
96 1100000 {2,5} { , ...} A?????? { }
97 1100001 {2,4,1} { , ...} A?????? { }
98 1100010 {2,3,1,1} { , ...} A?????? { }
99 1100011 {2,3,2} { , ...} A?????? { }
100 1100100 {2,2,1,2} { , ...} A?????? { }
101 1100101 {2,2,1,1,1} { , ...} A?????? { }
102 1100110 {2,2,2,1} { , ...} A?????? { }
103 1100111 {2,2,3} { , ...} A?????? { }
104 1101000 {2,1,1,3} { , ...} A?????? { }
105 1101001 {2,1,1,2,1} { , ...} A?????? { }
106 1101010 {2,1,1,1,1,1} { , ...} A?????? { }
107 1101011 {2,1,1,1,2} { , ...} A?????? { }
108 1101100 {2,1,2,2} { , ...} A?????? { }
109 1101101 {2,1,2,1,1} { , ...} A?????? { }
110 1101110 {2,1,3,1} { , ...} A?????? { }
111 1101111 {2,1,4} { , ...} A?????? { }
112 1110000 {3,4} { , ...} A?????? { }
113 1110001 {3,3,1} { , ...} A?????? { }
114 1110010 {3,2,1,1} { , ...} A?????? { }
115 1110011 {3,2,2} { , ...} A?????? { }
116 1110100 {3,1,1,2} { , ...} A?????? { }
117 1110101 {3,1,1,1,1} { , ...} A?????? { }
118 1110110 {3,1,2,1} { , ...} A?????? { }
119 1110111 {3,1,3} { , ...} A?????? { }
120 1111000 {4,3} { , ...} A?????? { }
121 1111001 {4,2,1} { , ...} A?????? { }
122 1111010 {4,1,1,1} { , ...} A?????? { }
123 1111011 {4,1,2} { , ...} A?????? { }
124 1111100 {5,2} { , ...} A?????? { }
125 1111101 {5,1,1} { , ...} A?????? { }
126 1111110 {6,1} { , ...} A?????? { }
127 1111111 {7} { , ...} A?????? { }

Sequences

A124734 Triangle read by rows, in which row
n
 sorts the compositions of
n
 by increasing length, and then by increasing lexicographic order. Similar to the “Abramowitz and Stegun” ordering of the partitions, A036036.

For example, the fifth row reads

(5),
(1, 4), (2, 3), (3, 2), (4, 1),
(1, 1, 3), (1, 2, 2), (1, 3, 1), (2, 1, 2), (2, 2, 1), (3, 1, 1),
(1, 1, 1, 2), (1, 1, 2, 1), (1, 2, 1, 1), (2, 1, 1, 1),
(1, 1, 1, 1, 1).
A296774 Triangle read by rows, in which row
n
 sorts the compositions of
n
 by increasing length, and then by decreasing lexicographic order.

For example, the fifth row reads

(5),
(4, 1), (3, 2), (2, 3), (1, 4),
(3, 1, 1), (2, 2, 1), (2, 1, 2), (1, 3, 1), (1, 2, 2), (1, 1, 3),
(2, 1, 1, 1), (1, 2, 1, 1), (1, 1, 2, 1), (1, 1, 1, 2),
(1, 1, 1, 1, 1).
A337243 Triangle read by rows, in which row
n
 sorts the compositions of
n
 by increasing length, and then by increasing colexicographic order.

For example, the fifth row reads

(5),
(4, 1), (3, 2), (2, 3), (1, 4),
(3, 1, 1), (2, 2, 1), (1, 3, 1), (2, 1, 2), (1, 2, 2), (1, 1, 3),
(2, 1, 1, 1), (1, 2, 1, 1), (1, 1, 2, 1), (1, 1, 1, 2),
(1, 1, 1, 1, 1).
A337259 Triangle read by rows, in which row
n
 sorts the compositions of
n
 by increasing length, and then by decreasing colexicographic order.

For example, the fifth row reads

(5),
(1, 4), (2, 3), (3, 2), (4, 1),
(1, 1, 3), (1, 2, 2), (2, 1, 2), (1, 3, 1), (2, 2, 1), (3, 1, 1),
(1, 1, 1, 2), (1, 1, 2, 1), (1, 2, 1, 1), (2, 1, 1, 1),
(1, 1, 1, 1, 1).
A296773 Triangle read by rows, in which row
n
 sorts the compositions of
n
 by decreasing length, and then by increasing lexicographic order.

For example, the fifth row reads

(1, 1, 1, 1, 1),
(1, 1, 1, 2), (1, 1, 2, 1), (1, 2, 1, 1), (2, 1, 1, 1),
(1, 1, 3), (1, 2, 2), (1, 3, 1), (2, 1, 2), (2, 2, 1), (3, 1, 1),
(1, 4), (2, 3), (3, 2), (4, 1),
(5).
A296772 Triangle read by rows, in which row
n
 sorts the compositions of
n
 by decreasing length, and then by decreasing lexicographic order.

For example, the fifth row reads

(1, 1, 1, 1, 1),
(2, 1, 1, 1), (1, 2, 1, 1), (1, 1, 2, 1), (1, 1, 1, 2),
(3, 1, 1), (2, 2, 1), (2, 1, 2), (1, 3, 1), (1, 2, 2), (1, 1, 3),
(4, 1), (3, 2), (2, 3), (1, 4),
(5).
A337260 Triangle read by rows, in which row
n
 sorts the compositions of
n
 by decreasing length, and then by increasing colexicographic order.

For example, the fifth row reads

(1, 1, 1, 1, 1),
(2, 1, 1, 1), (1, 2, 1, 1), (1, 1, 2, 1), (1, 1, 1, 2),
(3, 1, 1), (2, 2, 1), (1, 3, 1), (2, 1, 2), (1, 2, 2), (1, 1, 3),
(4, 1), (3, 2), (2, 3), (1, 4),
(5).
A108244 Triangle read by rows, in which row
n
 sorts the compositions of
n
 by decreasing length, and then by decreasing colexicographic order. Similar to the “Maple” ordering of the partitions, A080576.

For example, the fifth row reads

(1, 1, 1, 1, 1),
(1, 1, 1, 2), (1, 1, 2, 1), (1, 2, 1, 1), (2, 1, 1, 1),
(1, 1, 3), (1, 2, 2), (2, 1, 2), (1, 3, 1), (2, 2, 1), (3, 1, 1),
(1, 4), (2, 3), (3, 2), (4, 1),
(5).
A228369 Triangle read by rows, in which row
n
 sorts the compositions of
n
 by increasing lexicographic order.

For example, the fifth row reads

(1, 1, 1, 1, 1),
(1, 1, 1, 2), (1, 1, 2, 1), (1, 1, 3), (1, 2, 1, 1),
(1, 2, 2), (1, 3, 1), (1, 4), (2, 1, 1, 1), (2, 1, 2), (2, 2, 1),
(2, 3), (3, 1, 1), (3, 2), (4, 1),
(5).
A066099 Triangle read by rows, in which row
n
 sorts the compositions of
n
 by decreasing lexicographic order. This is the standard ordering for compositions in this database. Similar to the “Mathematica” ordering of the partitions, A080577.

For example, the fifth row reads

(5),
(4, 1),
(3, 2), (3, 1, 1),
(2, 3), (2, 2, 1), (2, 1, 2), (2, 1, 1, 1),
(1, 4), (1, 3, 1), (1, 2, 2), (1, 2, 1, 1), (1, 1, 3), (1, 1, 2, 1), (1, 1, 1, 2), (1, 1, 1, 1, 1).
A228525 Triangle read by rows, in which row
n
 sorts the compositions of
n
 by increasing colexicographic order.

For example, the fifth row reads

(1, 1, 1, 1, 1), (2, 1, 1, 1), (1, 2, 1, 1), (3, 1, 1), (1, 1, 2, 1), (2, 2, 1), (1, 3, 1), (4, 1),
(1, 1, 1, 2), (2, 1, 2), (1, 2, 2), (3, 2),
(1, 1, 3), (2, 3),
(1, 4),
(5).
A228351 Triangle read by rows, in which row
n
 sorts the compositions of
n
 by decreasing colexicographic order.

For example, the fifth row reads

(5),
(1, 4),
(2, 3), (1, 1, 3),
(3, 2), (1, 2, 2), (2, 1, 2), (1, 1, 1, 2),
(4, 1), (1, 3, 1), (2, 2, 1), (1, 1, 2, 1), (3, 1, 1), (1, 2, 1, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1).
A101211 Triangle read by rows:
n
-th row is length of run of leftmost 1’s, followed by length of run of 0’s, followed by length of run of 1’s, etc., in the binary representation of
n
, A007088. Row
n
 has A005811
 (n)
 elements.
{1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 3, 1, 4, 1, 4, 1, 3, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 3, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 3, 2, 3, 1, ...}


Notes

  1. ↑ We should consider having the empty sum here, the leading zeros not being normally represented (we put the leading zero for zero only to avoid an empty representation for zero, which would not be convenient).
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