# Lexicographic and colexicographic order

Lexicographic (Lex) and colexicographic (CoLex) order are probably the most important ways to order tuples in mathematics.

Lex order is that of a dictionary.

CoLex order is obtained by reflecting all tuples, applying Lex order, and reflecting the tuples again.

CoLex can also be obtained from Lex by reflecting all tuples and then reversing the order, or vice versa.

Lex order is more intuitive for most people.

CoLex order is more practical when the *finite sets of tuples* to be ordered shall be generalized to *infinite sets of sequences*. CoLex can be extended to natural numbers in a way that Lex cannot. Each *r*-subset of N has a finite rank under CoLex, whereas this is not the case under Lex. For example, ordering sets of natural numbers of size 3 by Lex, {2,3,4} comes after *all* sets whose lowest element is 1, which are infintely many.

Both orderings can be reversed, so there are actually four different orderings.

They can also be reflected (see **here**), but that's not a different ordering of the set of tuples, but just a certain ordering written in a different way.

When the set of tuples contains all reflections, each reflected order is equal to some of the four others (see **below**).

## Combinations[edit]

The sequence of the *k*-subsets of in CoLex order

is the beginning of the infinite sequence of *k*-subsets of in CoLex order.

This corresponds to the increasing sequence A014311 = 7,11,13,14,19,21...

In the file on the left it can be seen that the blue patterns are horizontally reflected.

However, this is not the case in the file on the right, which shows only some of the subsets.

Combinations with repetition |
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The triangle on the right shows 2-subsets of the integers.

This corresponds to the sequence A018900 as a square array.

The sequence of numbers (3,5,6,9,10,12...) corresponds to the CoLex ordering of 2-subsets:

1 2 1 3 2 3 1 4 2 4 3 4 1 5 2 5 3 5 4 5 1 6 2 6 3 6 4 6 5 6 1 7 2 7 3 7 4 7 5 7 6 7 1 8 2 8 3 8 4 8 5 8 6 8 7 8

## Permutations[edit]

Finite sets of permutations are often shown in lex order, but rev colex order has the advantage that it can be used to order an infinite number of permutations.

This above all means the finitary symmetric group on , which is the union of all finite symmetric groups, and thus contains each finitary permutation of .

Its *n*-th element (counted from 0) is found in row *n* of A055089.

This makes it possible to assign a number to a permutation that is only given in cycle notation, whithout seeing it as a permutation of a particular number of elements.

The reverse colex rank is its left inversion count interpreted as a (little-endian) factorial number.

The reverse colex order of the permutations corresponds to the colex order of these left inversion counts, as seen in the following illustrations.

Reflections are redundant |
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## Partitions[edit]

Infinite orderings of integer partitions and set partitions can be defined using CoLex ordering.

Partitions of a 5-set | |
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The rows of the binary matrices are in CoLex order. (The white fields are 0s, colored fields are 1s. | |

## Walsh functions[edit]

The rows of binary Walsh matrices in Lex and CoLex order give symmetric matrices.

For normal Walsh matrices (with 1 and −1 instead of 0 and 1) RevLex and RevCoLex would give this result.

## Subsets[edit]

The following Python code shows the 16 subsets of the set {a,b,c,d} in lexicographic order.

The second part shows that Lex order is the same as binary CoLex order, when the subsets themselves are reversed.

```
>>> asc = ['', 'a', 'b', 'ab', 'c', 'ac', 'bc', 'abc', 'd', 'ad', 'bd', 'abd', 'cd', 'acd', 'bcd', 'abcd']
>>> sorted(asc)
['', 'a', 'ab', 'abc', 'abcd', 'abd', 'ac', 'acd', 'ad', 'b', 'bc', 'bcd', 'bd', 'c', 'cd', 'd']
>>> desc = ['', 'a', 'b', 'ba', 'c', 'ca', 'cb', 'cba', 'd', 'da', 'db', 'dba', 'dc', 'dca', 'dcb', 'dcba']
>>> print desc == sorted(desc)
True
```

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