# 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.

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*.

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]

Permutations are often shown in Lex order (see **here**),

but RevCoLex is the order that works for an infinite number of permutations

and corresponds to the CoLex order of the permutations' inversion vectors (shown in red in the following files, except the one on the right).

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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## 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.

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