Flatten

The collection with all levels of nested lists merged into one.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageFlatten✓↗

Flatten(collection)every level of nested lists merged into one.

Flatten(collection, n)flattening limited to the top n levels.

Flatten(collection, PositiveInfinity)an explicit infinite depth, the same as the default.

Flatten(collection, {{p1}, {p2}, …})a permutation-shaped list of levels regroups the array's dimensions — transposing a matrix, say.

Flatten(expr)nested calls of any one head flatten, not just $List$'s.

Typed signatures

Type	Package	Applies
(value, integer?) -> list	compute-engine	always
(any, any?) -> any	combinatoricsreplaces compute-engine	always

Domain: Collections

Details

• Undoes Partition: chunking and then re-flattening recovers the original list.
• Simply deletes inner braces/levels; it doesn't otherwise reorder or transform elements.
• A depth argument limits flattening to that many levels, leaving deeper nesting intact.
• Any single head's nested calls flatten, not just $List$'s.

Examples

1 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{Flatten}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack1, 2, 3, 4\bigr\rbrack$

live ↗

In$\mathrm{Flatten}(\bigl\lbrack\bigl\lbrack1, \bigl\lbrack2, 3\bigr\rbrack\bigr\rbrack, \bigl\lbrack4\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack1, 2, 3, 4\bigr\rbrack$

live ↗

In$\mathrm{Flatten}(\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, \bigl\lbrack d\bigr\rbrack, f\bigr\rbrack, \bigl\lbrack g, \bigl\lbrack h, k\bigr\rbrack\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack a, b, c, d, f, g, h, k\bigr\rbrack$

live ↗

In$\mathrm{Flatten}(\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, \bigl\lbrack d\bigr\rbrack, k\bigr\rbrack, \bigl\lbrack f, \bigl\lbrack g, h\bigr\rbrack\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack a, b, c, d, k, f, g, h\bigr\rbrack$

live ↗

In$\mathrm{Flatten}(\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, \bigl\lbrack d\bigr\rbrack, k\bigr\rbrack, \bigl\lbrack f, \bigl\lbrack g, h\bigr\rbrack\bigr\rbrack\bigr\rbrack, 1)$Out$\bigl\lbrack a, b, c, \bigl\lbrack d\bigr\rbrack, k, f, \bigl\lbrack g, h\bigr\rbrack\bigr\rbrack$

Scope #

A depth argument limits flattening to just that many levels, leaving deeper nesting intact

live ↗

In$\mathrm{Flatten}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, \bigl\lbrack4\bigr\rbrack\bigr\rbrack\bigr\rbrack, 1)$Out$\bigl\lbrack1, 2, 3, \bigl\lbrack4\bigr\rbrack\bigr\rbrack$

Flatten only the first level

live ↗

In$\mathrm{Flatten}(\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, \bigl\lbrack d\bigr\rbrack, f\bigr\rbrack, \bigl\lbrack g, \bigl\lbrack h, k\bigr\rbrack\bigr\rbrack\bigr\rbrack, 1)$Out$\bigl\lbrack a, b, c, \bigl\lbrack d\bigr\rbrack, f, g, \bigl\lbrack h, k\bigr\rbrack\bigr\rbrack$

Depth 0 leaves the list unchanged

live ↗

In$\mathrm{Flatten}(\bigl\lbrack0, \bigl\lbrack1\bigr\rbrack, \bigl\lbrack\bigl\lbrack2, -2\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack3\bigr\rbrack, \bigl\lbrack-3\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack4\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack, 0)$Out$\bigl\lbrack0, \bigl\lbrack1\bigr\rbrack, \bigl\lbrack\bigl\lbrack2, -2\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack3\bigr\rbrack, \bigl\lbrack-3\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack4\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack$

Depth 2 removes two levels of nesting

live ↗

In$\mathrm{Flatten}(\bigl\lbrack0, \bigl\lbrack1\bigr\rbrack, \bigl\lbrack\bigl\lbrack2, -2\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack3\bigr\rbrack, \bigl\lbrack-3\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack4\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack, 2)$Out$\bigl\lbrack0, 1, 2, -2, \bigl\lbrack3\bigr\rbrack, \bigl\lbrack-3\bigr\rbrack, \bigl\lbrack\bigl\lbrack4\bigr\rbrack\bigr\rbrack\bigr\rbrack$

Depth 3

live ↗

In$\mathrm{Flatten}(\bigl\lbrack0, \bigl\lbrack1\bigr\rbrack, \bigl\lbrack\bigl\lbrack2, -2\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack3\bigr\rbrack, \bigl\lbrack-3\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack4\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack, 3)$Out$\bigl\lbrack0, 1, 2, -2, 3, -3, \bigl\lbrack4\bigr\rbrack\bigr\rbrack$

Depth 4 reaches the deepest level

live ↗

In$\mathrm{Flatten}(\bigl\lbrack0, \bigl\lbrack1\bigr\rbrack, \bigl\lbrack\bigl\lbrack2, -2\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack3\bigr\rbrack, \bigl\lbrack-3\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack4\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack, 4)$Out$\bigl\lbrack0, 1, 2, -2, 3, -3, 4\bigr\rbrack$

An explicit infinite depth flattens every level, like the default

live ↗

In$\mathrm{Flatten}(\bigl\lbrack0, \bigl\lbrack1\bigr\rbrack, \bigl\lbrack\bigl\lbrack2, -2\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack3\bigr\rbrack, \bigl\lbrack-3\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack4\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack, ∞)$Out$\bigl\lbrack0, 1, 2, -2, 3, -3, 4\bigr\rbrack$

A list of levels regroups the dimensions, here into a transpose

live ↗

In$\mathrm{Flatten}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack2\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, 3\bigr\rbrack, \bigl\lbrack2, 4\bigr\rbrack\bigr\rbrack$

Nested calls of any one head flatten, not just lists

live ↗

In$\mathrm{Flatten}(f(a, f(b, f(c))))$Out$f(a, b, c)$

A depth limit of 1 removes just one layer of nesting

live ↗

In$\mathrm{Flatten}(\bigl\lbrack0, \bigl\lbrack1\bigr\rbrack, \bigl\lbrack\bigl\lbrack2, -2\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack3\bigr\rbrack, \bigl\lbrack-3\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack4\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack, 1)$Out$\bigl\lbrack0, 1, \bigl\lbrack2, -2\bigr\rbrack, \bigl\lbrack\bigl\lbrack3\bigr\rbrack, \bigl\lbrack-3\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack4\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack$

With no depth given, every level of nesting is removed

live ↗

In$\mathrm{Flatten}(\bigl\lbrack0, \bigl\lbrack1\bigr\rbrack, \bigl\lbrack\bigl\lbrack2, -2\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack3\bigr\rbrack, \bigl\lbrack-3\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack4\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack0, 1, 2, -2, 3, -3, 4\bigr\rbrack$

Properties #

$Flatten$ undoes Partition: chunking and re-flattening recovers the original list

live ↗

In$\mathrm{Flatten}(\mathrm{Partition}(\bigl\lbrack1, 2, 3, 4, 5, 6\bigr\rbrack, 2))=\bigl\lbrack1, 2, 3, 4, 5, 6\bigr\rbrack$Out$\mathrm{True}$

A list with no nesting is unchanged

live ↗

In$\mathrm{Flatten}(\bigl\lbrack1, 2, 3\bigr\rbrack)$Out$\bigl\lbrack1, 2, 3\bigr\rbrack$

Flattening an $m \times n$ matrix gives $mn$ elements. See Length

live ↗

In$\mathrm{Length}(\mathrm{Flatten}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack4, 5, 6\bigr\rbrack\bigr\rbrack))$Out$6$

A $3\times4$ array of symbolic entries flattens row by row

live ↗

In$\mathrm{Flatten}(\mathrm{Array}(a, \bigl\lbrack3, 4\bigr\rbrack))$Out$\bigl\lbrack a(1, 1), a(1, 2), a(1, 3), a(1, 4), a(2, 1), a(2, 2), a(2, 3), a(2, 4), a(3, 1), a(3, 2), a(3, 3), a(3, 4)\bigr\rbrack$

A three-index array flattens in lexicographic index order, so the digits count up like an odometer

live ↗

In$\mathrm{Flatten}(\mathrm{Array}((\operatorname{\_1}, \operatorname{\_2}, \operatorname{\_3})↦100\operatorname{\_1}+10\operatorname{\_2}+\operatorname{\_3}, \bigl\lbrack3, 3, 3\bigr\rbrack))$Out$\bigl\lbrack111, 112, 113, 121, 122, 123, 131, 132, 133, 211, 212, 213, 221, 222, 223, 231, 232, 233, 311, 312, 313, 321, 322, 323, 331, 332, 333\bigr\rbrack$

live ↗

In$\mathrm{Flatten}(\mathrm{array})$Out$\mathrm{Flatten}(\mathrm{array})$

About

Examples	23
Checked against	Wolfram Language 23 agreemathlib4 2 agree10 error
Crosswalk	Wolfram Language

Implementation

mappedmathlib4packages/oracle/src/mappings.ts

A list of lists of numbers, merged one level, which is every level there is. Deeper or ragged nesting and a flat list fail to elaborate, since Lean types them differently from ours.

mappedmathlib4packages/oracle/src/mappings.ts

A depth of one or more over a list of lists of numbers, where that is the same as flattening fully. Depth zero or a negative depth fails to elaborate, as does any nesting deeper than two levels.

See also: Join