Join

The concatenation of the argument collections, in order.

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

Join(a, b, …)the concatenation of the collections, in order.

Join(a, b, …, n)corresponding sublists n-1 levels down joined pairwise, instead of the top level.

Typed signatures

Type	Package	Applies
((T+) -> T where T: string) & ((collection<any>*) -> collection)	combinatoricsreplaces compute-engine	always
((T+) -> T where T: string) & ((collection<any>*) -> collection)	compute-engine	always

Domain: Collections

Details

• $Join(A, B) = Flatten(\{A, B\}, 1)$. See Flatten.
• The argument collections don't need to be $List$, but must all share the same head.
• A trailing integer argument joins at that level instead of the top level.
• See Append for adding a single element rather than concatenating collections.

Examples

3 more kept as data, checked against the oracles

Basic #

live ↗

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

live ↗

In$\mathrm{Join}(\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack, \bigl\lbrack5\bigr\rbrack)$Out$\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack$

live ↗

In$\mathrm{Join}(\bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack x, y\bigr\rbrack, \bigl\lbrack u, v, w\bigr\rbrack)$Out$\bigl\lbrack a, b, c, x, y, u, v, w\bigr\rbrack$

Scope #

A trailing integer argument joins at that level rather than the top level

live ↗

In$\mathrm{Join}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack5, 6\bigr\rbrack, \bigl\lbrack7, 8\bigr\rbrack\bigr\rbrack, 2)$Out$\bigl\lbrack\bigl\lbrack1, 2, 5, 6\bigr\rbrack, \bigl\lbrack3, 4, 7, 8\bigr\rbrack\bigr\rbrack$

Joining matrices stacks their rows

live ↗

In$\mathrm{Join}(\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, d\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, d\bigr\rbrack, \bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack$

Sets as well as lists, as long as all arguments share the head

live ↗

In$\mathrm{Join}(\lbrace1, 2\rbrace, \lbrace3\rbrace)$Out$\lbrace1, 2, 3\rbrace$

A lazy Range is spliced in

live ↗

In$\mathrm{Join}(\bigl\lbrack1, 2\bigr\rbrack, 3..5)$Out$\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack$

With no arguments, the empty list

live ↗

In$\mathrm{Join}()$Out$\bigl\lbrack\bigr\rbrack$

At level 2, matrices are joined side by side

live ↗

In$\mathrm{Join}(\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, d\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack, 2)$Out$\bigl\lbrack\bigl\lbrack a, b, 1, 2\bigr\rbrack, \bigl\lbrack c, d, 3, 4\bigr\rbrack\bigr\rbrack$

Ragged rows join at level 2 too

live ↗

In$\mathrm{Join}(\bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack5, 6\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack2, 3\bigr\rbrack, \bigl\lbrack7\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack4\bigr\rbrack, \bigl\lbrack8\bigr\rbrack\bigr\rbrack, 2)$Out$\bigl\lbrack\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \bigl\lbrack5, 6, 7, 8\bigr\rbrack\bigr\rbrack$

A row missing from the shorter array simply isn't merged in

live ↗

In$\mathrm{Join}(\bigl\lbrack\bigl\lbrack x\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack, 2)$Out$\bigl\lbrack\bigl\lbrack x, 1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack$

Any head, as long as all the arguments share it

live ↗

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

live ↗

In$\mathrm{Join}(\mathrm{aa}, \mathrm{bb})$Out$\mathrm{Join}(\mathrm{aa}, \mathrm{bb})$

Properties #

$Join(A, B) = Flatten(\{A, B\}, 1)$. See Flatten

live ↗

In$\mathrm{Join}(\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack)=\mathrm{Flatten}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack, 1)$Out$\mathrm{True}$

$Join$ is associative

live ↗

In$\mathrm{Join}(\mathrm{Join}(\bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack x, y\bigr\rbrack), \bigl\lbrack u, v, w\bigr\rbrack)=\mathrm{Join}(\bigl\lbrack a, b, c\bigr\rbrack, \mathrm{Join}(\bigl\lbrack x, y\bigr\rbrack, \bigl\lbrack u, v, w\bigr\rbrack))$Out$\mathrm{True}$

Neat examples #

Joining a list to itself doubles it: lengths $1, 2, 4, 8, 16$

live ↗

In$\mathrm{NestList}(\operatorname{\_1}↦\mathrm{Join}(\operatorname{\_1}, \operatorname{\_1}), \bigl\lbrack x\bigr\rbrack, 4)$Out$\bigl\lbrack\bigl\lbrack x\bigr\rbrack, \bigl\lbrack x, x\bigr\rbrack, \bigl\lbrack x, x, x, x\bigr\rbrack, \bigl\lbrack x, x, x, x, x, x, x, x\bigr\rbrack, \bigl\lbrack x, x, x, x, x, x, x, x, x, x, x, x, x, x, x, x\bigr\rbrack\bigr\rbrack$

live ↗

In$\mathrm{NestList}(\operatorname{\_1}↦\bigl\lbrack\operatorname{\_1}, 1-\operatorname{\_1}\bigr\rbrack, \bigl\lbrack1\bigr\rbrack, 5)$Out$\bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack\bigl\lbrack0\bigr\rbrack, \bigl\lbrack1\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack1\bigr\rbrack, \bigl\lbrack0\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack\bigr\rbrack$

About

Examples	20
Checked against	Wolfram Language 20 agreemathlib4 5 agree2 error
Crosswalk	Wolfram Language

Implementation

mappedmathlib4packages/oracle/src/mappings.ts

Two lists of one element type concatenated; a trailing level integer, or a non-list operand, fails to elaborate.

mappedmathlib4packages/oracle/src/mappings.ts

Three lists of one element type concatenated; a trailing level integer, or a non-list operand, fails to elaborate.

See also: Flatten, Append