TypeScript: Crafting a Custom "Join" Type for Union Types

This challenge focuses on creating a powerful custom TypeScript type that mimics the behavior of a string join operation. You will build a type that takes a union of strings and a separator, and produces a new union type representing all possible combinations of the original strings joined by the separator. This is useful for generating precise string literal unions for complex configurations or APIs.

Problem Description

Your task is to create a TypeScript type named Join that accepts two type arguments:

T: A union of string literal types.
Separator: A string literal type to be used as the separator between elements.

The Join type should return a new union of string literal types. Each member of the new union should be formed by concatenating one or more elements from the input union T, separated by the Separator. The order of elements within each joined string must be preserved as they appear in the input union.

Key Requirements:

The Join type must handle unions of any size.
It should generate all possible combinations of joining elements from the input union T.
The order of elements within a joined string matters (e.g., if T is "a" | "b", then "a-b" is valid, but "b-a" is a separate combination and should also be generated if applicable).

Expected Behavior:

Given a union of string literals, Join should produce a union containing:

Each individual string literal from the original union.
All possible combinations of two or more string literals joined by the Separator.

Edge Cases to Consider:

An empty union: What should Join return if T is an empty union?
A union with a single element: Join should still return that element.

Examples

Example 1:

type Elements = "a" | "b";
type Separator = "-";

type Joined = Join<Elements, Separator>;
// Expected type: "a" | "b" | "a-b" | "b-a"

Explanation: The input union Elements contains "a" and "b". The separator is "-". The Join type should produce:

The individual elements: "a", "b".
Combinations of two elements: "a-b", "b-a".

Example 2:

type Words = "hello" | "world";
type Punctuation = "!";

type JoinedPhrases = Join<Words | Punctuation, " ">;
// Expected type: "hello" | "world" | "!" | "hello world" | "world hello" | "hello !" | "! hello" | "world !" | "! world"

Explanation: The input union is "hello" | "world" | "!". The separator is " ". The Join type should produce:

Individual elements: "hello", "world", "!".
Two-element combinations: "hello world", "world hello", "hello !", "! hello", "world !", "! world".
(Note: For simplicity in this challenge, we will focus on combinations of up to two elements for the general case of Join. A more complex Join would handle arbitrary lengths, but this example illustrates the core concept.)

Example 3:

type Single = "only";
type JoinedSingle = Join<Single, "-">;
// Expected type: "only"

Explanation: When the input union has only one element, Join should return that element as is.

Constraints

The Join type should be purely declarative and rely on TypeScript's type system.
The implementation should be as efficient as possible within the constraints of the type system. Avoid overly complex recursive types if simpler alternatives exist for the specified scope.
Assume T will be a union of string literal types.

Notes

This challenge is designed to test your understanding of advanced TypeScript features like conditional types, mapped types, and recursive type definitions (if needed). Consider how you can iterate over the elements of a union type and combine them. Think about how to handle the base cases (single elements) and the recursive step (combining elements).