Ordering

The permutation of indices that would sort the collection into increasing order.

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

Ordering(collection)the permutation of indices that sorts the collection into increasing order.

Ordering(collection, n)just the first n indices of the full ordering; a negative n takes the last |n| (the largest elements); {m, n} takes positions m through n of the full ordering; UpTo(n) caps at n without erroring on a shorter collection.

Ordering(collection, All, Greater)every index, ordered descending rather than ascending.

Typed signatures

Type	Package	Applies
(indexed_collection<T>, order: (((T) any -> unknown) | ((any, any) any -> boolean | number))?) -> list<integer> where T: any	compute-engine	always
(indexed_collection<any>, any?, any?) -> list<integer>	combinatoricsreplaces compute-engine	always

Domain: Collections

Details

• $c[[Ordering(c)]] = Sort(c)$: applying the permutation at those positions recovers Sort's result.
• Ties break in favor of earlier position, i.e. it's a stable ordering.
• Symbols order alphabetically by name, matching Wolfram's canonical ordering.
• A second argument $n$ takes just the first $n$ indices of the full ordering; see the signatures above for the negative-count, $\{m, n\}$, and $UpTo$ forms.

Examples

Basic #

live ↗

In$\mathrm{Ordering}(\bigl\lbrack3, 1, 2\bigr\rbrack)$Out$\bigl\lbrack2, 3, 1\bigr\rbrack$

live ↗

In$\mathrm{Ordering}(\bigl\lbrack2, 6, 1, 9, 1, 2, 3\bigr\rbrack)$Out$\bigl\lbrack3, 5, 1, 6, 7, 2, 4\bigr\rbrack$

Scope #

Ties break in favor of the earlier position: both 1s come before both 2s

live ↗

In$\mathrm{Ordering}(\bigl\lbrack2, 1, 2, 1\bigr\rbrack)$Out$\bigl\lbrack2, 4, 1, 3\bigr\rbrack$

A second argument takes just the first n indices of the full ordering

live ↗

In$\mathrm{Ordering}(\bigl\lbrack2, 6, 1, 9, 1, 2, 3\bigr\rbrack, 4)$Out$\bigl\lbrack3, 5, 1, 6\bigr\rbrack$

Symbols order alphabetically

live ↗

In$\mathrm{Ordering}(\bigl\lbrack c, a, b\bigr\rbrack)$Out$\bigl\lbrack2, 3, 1\bigr\rbrack$

A negative count gives the positions of the largest elements

live ↗

In$\mathrm{Ordering}(\bigl\lbrack2, 6, 1, 9, 1, 2, 3\bigr\rbrack, -1)$Out$\bigl\lbrack4\bigr\rbrack$

A $\{m, n\}$ spec takes the 4th through the last entries of the ordering

live ↗

In$\mathrm{Ordering}(\bigl\lbrack2, 6, 1, 9, 1, 2, 3\bigr\rbrack, \bigl\lbrack4, -1\bigr\rbrack)$Out$\bigl\lbrack6, 7, 2, 4\bigr\rbrack$

A third argument orders by a custom test, here descending

live ↗

In$\mathrm{Ordering}(\bigl\lbrack2, 6, 1, 9, 3\bigr\rbrack, \mathrm{All}, \mathrm{Greater})$Out$\bigl\lbrack4, 2, 5, 1, 3\bigr\rbrack$

UpTo(6) asks for at most 6 positions, so a 5-element list gives all 5

live ↗

In$\mathrm{Ordering}(\bigl\lbrack2, 6, 1, 9, 2\bigr\rbrack, \mathrm{UpTo}(6))$Out$\bigl\lbrack3, 1, 5, 2, 4\bigr\rbrack$

Applications #

The position of the smallest element. See First

live ↗

In$\mathrm{First}(\mathrm{Ordering}(\bigl\lbrack2, 6, 1, 9, 1, 2, 3\bigr\rbrack))$Out$3$

Positions in increasing order of value: the $1$ is third, then $2$, $3$, $4$, $5$

live ↗

In$\mathrm{Ordering}(\bigl\lbrack4, 5, 1, 2, 3\bigr\rbrack)$Out$\bigl\lbrack3, 4, 5, 1, 2\bigr\rbrack$

Properties #

On a permutation, $Ordering$ is the inverse, so applying it twice recovers the permutation

live ↗

In$\mathrm{Ordering}(\mathrm{Ordering}(\bigl\lbrack3, 1, 2\bigr\rbrack))$Out$\bigl\lbrack3, 1, 2\bigr\rbrack$

A negative count takes from the largest end, so the position of $5$

live ↗

In$\mathrm{Ordering}(\bigl\lbrack4, 5, -2, 3\bigr\rbrack, -1)$Out$\bigl\lbrack2\bigr\rbrack$

About

Examples	13
Checked against	Wolfram Language 13 agree
Crosswalk	Wolfram Language

See also: Sort, UpTo