Indices

This page describes a primitive whose result is a list of indices. See Index for the page on indices themselves. See Index of for the index generator.
 `⍸`

Indices (`⍸`) or Where, is a monadic primitive function which returns the indices of all ones in a Boolean array. More generally, Indices accepts an array of non-negative integers and copies each index the corresponding number of times.

Indices was introduced in J as `I.` and is present in K as `&`.

Examples

In all implementations, Indices gives the indices of ones in a Boolean vector.

```      ⍸ 0 0 1 0 0 0 1 0
3 7
```

In nested APLs it returns nested indices when passed a matrix or higher-dimensional array.

```      ⍸ 3 3⍴0 0 1 0 0 0 1 0
┌───┬───┐
│1 3│3 1│
└───┴───┘
⍸ 1   ⍝ An index into a scalar is empty!
┌┐
││
└┘
```

If numbers higher than 1 are allowed, they indicate that the index of the number is repeated. Negative numbers are never allowed.

```      ⍸ 3 0 2
1 1 1 3 3
```
Works in: dzaima/APL, NARS2000

Description and APL model

Indices replicates each index in the argument by the number of times it appears. It is identical to the APL function:

```Where ← {(,⍵)⌿,⍳⍴⍵}
```

The argument is restricted to be an array of non-negative integers, or, in Dyalog APL, Booleans.

Because Indices returns indices (like Iota), it is subject to index origin.

The only flat array language which implements Indices is J. Because J's Iota does not return multidimensional indices, J defines Indices to have function rank 1 so that only vector indices are used.

Mathematical interpretation

Indices may be viewed as a way to convert between two ways of writing multisets of array indices. The argument uses a dense representation indicating for each index the number of times it appears, and the result uses a sparse representation which lists all the indices contained in the set.

Other resources

APL built-ins 
Primitive functions
Scalar