As with most APL symbols, ⍉ has different meanings when called with one argument (transpose) versus two arguments (dyadic transpose / reorder dimensions). This challenge concerns the latter, which acts similarly to numpy.moveaxis in Python or permute in MATLAB, but is more powerful.

## order ⍉ A when order has distinct entries

When all members of order are distinct, order ⍉ A is equivalent to:

• numpy.moveaxis(A, tuple(range(len(A.shape)), order) in Python, or
• permute(A,order) in MATLAB. Quoting from documentation of the latter:

B = permute(A,order) rearranges the dimensions of A so that they are in the order specified by the vector order. The resulting array B has the same values as A but the order of the subscripts needed to access any particular element is rearranged as specified by order.

For example, suppose A is a 3D array, and let B ← (2 0 1)⍉A. Then B is such that B[x0,x1,x2] = A[x2,x0,x1] for all x2,x0,x1

## order ⍉ A when order has repeated entries

When order has repeated entries, we take a diagonal slice of the array. For example, let A be a 2x3x4 array. B ← (0 0 1)⍉A takes a diagonal slice along A to create B such that B[x0,x1] = A[x0,x0,x1]. Note that B is a 2x4 array: if it were 3x4, we would need to set B[2, x1] = A[2, 2, x1] which would be out of bounds of A. In general the kth dimension of B will be the minimum of all A.shape[i] such that order[i] = k.

## Example

Consider the dyadic transpose order⍉A where order = [2, 1, 0] and A is 3x4x5

    A =
[[[ 0  1  2  3  4]
[ 5  6  7  8  9]
[10 11 12 13 14]
[15 16 17 18 19]]

[[20 21 22 23 24]
[25 26 27 28 29]
[30 31 32 33 34]
[35 36 37 38 39]]

[[40 41 42 43 44]
[45 46 47 48 49]
[50 51 52 53 54]
[55 56 57 58 59]]]


The result is the 5x4x3 array B =

[[[ 0 20 40]
[ 5 25 45]
[10 30 50]
[15 35 55]]

[[ 1 21 41]
[ 6 26 46]
[11 31 51]
[16 36 56]]

[[ 2 22 42]
[ 7 27 47]
[12 32 52]
[17 37 57]]

[[ 3 23 43]
[ 8 28 48]
[13 33 53]
[18 38 58]]

[[ 4 24 44]
[ 9 29 49]
[14 34 54]
[19 39 59]]]


Note that when, for example, (x0, x1, x2) = (4,1,2) we have B[x0,x1,x2] = A[x2, x1, x0] = A[2,1,4] = 49.

If instead order = [0, 0, 0] and A as above, then we would have the output B be the 1-dimensional size-3 array B = [0, 26, 52] so that B[1] = B[x0] = A[x0,x0,x0] = A[1,1,1] = 26

### Input

Here we use 0-indexing, but you may also use 1-indexing as is the APL default.

• A multidimensional or nested array A, of dimension n ≥ 1.

• A list order of n positive integers consisting of the integers {0,1,...,k} (or {1,...,k+1} for 1-index) for some k < n, in any order, possibly with repetitions.

### Output

• A multidimensional or nested array representing the result of applying the dyadic transpose with those arguments. (The output will have dimension k+1.)

You may write a full program, function, etc. as allowed by the current standard on meta.

If your language has a builtin, it is encouraged to also write a solution without the builtin for the sake of an interesting answer.

### Test cases

Test cases on TIO

Reference Python implementation coming soon.

Note for reading test cases: in APL, the penultimate and ultimate axes of an array are along columns and rows in that order.

• APL, 1 byte: ⍉ :P Commented Nov 17, 2018 at 23:01
• Actually, many APL symbols just use a default second argument when called with one argument. This includes ⍉ which uses the reversed axis indices as default, so ⍉A is the same as (2 1 0)⍉A if A is a 3-dimensional array and in general ⍉A is (⌽⍳≢⍴A)⍉A.
Commented Nov 18, 2018 at 10:04
• @lirtosiast question about i/o: can a multidimensional array be represented as a pair of shape (list of dimensions) and content (all elements in lexicographical order of their indices)?
– ngn
Commented Nov 19, 2018 at 13:26
• @ngn I would say no for now, but you should ask on meta whether that format is acceptable by default. Commented Nov 19, 2018 at 18:55
• @lirtosiast Anecdotally, Dyalog APL internally stores arrays as [number-of-dimensions,first-dimension-length,second-dimension-length,…,last-dimension-length,first-element,second-element,…,last-element].
Commented Nov 19, 2018 at 19:15

# APL (Dyalog Unicode), 34 bytesSBCS

This is my colleague's code (slightly modified from Roger Hui: A History of APL in 50 Functions, chapter 30), posted with explicit permission.

Anonymous tacit infix lambda (can be used as drop-in for ⍉).

{⍵[(⊂⊂⍺)⌷¨,¨⍳⍺[⍋⍺]{⌊/⍵}⌸(⍴⍵)[⍋⍺]]}


Try it online!

{} dfn; ⍺ is left argument (axes), ⍵ is right argument (array)
E.g. [2,2,1] and [[[1,2],[3,4]]]

⍵[] index the array with:

(⍴⍵)[] the shape (lengths of axes) of the array, indexed by:
[1,2,2]

⍋⍺ the grading vector (the indices that would sort them) of the axes
[3,1,2]
[2,1,2]

⍺[⍋⍺]{}⌸ use the sorted axes as keys to group that, and for each group:
[1,2,2]{"1":[2],"2":[1,2]}

{⌊/⍵} find the lowest axis length
{"1":2,"2":1}[2,1]

⍳ generate the indices in a Cartesian system of those dimensions
[[[1,1]],[[2,1]]]

,¨ ensure that the indices of each coordinate is a vector (would be scalar if ⍵ is a vector)
[[[1,1]],[[2,1]]]

()⌷¨ index into each of those with the following:

⊂⊂⍺ the axes (doubly enclosed; once for ⌷ to select those cells along the first and only axis, and once for ¨ to pair up each vector on the right with the entire set of axes on the left)
2 1 2
[[[1,1,1]],[[1,2,1]]]
[[1],[3]]