flattens out nested lists.


flattens to level n.


flattens subexpressions with head h.


flattens list by combining all levels sij to make each level i in the result.


  • Flatten "unravels" lists, effectively just deleting inner braces.
  • Flatten[list,n] effectively flattens the top level in list n times.
  • Flatten[f[e,]] flattens out subexpressions with head f.
  • If the mij are matrices, Flatten[{{m11,m12},{m21,m22}},{{1,3},{2,4}}] effectively constructs a single matrix from the "blocks" mij.
  • Flatten[list,{{i1},{i2},}] effectively transposes levels in list, putting level ik in list at level k in the result. Note that the function Transpose in effect uses an inverse of this specification.
  • Flatten flattens out levels in SparseArray objects just as in the corresponding ordinary arrays. »


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Basic Examples  (3)

Flatten out lists at all levels:

Flatten only at level 1:

Flatten levels 2 and 3 of an array of depth 3 into the first level of the resulting matrix:

Scope  (5)

Level Specification  (5)

No flattening:

Flatten to level 1:

Flatten to level 2:

Flatten to level 3:

Flatten to level 4:

This is the same as using level :

And the same as not specifying a level:

Generalizations & Extensions  (4)

Flatten a sparse array:

Flatten works with any head:

Flatten all levels with respect to g:

Flatten all levels with respect to f:

Here is a matrix:

Flatten an array of blocks with the shape of u into a single matrix:

Flatten into a single matrix, effectively using the transpose of the blocks:

Applications  (5)

Join lists and individual elements:

Unravel a matrix:

Make a flattened list of rules:

Do a "transpose" on a ragged array:

Contract three levels of arrays in a single Dot operation by flattening them first:

Obtain the same result by explicit contraction of three pairs of levels:

Properties & Relations  (5)

Flatten acts as an inverse of Partition:

ArrayReshape acts as an inverse for Flatten on rectangular arrays:

For a rectangular array a, ArrayFlatten[a,r] is equivalent to Flatten[a,{{1,r+1},{2,r+2},,{r,2r}}]:

Flatten effectively arranges elements in the lexicographic order of their indices:

For a permutation p with inverse , Flatten[a,List/@p-1]==Transpose[a,p]:

A random permutation:

Get its inverse:

Neat Examples  (1)

Peel off successive layers of Framed:

Introduced in 1988
Updated in 2003