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Power u^:v  _ _ _  

The case of ^: with a verb right argument is defined in terms of the noun right argument case (u ^: n) as follows:                    

x u ^: v y ↔ x u^:(x v y) y
  u ^: v y ↔   u^:(  v y) y

For example:
   x=: 1 3 3 1
   y=: 0 1 2 3 4 5 6
   x p. y
1 8 27 64 125 216 343

   x p. ^: (]>3:)"1 0 y
0 1 2 3 125 216 343

   a=: _3 _2 _1 0 1 2 3
   %: a
0j1.73205 0j1.41421 0j1 0 1 1.41421 1.73205

   * a
_1 _1 _1 0 1 1 1

   %: ^: * " 0 a
9 4 1 0 1 1.41421 1.73205

   *: a
9 4 1 0 1 4 9
The following monads are equivalent. (See the example of ^ T. _ in the definition of Taylor Approximation (T.) .)
   g=: u ^: p ^: _
   h=: 3 : 't=. y while. p t do. t=. u t end.'

   u=: -&3 
   p=: 0&<
   (g"0 ; h"0) i. 10
+-------------------------+-------------------------+
|0 _2 _1 0 _2 _1 0 _2 _1 0|0 _2 _1 0 _2 _1 0 _2 _1 0|
+-------------------------+-------------------------+

All partitions of an integer, based on an algorithm by R.E. Boss on 2005-07-05:

partition=: [: final ] (,new)@]^:[ (<i.1 0)"_
final    =: 0 <@-."1~ >@{: 
new      =: [: <@; (-i.)@# cat&.> ]
cat      =: [ ,. (>: {."1) # ]

   partition 5
+-+---+---+-----+-----+-------+---------+
|5|4 1|3 2|3 1 1|2 2 1|2 1 1 1|1 1 1 1 1|
+-+---+---+-----+-----+-------+---------+

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