The left argument is a list of the cycles in the permutation. A cycle
is a list of elements that move one to the next during application of
the permutation, with the last wrapping back around to the first.
(<0 1) C. 'abcde'
bacde
(<0 2 1) C. 'abcde'
cabde
(0 2 1;3 4) C. 'abcde'
cabed
Any permutation can be put into a canonical form: in canonical form,
each cycle is rotated until the biggest number is first, then the cycles
are put in ascending order of largest elements.
There are many ways to express the same permutation. Applying C. twice
will produce the canonical form:
C. C. 3 4 ; 0 1 2
+-----+---+
|2 0 1|4 3|
+-----+---+
C. C. 3 4 ; 2 1 0
+-----+---+
|2 1 0|4 3|
+-----+---+
C. C. 2 1 0 ; 4 3
+-----+---+
|2 1 0|4 3|
+-----+---+
What you are finding is that there are only two different permutations
in your list:
((i...@! <@A. i.) 3)
+-----+-----+-----+-----+-----+-----+
|0 1 2|0 2 1|1 0 2|1 2 0|2 0 1|2 1 0|
+-----+-----+-----+-----+-----+-----+
C."0 ((i...@! <@A. i.) 3)
1 2 0
2 0 1
2 0 1
1 2 0
1 2 0
2 0 1
If the elements are not distinct, that means that some of the 'cycles'
are chained together, and they actually form one big cycle. Apparently
Roger doesn't audit for this, but just lets it happen:
C. 3 4 ; 0 3 2
3 1 0 4 2
C. C. 3 4 ; 0 3 2
+-+-------+
|1|4 2 0 3|
+-+-------+
Raul Miller wrote:
> According ot the dictionary, for dyadic C.
>
> If q is boxed, the elements of (#b)|>j{q must be distinct for each
> j , and the boxes are applied in succession.
>
> But what does this mean?
>
> In particular, why must the residue of the elements of q be distinct?
>
> For example, consider:
>
> #((i...@! <@A. i.) 3) C."0 1 i.9
> 6
> #~.((i...@! <@A. i.) 3) C."0 1 i.9
> 2
>
> Here, we have six distinct, valid left arguments to dyadic C.
>
> But they produce only two distinct results.
>
> How is this useful?
>
> What is the meaning of the left argument here?
>
> Thanks,
>
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