Permutations  

Permutations or the interchange of numbers play an important part in the realm of magic figures in fact any magic figure can be seen as a defining "isomorphism" from a permutation of partaking numbers onto the figure itself. This article won't go into constructing these kinds of templates but defines terms used by this author, in the realm of magic hypercubes (and stars) 

Permutation factor: m! 
a reordering of m symbols (usualy numbers)  
the reordering of m symbols is well known in mathematical context, this author uses backend ordering of permutations when he is numbering the permutations pe. 0: [0,1,2], 1: [0,2,1], 2: [1,0,2], 3: [1,2,0], 4: [2,0,1], 5: [2,1,0] thus permutation #0 is natural ordering and permutation #(n!1) is reverse ordering (the author has need for a general formula back and from permutation number and the actual permutation (yet not known whether suck formula exists)) 

Symmetric Permutation factor: (m  odd(m))!! 
a reordering of [0 .. m1] such that P_{i} + P_{m1i} = m1.  
The symmetric permutation is defined by the above condition on the permuted numbers This type of permutation comes into play when one needs the sub nagonals of a hypercube to be merely permuted when the main nagonal of an hypercube is permuted (see the isomorphisms article: "main nagonal permutation") 

pMultiMagic Permutation factor: (unknown) 
a reordering of [0 .. m1] such that
_{i=1}∑^{m}(i m + P_{i})^{k} = ^{k}t_{m};
k = 1 .. p ^{k}t_{m} is the squares sum (when numbers are raised to the kth power) 

The kind of permutation is encountered when one studies the pMultimagic square. in fact any line in a pMultimagic square forms a pMultimagic permutation. bimagic and trimagic permutation are of course terms used for p=2 and p=3 resp. 

pMultiMagic Permutation multiplet factor: (unknown) 
pMultimagic condition on a set of permutations  
Studying higher dimensioned pMultimagic hypercubes each line in the figure consists of numbers whose base m digits form permutations of [0 .. m1] these digits thus forms the defined multiplets studying the first bimagic cubes this author found that bimagic pairs are combined bimagic permutations (as yet the reverse form an unproven hypothesis) 

Multiplet permutation factor _{i=0}∏^{q} [^{m  i p}_{p}; ∑ p(pq1)/2] (symbolically) 
A permutation of say p*q numbers which split into q sets os p numbers summing to the same sum  
These kinds of permutaton play part in the multiplet stacking of panmagic squares with consecutive numbers 0..pq1 the sum can be calculated to be p(pq1)/2 the amount of these permutation I symbolically denote by _{i=0}∏^{q} [^{m  i p}_{p}; ∑ p(pq1)/2] where [^{p}_{q}; ∑ s] denotes the amount of random selections of p numbers summing s out of q numbers currently I know of no formula to culculate this amount. 

Special uses of permutations  
Component permutation notation: #[perm] 
The use of a permutation to permute the components of a hypercube  
Numbering the components of the hypercube from 0 to n1 the ordering of those components can be depicted by a permutation of the n involved numbers. 

The notation uses a permutation of the component numbers 0..n1  
Transposition notation: ^[perm] 
The use of a permutation to to interchange the axes of a hypercube  
Numbering the axes of the hypercube from 0 to n1 the interchange of axes can be depicted by a permutation of the n involved numbers. 

The notation uses a permutation of the axial numbers 0..n1  
Component Transposition notation: ^R[perm] 
The use of a permutation to to interchange the axes of a hypercube component  
Numbering the axes of the hypercube from 0 to n1 the interchange of axes can be depicted by a permutation of the n involved numbers. note: ^[perm] = ^(2^{n}1)[perm] (on mod mcomponents) 

The notation uses a permutation of the axial numbers 0..n1 R is the bitwise sum of the involved components 

Digit changing Permutation notation: =[perm] 
The use of a permutation to change digit  
Most commonly the digits in an obtained latin hypercube are replaced by another digit this can be depicted by a permutation of the used digits. 

The notation uses a permutation of the digits 0..m1  
Component Digit changing Permutation notation: =R[perm] 
The use of a permutation to change digit  
Most commonly the digits in an obtained latin hypercube are replaced by another digit this can be depicted by a permutation of the used digits. note: =[perm] = =(2^{n}1)[perm] (on mod mcomponents) 

The notation uses a permutation of the digits 0..m1 R is the bitwise sum of the involved components 

Main nagonal Permutation notation: _[perm] 
The use of a permutation to reorder the main nagonal  
The permutation of a main nagonal and all it associated 1agonals is depicted by a permutation. (thus this involves more than a mere permutation regularly depicts) 

The notation uses a permutation of the digits 0..m1  
ragonal Permutation notation: _R[perm] 
The use of a permutation to reorder any ragonal  
The permutation of a ragonal and all it associated 1agonals is depicted by a permutation. (thus this involves more than a mere permutation regularly depicts) note: _[perm] = _(2^{n}1)[perm] and ~R = _R[m1,..,0] 

The notation uses a permutation of the digits 0..m1 R is the bitwise sum of the involved axes 