The Magic Encyclopedia ™

Permutations
(by Aale de Winkel)

This article deals with the permutations as they partake in the realm of magic figures
Permutations are one of the most basic isomorphisms (see article)

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 back-end 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 .. m-1] such that P_i + P_m-1-i = m-1.
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 n-agonals of a hypercube
to be merely permuted when the main n-agonal of an hypercube is permuted
(see the isomorphisms article: "main n-agonal permutation")

p-MultiMagic Permutation
factor: (unknown) a reordering of [0 .. m-1] such that _i=1∑^m(i m + P_i)^k = ^kt_m; k = 1 .. p
^kt_m is the squares sum (when numbers are raised to the k-th power)
The kind of permutation is encountered when one studies the p-Multimagic square.
in fact any line in a p-Multimagic square forms a p-Multimagic permutation.
bimagic and trimagic permutation are of course terms used for p=2 and p=3 resp.

p-MultiMagic Permutation
multiplet
factor: (unknown) p-Multimagic condition on a set of permutations
Studying higher dimensioned p-Multimagic hypercubes each line in the figure
consists of numbers whose base m digits form permutations of [0 .. m-1]
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(pq-1)/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..pq-1 the sum can be calculated to be p(pq-1)/2
the amount of these permutation I symbolically denote by
_i=0∏^q [^{m - i p}_p; ∑ p(pq-1)/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

language element:
#[perm] The use of a permutation to permute the components of a hypercube
Numbering the components of the hypercube from 0 to n-1 the ordering of those
components can be depicted by a permutation of the n involved numbers.

The hypercube language element uses a permutation of the component numbers 0..n-1

Transposition

language element:
^[perm] The use of a permutation to to interchange the axes of a hypercube
Numbering the axes of the hypercube from 0 to n-1 the interchange of axes can be
depicted by a permutation of the n involved numbers.

The hypercube language element uses a permutation of the axial numbers 0..n-1

Digit changing Permutation

language element:
=[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 hypercube language element uses a permutation of the digits 0..m-1

Main n-agonal Permutation

language element:
_[perm] The use of a permutation to reorder the main n-agonal
The permutation of a main n-agonal and all it associated 1-agonals is depicted by a
permutation. (thus this involves more than a mere permutation regularly depicts)

The hypercube language element uses a permutation of the digits 0..m-1

r-agonal Permutation

language element:
_R[perm] The use of a permutation to reorder any r-agonal
The permutation of a r-agonal and all it associated 1-agonals is depicted by a
permutation. (thus this involves more than a mere permutation regularly depicts)
R is the bitwise sum (ie &Sum 2^axis with axis the involved monagonal direction)
note: _[perm] = _(2ⁿ-1)[perm] and ~R = _R[m-1,..,0]

The hypercube language element uses a permutation of the digits 0..m-1

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)
Permutation factor: m!	the reordering of m symbols is well known in mathematical context, this author uses back-end 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 .. m-1] such that P_i + P_m-1-i = m-1.
Symmetric Permutation factor: (m - odd(m))!!	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 n-agonals of a hypercube to be merely permuted when the main n-agonal of an hypercube is permuted (see the isomorphisms article: "main n-agonal permutation")
p-MultiMagic Permutation factor: (unknown)	a reordering of [0 .. m-1] such that _i=1∑^m(i m + P_i)^k = ^kt_m; k = 1 .. p ^kt_m is the squares sum (when numbers are raised to the k-th power)
p-MultiMagic Permutation factor: (unknown)	The kind of permutation is encountered when one studies the p-Multimagic square. in fact any line in a p-Multimagic square forms a p-Multimagic permutation. bimagic and trimagic permutation are of course terms used for p=2 and p=3 resp.
p-MultiMagic Permutation multiplet factor: (unknown)	p-Multimagic condition on a set of permutations
p-MultiMagic Permutation multiplet factor: (unknown)	Studying higher dimensioned p-Multimagic hypercubes each line in the figure consists of numbers whose base m digits form permutations of [0 .. m-1] 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(pq-1)/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..pq-1 the sum can be calculated to be p(pq-1)/2 the amount of these permutation I symbolically denote by _i=0∏^q [^{m - i p}_p; ∑ p(pq-1)/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 language element: #[perm]	The use of a permutation to permute the components of a hypercube
	Numbering the components of the hypercube from 0 to n-1 the ordering of those components can be depicted by a permutation of the n involved numbers.
	The hypercube language element uses a permutation of the component numbers 0..n-1
Transposition language element: ^[perm]	The use of a permutation to to interchange the axes of a hypercube
	Numbering the axes of the hypercube from 0 to n-1 the interchange of axes can be depicted by a permutation of the n involved numbers.
	The hypercube language element uses a permutation of the axial numbers 0..n-1
Digit changing Permutation language element: =[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 hypercube language element uses a permutation of the digits 0..m-1
Main n-agonal Permutation language element: _[perm]	The use of a permutation to reorder the main n-agonal
	The permutation of a main n-agonal and all it associated 1-agonals is depicted by a permutation. (thus this involves more than a mere permutation regularly depicts)
	The hypercube language element uses a permutation of the digits 0..m-1
r-agonal Permutation language element: _R[perm]	The use of a permutation to reorder any r-agonal
	The permutation of a r-agonal and all it associated 1-agonals is depicted by a permutation. (thus this involves more than a mere permutation regularly depicts) R is the bitwise sum (ie &Sum 2^axis with axis the involved monagonal direction) note: _[perm] = _(2ⁿ-1)[perm] and ~R = _R[m-1,..,0]
	The hypercube language element uses a permutation of the digits 0..m-1