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1 the permutation number. For a regions of the same extension we 
shall write here a.ss. A permutation number is higher than an 
other one, when its first region is greater or, in the case of equality, 
when the second region is greater etc. FROBENIUS introduced the name 
of permutation class for all permutations with the same permutation 
number. In the same way we shall call the sum of all isomers 
with the same permutation number divided by p/ the tsomer class 
with that number. The number of classes is therefore equal to the 
number & of the whole positive solutions of the equation 
eo, +2a,+3ea@,4+... nh ae) 
The classes are arranged in ascending order and written: 
ey eee Gi p p 
,4u——Uu ’ K, u ’ BONG vee ’ Kj. u 
p! 
The permutation number may be added as index on the left side 
eg. for p= 6: 
Et ’ A ’ sees ’ wah, ’ aie ’ SA u te Ree EN is re 
A class is called even or odd according as it consists of even or 
odd permutations. 
7 . . " Pp 
Alternations and mixings. The affinor that is found from u by 
replacing each of ¢ definite groups of s,,...,s, factors (without 
displacing them) by the ideal factors of their alternating or symme- 
. . . . Pp . 
trical product is called a simple alternation resp. miving of u with 
the permutation number s,,...., 5, and is written as: 
(A) p Sirs (A) p 
A au resp. M u 
By yeyS 
t 
The index (2) at the top on the right indicates the choice of the 
permutation regions. The affinor is called, locally alternating resp. 
symmetrical in these regions and in general locally permutable. 
The sum of all simple alternations resp. mixings with the same 
permutation number divided by their number is called the general 
alternation resp. mixing with that number. The general alternations 
and mixings are arranged in ascending order and written as 
= p ED eer) 
Aya Ae den 
ED MP AE Zep 
Mum hae Mel ee a 
eventually, when desirable, with the permutation number as index on 
the left e.g. for p= 6: 
