Cycle Index Count of a Permutation

1. Any given Permutation of N Items can consist of 1 or more Permutation Cycles, with each having a Cycle Length from 1 to N. When more than 1 Permutation Cycle is present, they can be of Different Cycle Lengths.
   
   Cycle Index Count of any given Permutation is a List Containing the Count of Permutation Cycles of Each Distinct Cycle Length in the Permutation.
2. As an example consider the following Permutation \(P_1\)
   
   \(P_1=\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\4 & 2 & 1 & 6 & 5 & 3 & 8 & 7 & 9\end{pmatrix}=(1\hspace{.1cm}4\hspace{.1cm}6\hspace{.1cm}3)\hspace{.1cm}(7\hspace{.1cm}8)\hspace{.1cm}(9)\)   ...(1)
   
   The Cycle Index Count for Permutation \(P_1\) above is calculated as follows
   
   The Number of Permutation Cycles of Length 1 \(c_1 = 1\) = \((9)\)
   
   The Number of Permutation Cycles of Length 2 \(c_2 = 1\) = \((7\hspace{.1cm}8)\)
   
   The Number of Permutation Cycles of Length 4 \(c_4 = 1\) = \((1\hspace{.1cm}4\hspace{.1cm}6\hspace{.1cm}3)\)
   
   Hence the Cycle Index Count for Permutation \(P_1\) is given as \(c_1=1,c_2=1,c_4=1\)
   
   As another example consider the following Permutation \(P_2\)
   
   \(P_2=\begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\3 & 4 & 1 & 2 & 7 & 6 & 9 & 8 & 5\end{pmatrix}=(1\hspace{.1cm}3)\hspace{.1cm}(2\hspace{.1cm}4)\hspace{.1cm}(5\hspace{.1cm}7\hspace{.1cm}9)\hspace{.1cm}\hspace{.1cm}(6)\hspace{.1cm}(8)\)   ...(2)
   
   The Cycle Index Count for Permutation \(P_2\) above is calculated as follows
   
   The Number of Permutation Cycles of Length 1 \(c_1 = 2\) = \((6)\) and \((8)\)
   
   The Number of Permutation Cycles of Length 2 \(c_2 = 2\) = \((1\hspace{.1cm}3)\) and \((2\hspace{.1cm}4)\)
   
   The Number of Permutation Cycles of Length 3 \(c_3 = 1\) = \((5\hspace{.1cm}7\hspace{.1cm}9)\)
   
   Hence the Cycle Index Count for Permutation \(P_2\) is given as \(c_1=2,c_2=2,c_3=1\)
Related Topics and Calculators
Decomposition of Permutation/Permutation Cycles into Transpositions,    Product of Permutations, Permutation Cycles and Transpositions,    Inverse and Order of a Permutation,    Permutations and Permutation Matrices,    Permutations from Permutation Matrix Calculator,    Permutation Matrices from Permutation Calculator
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