Circular Permutations With Unrestricted Repeatition

1. Circular Permutations With Unrestricted Repeatition refer to All Distinct Cyclic Arrangements having Cycle Length of \(R\) of \(N\) Number of Distinct Type of Items such that Any Item can occur any number of times (between \(0\) and \(R\)) in any given Cyclic Arrangement.
2. Given \(N\) Number of Distinct Type of Items, the formula for Count of Circular Permutations With Unrestricted Repeatition having a Cycle Length of \(R\) is defined recursively as follows
   
   \(C_{NR}=\frac{N^R\hspace{.1cm}+\hspace{.1cm}(R-1)N\hspace{.1cm}+\hspace{.1cm}(\sum_{i=1}^n(R- F_i)\times F_{i}Count(C_{NF_i}))}{R}\)   ...(1)
   
   where
   
   \(C_{NR} =\) Count of Circular Permutations With Unrestricted Repeatition of \(N\) Distinct Type of Items having Cycle Length \(R\)
   
   \(n =\) Number of Factors of \(R\) Greater than 1 but Less than R
   
   \(F_i =\) \(i^{th}\) Factor of \(R\) Greater than 1 but Less than R (i.e \(1 < F_i < R\))
   
   \(C_{NF_i} =\) Count of Circular Permutations With Unrestricted Repeatition of \(N\) Distinct Type of Items having Cycle Length \(F_i\)
   
   \(F_{i}Count(C_{NF_i}) =\) Count of \(F_i\) Cycles in Circular Permutations With Unrestricted Repeatition of \(N\) Distinct Type of Items having Cycle Length \(F_i\)
   
   If \(R\) is Prime, the formula given in equation (1) above reduces to
   
   \(C_{NR}=\frac{N^R + (R-1)N}{R}\)   ...(2)
   
   Also, if any factor \(F_i\) of \(R\) is Prime, then
   
   \(F_{i}Count(C_{NF_i}) =C_{NF_i}-N\)    ...(3)