Clebcsh-Gordan coefficients in the symmric group $S_n$

The coefficients of fractional parentage (CFP) or Clebcsh-Gordan coefficients of the outer product of representations of the symmetric group $S_n$ are evaluated using an build up algorithm defined in terms of the chain involving the chain $S_{n-1}\subset S_n$ and $S_{n-2}\subset S_n$ chains. Some applications in mathematical physics are considered by combining them with the Clebcsh-Gordan (C-G) coefficients of the inner product of representations of the symmetric group $S_n$