Abstract
The mathematical properties of isomers numbers of substituted \(C_{2v}\)-based compounds presented in this paper includes : -(1) the formulation of 4-dimensional permutomers count vectors \(PCV =( N_{E},N_{C_{2}},N_{\sigma_{v_{1}}},N_{\sigma_{v_{2}}})\)
and 5-dimensional itemized isomers count vectors IICV=\(a_{c_{1}},a_{c_{2}},a_{c_{s}},a_{c'_s},a_{c_{2v}}\) which satisfy the dot product PCV =\(IICV\times W_{C_{2v}}\) . (2)- The expansion of this equation to obtain the denumerants of
type \(N_{g_{i}\epsilon C_{2v}}=\sum_{g_{i}\epsilon C_{2v}} a_{G_{j}\epsilon C_{2v}}W_{G_j},{g_i}\) mapping permutomers numbers as sum of symmetry itemized isomers numbers \(a_{G_{j}\epsilon C_{2v}}\) scaled by \(W_{G_j},{g_i}\) the markaracters
of \(C_{2v}\). (3)-The collection of 4 and 5 entries \(PCV_s\) and \(IICV_s\) generating respectively, permutomers count matrices \((PCM_s)\) and itemized isomers count matrices \((IICM_s)\) that construct two associated vector spaces
of isomers numbers for such series of molecules.