The combinatorial identities are easily verified by induction.
The dimension of S14 is known completely. For details consult Peter Alfeld, Bruce Piper, and L.L. Schumaker, An Explicit Basis for C 1 Quartic Bivariate Splines, SIAM J. Num. Anal. 24 (1987), pp. 891-911.
However, I use the above much simpler statement as a vehicle
to explain the concepts in this talk.