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.

[21-Apr-1997]