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We obtain a combinatorial formula related to the shear transformation forsemi-invariants of binary forms, which implies the classical characterizationof semi-invariants in terms of a differential operator. Then, we present acombinatorial proof of an identity of Hilbert, which leads to a relation ofCayley on semi-invariants. This identity plays a crucial role in the originalproof of Sylvester's theorem on semi-invariants in connection with the Gaussiancoefficients. Moreover, we show that the additivity lemma of Pak and Panovawhich yields the strict unimodality of the Gaussian coefficients for $n,k \geq8$ can be deduced from the ring property of semi-invariants.

Semi-invariants of Binary Forms and Sylvester's Theorem