Semi-invariants of binary forms and symmetrized graph-monomials

This article provides a method for constructing invariants and semi-invariants of a binary $N$-ic form over a field $k$ characteristics $0$ or $p > N$. A practical and broadly applicable sufficient condition for ensuring nontriviality of the symmetrization of a graph-monomial is established. This allows construction of infinite families of invariants (especially, skew-invariants) and families of $k$-linearly independent semi-invariants. These constructions are very useful in the quantum physics of Fermions. Additionally, they permit us to establish a new polynomial-type lower bound on the coefficient of $q^{w}$ in $(q - 1) {N + d \choose d}_{q}$ for all sufficiently large integers $d$ and $w \leq N d / 2$.