  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 $w \leq N d / 2$.
