  We give a new Jacobi--Trudi-type formula for characters of finite-dimensional
irreducible representations in type $C_n$ using characters of the fundamental
representations and non-intersecting lattice paths. We give equivalent
determinant formulas for the decomposition multiplicities for tensor powers of
the spin representation in type $B_n$ and the exterior representation in type
$C_n$. This gives a combinatorial proof of an identity of Katz and equates such
a multiplicity with the dimension of an irreducible representation in type
$C_n$. By taking certain specializations, we obtain identities for $q$-Catalan
triangle numbers, the $q,t$-Catalan number of Stump, $q$-triangle versions of
Motzkin and Riordan numbers, and generalizations of Touchard's identity. We use
(spin) rigid tableaux and crystal base theory to show some formulas relating
Catalan, Motzkin, and Riordan triangle numbers.
