  It is shown that there are infinitely many formulas to calculate
multiplicities of weights participating in irreducible representations of $A_N$
Lie algebras. On contrary to recursive character of Kostant and Freudenthal
multiplicity formulas, they provide us systems of linear algebraic equations
with N-dependent polinomial coefficients. These polinomial coefficients are in
fact related with polinomials which represent eigenvalues of Casimir operators.
