Though I can't answer the question directly, it may be helpful to clarify some of the issues here. First, it's usually best to focus on the case when $W$ (or the root system of the Lie algebra) is irreducible. For any irreducible finite reflection group (such as $W$), Coxeter found a remarkable conjugacy class consisting of elements now called Coxeter elements (of order $h$ equal to the unique largest degree of the $W$-invariant polynomials in the reflection representation of $W$. These are just products in any order of the simple reflections in $W$ (for any simple system). Recall that the order of $W$ is just the product of the degrees, which in the example $W=S_n$ are $2, \dots, n$, the Lie rank being $n-1$.

What is the number of Coxeter elements in $W$? By basic finite group theory, this number is the index in $W$ of the centralizer of any fixed Coxeter element $c$ (the size of the conjugacy class of $c$). In turn, the centralizer is just the cyclic subgroup of $W$ generated by $c$, whose order is $h$. This assertion is worked out more generally for all "regular" elements in a finite real or complex reflection group, in a 1974 (Invent. Math. 25) paper by T.A. Springer here: see especially 4.2 and 4.4.

For example, when $W = S_n$ there are $(n-1)!$ Coxeter elements.

Concerning finite dimensional irreducible representations of a simple Lie algebra (say in characteristic 0), these are nicely parametrized by the dominant weights. But it's nontrivial as a rule to work out the explicit weight multiplicities, though it's enough to do this for dominant weights (and then apply $W$-conjugacy). Typically computer computations rely on some version of Freudenthal's recursive formula. While Kostant's formula is more explicit (and equivalent to Weyl's character formula), it isn't usually useful for computations.

Though I can't answer the question directly, it may be helpful to clarify some of the issues here. First, it's usually best to focus on the case when $W$ (or the root system of the Lie algebra) is irreducible. For any irreducible finite reflection group (such as $W$), Coxeter found a remarkable conjugacy class consisting of elements now called Coxeter elements (of order $h$ equal to the unique largest degree of the $W$-invariant polynomials in the reflection representation of $W$. These are just products in any order of the simple reflections in $W$ (for any simple system). Recall that the order of $W$ is just the product of the degrees, which in the example $W=S_n$ are $2, \dots, n$, the Lie rank being $n-1$.

What is the number of Coxeter elements in $W$? By basic finite group theory, this number is the index in $W$ of the centralizer of any fixed Coxeter element $c$ (the size of the conjugacy class of $c$). In turn, the centralizer is just the cyclic subgroup of $W$ generated by $c$, whose order is $h$. This assertion is worked out more generally for all "regular" elements in a finite real or complex reflection group, in a 1974 (Invent. Math. 25) paper by T.A. Springer here: see especially 4.2 and 4.4.

For example, when $W = S_n$ there are $(n-1)!$ Coxeter elements.

Concerning finite dimensional irreducible representations of a simple Lie algebra (say in characteristic 0), these are nicely parametrized by the dominant weights. But it's nontrivial as a rule to work out the explicit weight multiplicities, though it's enough to do this for dominant weights (and then apply $W$-conjugacy). Typically computer computations rely on some version of Freudenthal's recursive formula. While Kostant's formula is more explicit (and equivalent to Weyl's character formula), it isn't usually useful for computations.

[ADDED] To make the details and references more explicit, I've written up some notes here (comments welcome).

Though I can't answer the question directly, it may be helpful to clarify some of the issues here. First, it's usually best to focus on the case when $W$ (or the root system of the Lie algebra) is irreducible. For any irreducible finite reflection group (such as $W$), Coxeter found a remarkable conjugacy class consisting of elements now called Coxeter elements (of order $h$ equal to the unique largest degree of the $W$-invariant polynomials in the reflection representation of $W$. These are just products in any order of the simple reflections in $W$ (for any simple system). Recall that the order of $W$ is just the product of the degrees, which in the example $W=S_n$ are $2, \dots, n$, the Lie rank being $n-1$.

What is the number of Coxeter elements in $W$? By basic finite group theory, this number is the index in $W$ of the centralizer of any fixed Coxeter element $c$ (the size of the conjugacy class of $c$). In turn, the centralizer is just the cyclic subgroup of $W$ generated by $c$, whose order is $h$. This assertion is worked out more generally for all "regular" elements in a finite real or complex reflection group, in a 1974 (Invent. Math. 25) paper by T.A. Springer: see especially 4.2 and 4.4.

For example, when $W = S_n$ there are $(n-1)!$ Coxeter elements.

Concerning finite dimensional irreducible representations of a simple Lie algebra (say in characteristic 0), these are nicely parametrized by the dominant weights. But it's nontrivial as a rule to work out the explicit weight multiplicities, though it's enough to do this for dominant weights (and then apply $W$-conjugacy). Typically computer computations rely on some version of Freudenthal's recursive formula. While Kostant's formula is more explicit (and equivalent to Weyl's character formula), it isn't usually useful for computations.

Though I can't answer the question directly, it may be helpful to clarify some of the issues here. First, it's usually best to focus on the case when $W$ (or the root system of the Lie algebra) is irreducible. For any irreducible finite reflection group (such as $W$), Coxeter found a remarkable conjugacy class consisting of elements now called Coxeter elements (of order $h$ equal to the unique largest degree of the $W$-invariant polynomials in the reflection representation of $W$. These are just products in any order of the simple reflections in $W$ (for any simple system). Recall that the order of $W$ is just the product of the degrees, which in the example $W=S_n$ are $2, \dots, n$, the Lie rank being $n-1$.

What is the number of Coxeter elements in $W$? By basic finite group theory, this number is the index in $W$ of the centralizer of any fixed Coxeter element $c$ (the size of the conjugacy class of $c$). In turn, the centralizer is just the cyclic subgroup of $W$ generated by $c$, whose order is $h$. This assertion is worked out more generally for all "regular" elements in a finite real or complex reflection group, in a 1974 (Invent. Math. 25) paper by T.A. Springer here: see especially 4.2 and 4.4.

For example, when $W = S_n$ there are $(n-1)!$ Coxeter elements.

Concerning finite dimensional irreducible representations of a simple Lie algebra (say in characteristic 0), these are nicely parametrized by the dominant weights. But it's nontrivial as a rule to work out the explicit weight multiplicities, though it's enough to do this for dominant weights (and then apply $W$-conjugacy). Typically computer computations rely on some version of Freudenthal's recursive formula. While Kostant's formula is more explicit (and equivalent to Weyl's character formula), it isn't usually useful for computations.