Is there any known formula for the number of compositions of an integer k (partitions with considering the order of the parts) of length m (exactly m parts) where the parts do not exceed a given integer n? 
Without limitation of the parts there is, of course, a well-known formula (binomial k-1 over m-1). Introducing the limitation I worked out a formula but I don´t know whether it´s already published anywhere ... 



Explicit formula for the number of compositions with m strictly positive parts bounded by n?
If I understand well, you consider the number $a(k,n,m)$ of multi-indices $a=(a\_1,\dots,a\_m)\in\{1,\dots,n\}^m$ with weight $\sum\_{i=1}^m a\_i=k$. This is therefore the coefficient of $x^k$ in $$\left (\sum \_ {j=1}^n x^j \right)^m = x^m(1-x^n)^m (1-x)^{-m}\, .$$
Since the above generating function is the product of functions with elementary power series expansion, a formula for $a(k,n,m)$ is available as a convolution of binomial coefficients. Is this what you mean? This is certainly in any text on the subject .


