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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

3 votes

Sum over integer compositions

Yes, there are closed-form expressions. The number $S(n,k)$ you are looking for is the number of weighted integer compositions with weighting function $f(a)=a^p$. Many recursions and other representat …
mathse's user avatar
  • 171
1 vote

Lower bound for constrained ordered partitions (i.e., compositions)?

For my answer, I assume that compositions are on non-negative numbers (allowing only positive numbers does not change my statements, qualitatively). First, observe that $p(N,m,n)=\binom{m}{n}_{N+1}$, …
mathse's user avatar
  • 171