Question

Number of ways to represent n cell repetition

Answer 1

I know how to find the number of noncommutative ways to form the sum: Imagine a line of n+k−1n+k−1 positions, where each position can contain either a cat or a divider. If you have nn(nameless) cats and k−1k−1 dividers, you can split the cats in to kk groups by choosing positions for the dividers: (n+k−1k−1)(n+k−1k−1). The size of each group of cats corresponds to one of the nonnegative integers in the sum.

Answer 2

I know how to find the number of noncommutative ways to form the sum: Imagine a line of n+k−1n+k−1 positions, where each position can contain either a cat or a divider. If you have nn(nameless) cats and k−1k−1 dividers, you can split the cats in to kk groups by choosing positions for the dividers: (n+k−1k−1)(n+k−1k−1). The size of each group of cats corresponds to one of the nonnegative integers in the sum.