$N$ birds are distributed on a telephone wire

Question

$N$ birds are distributed on a telephone wire that can fit a maximum of $2N$ birds. The spacings between birds form a sequence $S$. The minimum space between birds is $1$ unit. The sequence is ordered from least to greatest. For $N=4$ birds, one possible sequence is $S=\{1,1,3\}.$

Generate $2$ candidate step functions for $N=100,$ and provide a plot.

The sequence $S$ is a finite abstract counting function, which doesn’t count primes but may encode similar statistical properties as a finite sequence of primes.

I think analyzing these types of finite abstract counting functions is important because they can provide insights to finite distribution patterns over the integers.

Which function best approximates each of the $2$ candidates for $N=100$?

Answer 1

$S$ has to be a sequence (not a set-you need it ordered) of $N-1$ positive integers that sum to no more than $2N$, assuming the birds are allowed to be at the end of the wire. There is nothing in the question to choose one such sequence, so you can pick any two. $100$ is a lot of numbers to pick, so I would pick simple ones, like all $1$'s and all $2$'s, and call it good. Finding an approximating function is easy for these.