On a tangent to a problem I've been working on, I've run into a combinatorial/partition-theoretic problem that I wondered if anyone had run into before.

Let $N$ be a positive integer, and *ad-hoc*-ly call an (ordered) non-negative partition of $N$ into exactly $N$ parts
$$
N=n_1+n_2+\cdots+n_N
$$
*valid* if

- $0\leq n_i\leq 2$ for all $i$; and

- $\sum\limits_{k=1}^i n_k<i$ for all $i<N$.

So these are something like partitions where the running total is always bounded by the number of terms added thus far. (So the running average of the elements of the partition is less than 1.) In particular, this forces $n_1=0$ and $n_N=2$.

I'm more interested in whether this notion of a "valid" partition has arisen previously in the literature than an explicit count of how many of them there are for a given $N$ (probably a reasonably straight-forward linear recurrence or something), so any such references would be appreciated.