Partitions of n that generate all numbers smaller than n Consider a  partition $$n=n_1+n_2+\cdots+n_k$$ such that each  number $1,\cdots, n$ can be obtained
by adding some of the numbers  $n_1,n_2,\cdots,n_k$. 
For example, $$9=4+3+1+1,$$ 
and every number $1,2,\cdots,9$ be ca written as a sum of some of the numbers $4,3,1,1$. This other  partition 
 $$9=6+1+1+1$$ 
fails the desired property, as $4$ (and $5$)  cannot be given by any sum of $6,1,1,1$.
Question: Can we charaterize which partitions of an arbitrary $n$ have
this property? We clearly need at least one $1$ among $n_1,n_2,\cdots,n_k$,
and intuitively we need  many small numbers $n_i$. But I can't see much beyond this. Any idea or reference  will be appreciated.
 A: Let $\lambda$ be a partition of $n$. The required condition is that $\lambda$ contain partitions $\lambda_i$ of each $1 \le i < n$. Clearly if $\lambda$ contains a partition of $j$ then it also contains a partition of $(n - j)$, being the multiset difference $\lambda - \lambda_j$.
Therefore the first thing to notice is that $\lambda$ cannot contain any element $a > \lceil \frac{n}{2} \rceil$, for if it did then $\{a\}$ cannot be part of a partition of $\lceil \frac{n}{2} \rceil$ and $\lambda - \{a\}$ is a partition of $(n - a) < (n - \lceil \frac{n}{2} \rceil) < \lceil \frac{n}{2} \rceil$ cannot contain a partition of $\lceil \frac{n}{2} \rceil$.
Now, suppose that the largest element of $\lambda$ is $m$. It is certainly sufficient that $\lambda - \{m\}$ should satisfy the corresponding condition of providing partitions for each $1 \le i < (n - m)$. Proof: $\lambda - \{m\}$ is a partition of $(n - m)$ and provides partitions for each smaller natural number, so it remains to construct partitions $\lambda_i$ for $(n - m) < i < n$. We can do this by taking partitions from $\lambda - \{m\}$ for $(n - 2m) < j < n - m$ and then adding $\{m\}$ to each one. This can only fail if $j < 0$, which can only happen if $(n - 2m) < -1$. Since $m \le \lceil \frac{n}{2} \rceil$ we have $n - 2m \ge n - 2\lceil \frac{n}{2} \rceil$, which is $0$ if $n$ is even and $-1$ if $n$ is odd, so all cases are covered.
The interesting question is whether it's necessary that $\lambda - \{m\}$ should satisfy the same condition. Clearly it must contain partitions of $1 \le i < m$, since $\{m\}$ doesn't participate in them. And by the simple principle of taking complements in $\lambda$ it's clear that for each $m \le i < n$ the remnant $\lambda - \{m\}$ must either contain partitions of $(i - m)$ and $n - i$; or $i$ and $n - m - i$. Is that sufficient?
My intuition is that it's necessary, and testing on small examples (up to $n = 30$) supports that, but I haven't proved it.
In the Online Encyclopedia of Integer Sequences it's A126796 and a comment claims the characterisation

A partition is complete iff each part is no more than 1 more than the sum of all smaller parts. (This includes the smallest part, which thus must be 1.) - Franklin T. Adams-Watters, Mar 22 2007

