Let $\Lambda :=\{\lambda\_1, \dots, \lambda\_n\}$ be a set of $n$ distinct real numbers.


For a given $p \in \mathbb N$, consider further the set


$$I\_p := \{ \{i\_1, i\_2, \dots, i\_p\} : i\_j \in \{1, \dots,n\} \text{ for all } j=1, \dots, p\}.$$


It is not hard to see that $|I\_p| = \binom{n+p-1}{p}$. For example, if $n=3$ and $p=2$, then


$$ I\_2 = \{ \{1,1\}, \{1,2\}, \{1,3\}, \{2,2\}, \{2,3\}, \{3,3\} \}.$$


Now for a given $p \in \mathbb N$ we write down all $\binom{n+p-1}{p}$ sums
$$\lambda\_{i\_1} + \lambda\_{i\_2} + \dots + \lambda\_{i\_p}. $$


Clearly it can happen that "different" sums have the same value, e.g. if


$$\Lambda = \{0,1,2\}$$


and $p=2$, then we get the sums $0, 1, 2, 2, 3, 4$ given by considering the elements of $I\_2$.


My question is the following: Is there in general an upper bound on how many "different" sums there are that lead to one particular value?


Another interesting question would be the following: Given a fixed upper bound $N^\*$, are there conditions on $\Lambda$ such that for all $p \in \mathbb N$, at most $N^\*$ different $p$'th order sums yield the same value. For $N^\* = 1$ one gets the condition that $\lambda\_1, \dots, \lambda\_n$ need to linearly independent over $\mathbb Q$.



Upper bound for different sums having the same value
I answer the first question in Corollary 4.1
of my paper at <http://math.mit.edu/~rstan/pubs/pubfiles/42.pdf>. The
maximum number of sums that have the same value is the middle
coefficient of the $q$-binomial coefficient $\left[ n+p-1\atop p
\right]$. This is achieved by taking $\Lambda=\{ 1,2,\dots,n\}$ and
the sum to be $\lfloor \frac 12 p(n+1)\rfloor$. A more elementary
proof follows from the work of Proctor and is given in my book
*Algebraic Combinatorics*. 


**Addendum.** For fixed $p$, it follows from <http://math.mit.edu/~rstan/papers/qbc.pdf> that the middle coefficient of $\left[ n+p-1\atop p\right]$ is asymptotic to 
 $$ \frac{A(p-1,\lceil p/2\rceil)n^{p-1}}{(p-1)!p!} $$
as $n\to \infty$, where $A(p-1,\lceil p/2\rceil)$ denotes an Eulerian number.


