# What is the probability of having similar summation values

Given a vector of non-increasing positive integers $A=[a_1, \cdots, a_n]$, where $\sum_{1 \leq i \leq n}{a_i}=n(n-1)$, what is the probability that a random non-increasing vector $B=[b_1, \cdots, b_n]$, where $\sum_{1 \leq i \leq n}{b_i}=n(n-1)$, has the following property $\sum_{1 \leq i \leq n}{i \cdot a_i} = \sum_{1 \leq i \leq n}{i \cdot b_i}$?

That would depend on the distribution of $B$. However with a wide range of distributions it is almost certain (having zero probability) that $B$ does not have those properties. The sums will require $B$ to lie in an affine space of dimension $n-2$ which normally is a set of zero measure in $\mathbb R^n$. Even the prerequisite that $B$ is non-increasing should not alter this.
$$b_j = \begin{cases} \sum_{k=1}^n a_k & \text{if } j = 0 \\ 0 & \text{otherwise} \end{cases}$$
• The condition is $\sum_{i} i \times b_i = \sum_{i} i \times a_i$, not just the sums of $a_i$ and $b_i$ must be equal. – Hetebrij Jan 9 '17 at 8:11
• I phrased my question in the wrong way. I meant what is the probability of finding such vector $B$. Furthermore, as pointed out by @Hetebrij, the condition is different with yours. – orezvani Jan 9 '17 at 12:27