# sum sets are equal

I have two sequence of positive integers say $$a_1, a_2, a_3, a_4, a_5, a_6$$ and $$b_1, b_2, b_3, b_4, b_5, b_6$$ (they may be repeated) such that the multisets $$\{a_1, a_1+a_2, \cdots , a_1+a_2+\cdots +a_6, a_6, a_6+a_5, \cdots ,a_6+a_5+\cdots +a_1\}=\{b_1, b_1+b_2, \cdots , b_1+b_2+\cdots +b_6, b_6, b_6+b_5, \cdots ,b_6+b_5+\cdots +b_1\}$$.

This means that the multiset containing all consecutive partial sums involving $$a_1$$ and $$a_6$$ is equal to the multiset containing all consecutive partial sums involving $$b_1$$ and $$b_6$$.

From here can we say that either $$a_i=b_i$$ for all $$i$$ or $$a_i=b_{7-i}$$ for all $$i$$ ? I have tried with some particular examples and it worked.

• Positivity is the key here. Smallest is $a_1$ on the left and $b_1$ on the right, so these match, and so on.
– lulu
Commented Jun 3 at 0:48
• No, $a_1$ can be $b_5+b_6$ for example.
– jack
Commented Jun 3 at 0:51
• Oh, I thought the terms were nested, but they aren't. So, right. Can't order them the way I said.
– lulu
Commented Jun 3 at 0:53
• So, what are the terms? Every possible subset sum?
– lulu
Commented Jun 3 at 0:54
• No, just all consecutive sums involving $a_1$ and $a_6$. (Similarly $b_1$ and $b_6$).
– jack
Commented Jun 3 at 0:57

It's false: $$a_i = 1, 2, 1, 1, 3, 2 \quad b_i = 1, 1, 3, 1, 1, 3$$ Set of partial sums for $$a_i$$ in order is $$1, 3, 4, 5, 8, 10, 2, 5, 6, 7, 9, 10$$ Set of partial sums for $$b_i$$ in order is $$1, 2, 5, 6, 7, 10, 3, 4, 5, 8, 9, 10$$

Edit: This is not true even for 5 elements...

$$a_i = 1, 2, 1, 1, 2 \quad b_i = 1, 1, 1, 1, 3$$ Set of partial sums for $$a_i$$ in order is $$1, 3, 4, 5, 7, 2, 3, 4, 6, 7$$ Set of partial sums for $$b_i$$ in order is $$1, 2, 3, 4, 7, 3, 4, 5, 6, 7$$

Edit2: This is not true even for 4 elements...

$$a_i = 1, 2, 1, 2 \quad b_i = 1, 1, 1, 3$$ Set of partial sums for $$a_i$$ in order is $$1, 3, 4, 6, 2, 3, 5, 6$$ Set of partial sums for $$b_i$$ in order is $$1, 2, 3, 6, 3, 4, 5, 6$$

Edit3: This is not true even for 3 elements...

$$a_i = 1, 2, 2 \quad b_i = 1, 1, 3$$ Set of partial sums for $$a_i$$ in order is $$1, 3, 5, 2, 4, 5$$ Set of partial sums for $$b_i$$ in order is $$1, 2, 5, 3, 4, 5$$

• On the bright side, it is true for 2 elements Commented Jun 3 at 14:29