is it true that ?
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Just thoughts.
L is left part, R is right part.
L = PL1 + PL2 + ... + PLk
R = PR1 + PR2 + ... + PRk
My theory is for all i = 1..k PLi > PRi.
![](https://espresso.codeforces.com/b55567fc19fb064be684025bbd4535f8dd93f59e.png)
![](https://espresso.codeforces.com/5b4a8a00d2291cf9ae759b4cec10df321d90e60b.png)
Induction. Base i = 1.
So for x > 0: PL1 > PR1
And I really don't know what's next (if I'm on the right path).
Yeah, it is true. But proving this is a bit painful. Here is the only way I know:
Reminder: generating function of sequence
is formal series
.
Consider amount of ways to split number
in sum of distinct numbers. Generating function for this sequence is
. Now consider number of ways to split number
in sum of not necessary distinct odd numbers. Generating function for this sequence is
. Let's multiply each functions by
. Then first one becomes
and the second one become
. Thus number of ways to split
in distinct numbers equals number of ways to split in in odd numbers (you can also prove it with some combinatorics, I guess).
Now consider
which is the number of ways to split
in
distinct numbers.
. Thus if
then
. Then
.
And with odd numbers part we have
. Since
, we have equation from the statement now.
Understanding of correctness of each transition in proof is left to reader as an excercise ;)