Permutation by Sum

2000ms

262144K

Description:

A permutation is a sequence of $$$n$$$ integers from $$$1$$$ to $$$n$$$, in which all the numbers occur exactly once. For example, $$$[1]$$$, $$$[3, 5, 2, 1, 4]$$$, $$$[1, 3, 2]$$$ are permutations, and $$$[2, 3, 2]$$$, $$$[4, 3, 1]$$$, $$$[0]$$$ are not.

Polycarp was given four integers $$$n$$$, $$$l$$$, $$$r$$$ ($$$1 \le l \le r \le n)$$$ and $$$s$$$ ($$$1 \le s \le \frac{n (n+1)}{2}$$$) and asked to find a permutation $$$p$$$ of numbers from $$$1$$$ to $$$n$$$ that satisfies the following condition:

$$$s = p_l + p_{l+1} + \ldots + p_r$$$.

For example, for $$$n=5$$$, $$$l=3$$$, $$$r=5$$$, and $$$s=8$$$, the following permutations are suitable (not all options are listed):

$$$p = [3, 4, 5, 2, 1]$$$;
$$$p = [5, 2, 4, 3, 1]$$$;
$$$p = [5, 2, 1, 3, 4]$$$.

But, for example, there is no permutation suitable for the condition above for $$$n=4$$$, $$$l=1$$$, $$$r=1$$$, and $$$s=5$$$.

Help Polycarp, for the given $$$n$$$, $$$l$$$, $$$r$$$, and $$$s$$$, find a permutation of numbers from $$$1$$$ to $$$n$$$ that fits the condition above. If there are several suitable permutations, print any of them.

Input:

The first line contains a single integer $$$t$$$ ($$$1 \le t \le 500$$$). Then $$$t$$$ test cases follow.

Each test case consist of one line with four integers $$$n$$$ ($$$1 \le n \le 500$$$), $$$l$$$ ($$$1 \le l \le n$$$), $$$r$$$ ($$$l \le r \le n$$$), $$$s$$$ ($$$1 \le s \le \frac{n (n+1)}{2}$$$).

It is guaranteed that the sum of $$$n$$$ for all input data sets does not exceed $$$500$$$.

Output:

For each test case, output on a separate line:

$$$n$$$ integers — a permutation of length $$$n$$$ that fits the condition above if such a permutation exists;
-1, otherwise.

If there are several suitable permutations, print any of them.

Sample Input:

Sample Output:

Informação

Provedor Codeforces

Código CF1512E