Water Storage in a Histogram

You are given an integer $N$ and an integer $X$. Consider a histogram consisting of $N$ vertical columns. You are to assign the heights of these columns to be a permutation of the integers $1,2,\dots,N$.

Once the permutation $h_1,h_2,\dots,h_N$ is fixed, we may add water on top of some columns. Let $v_i\ge 0$ be the amount of water added on column $i$, and define

$$s_i = h_i + v_i.$$

A water configuration (i.e. a choice of all $v_i$) is said to be (stable) if the following conditions hold:

\begin{enumerate} \item For every column with index $i\ge 2$, if $v_i>0$ then

$$s_i \le s_{i-1}.$$

\item For every column with index $i\le N-1$, if $v_i>0$ then

$$s_i \le s_{i+1}.$$

\item The boundary columns do not hold any water, i.e. $v_1=v_N=0$. \end{enumerate}

The capacity of the histogram is defined to be the maximum total water (\sum_{i=1}^N v_i) that can be stored in a (stable) configuration.

Your task is to find a permutation $h_1,h_2,\dots,h_N$ of $\{1,2,\dots,N\}$ so that the capacity of the histogram equals exactly $X$. If there is no such permutation, output -1. If there are multiple answers, output any one of them.

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Note that in any valid permutation the water is only allowed to be stored on a contiguous interval in the middle. One possible way to obtain a nonzero capacity is as follows. Suppose there exists an interval of indices $[2,r-1]$ (with $2\le r\le N-1$) that we intend to fill with water. We can enforce a \(stable\) water structure by requiring that
\(\bullet\) the left boundary at index 1 has height $h_1=H$;
\(\bullet\) the right boundary at index $r$ has a height at least $H$;
\(\bullet\) the basin (columns $2,3,\dots, r-1$) is assigned the smallest $r-2$ numbers (in increasing order).
Then, if we fill every basin cell with water up to level $H$, the amount of water stored on column $i$ (with $2\le i\le r-1$) is \(H-h_i\). In particular, if we let \(k=r-2\) (thus $1\le k\le N-2$) and choose $H$ with $k+1\le H\le N-1$, then the total water in the basin is
$$\;\;\; \sum_{j=1}^{k} (H - j) = k\cdot H - \frac{k(k+1)}{2}. $$
Your permutation must be constructed so that for some choice of $1\le k\le N-2$ and some $H$ with $k+1\le H\le N-1$, it holds that
$$ k\cdot H - \frac{k(k+1)}{2} = X. $$
In our intended solutions we use the following construction (which we call Construction A):

1. If \(X=0\), simply output the increasing permutation $1,2,\dots,N$ (note that the stability conditions then force no water to be added).
2. Otherwise, iterate over all possible values of \(k\) from $1$ to $N-2$ and over all possible $H$ (with $k+1\le H\le N-1$) until one is found satisfying $$ k\cdot H - \frac{k(k+1)}{2} = X. $$
3. Suppose such a pair \( (k,H) \) is found. Then construct a permutation of length $N$ as follows:
   • Set the left boundary: \(p_1 = H\).
   • For indices \(i=2,3,\dots,k+1\), assign \(p_i = i-1\) (i.e. the basin receives the smallest \(k\) numbers in increasing order).
   • Set the right boundary of the basin at position \(k+2\) to be \(N\) (note that \(N\) is larger than $H$, so the effective fill level is $\min(H,N)=H$).
   • Fill the remaining positions (if any) with the leftover numbers in descending order.
   This guarantees that the maximum water stored (by filling the basin up to level $H$) is exactly $$X=k\cdot H-\frac{k(k+1)}{2}.$$
4. If no pair \((k,H)\) can be found, output -1.

It can be shown that if a permutation exists with capacity $X$, then one of the above constructions (with a unique basin in the middle) will work.

Input/Output Requirements:

• The first line of input contains two space‐separated integers $N$ and $X$.
• If a valid permutation exists, output a single line containing $N$ space‐separated integers -- the permutation. Otherwise output -1.

Constraints:

• $2\le N\le 10^5$.
• $0\le X\le \; \frac{(N-2)(N-1)}{2}$ (which is the maximum capacity achievable by Construction A).

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Example

For example, let $N=5$ and $X=3$. One valid answer is

4 1 5 3 2

Explanation: Here we use (k=1) and (H=4) (since $1\cdot4-1=3$). The basin (just column 2) has water amount $4-1=3$, and the remainder of the permutation is filled appropriately.

inputFormat

The first line contains two integers $N$ and $X$, separated by a space.

outputFormat

If a valid permutation exists, output $N$ space‐separated integers denoting one such permutation. Otherwise, output -1.

sample

5 0

1 2 3 4 5

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