A13800 | Unshuffling a Deck

You are given a deck of $n$ cards numbered from $1$ to $n$ (not necessarily in this order in the deck). You have to sort the deck by repeating the following operation.

- Choose $2 \le k \le n$ and split the deck in $k$ nonempty contiguous parts $D_1, D_2,\dots, D_k$ ( $D_1$ contains the first $|D_1|$ cards of the deck, $D_2$ contains the following $|D_2|$ cards and so on). Then reverse the order of the parts, transforming the deck into $D_k, D_{k-1}, \dots, D_2, D_1$ (so, the first $|D_k|$ cards of the new deck are $D_k$ , the following $|D_{k-1}|$ cards are $D_{k-1}$ and so on). The internal order of each packet of cards $D_i$ is unchanged by the operation.

You have to obtain a sorted deck (i.e., a deck where the first card is $1$ , the second is $2$ and so on) performing at most $n$ operations. It can be proven that it is always possible to sort the deck performing at most $n$ operations.

Examples of operation: The following are three examples of valid operations (on three decks with different sizes).

- If the deck is \[3 6 2 1 4 5 7\] (so $3$ is the first card and $7$ is the last card), we may apply the operation with $k=4$ and $D_1=$ \[3 6\], $D_2=$ \[2 1 4\], $D_3=$ \[5\], $D_4=$ \[7\]. Doing so, the deck becomes \[7 5 2 1 4 3 6\].
- If the deck is \[3 1 2\], we may apply the operation with $k=3$ and $D_1=$ \[3\], $D_2=$ \[1\], $D_3=$ \[2\]. Doing so, the deck becomes \[2 1 3\].
- If the deck is \[5 1 2 4 3 6\], we may apply the operation with $k=2$ and $D_1=$ \[5 1\], $D_2=$ \[2 4 3 6\]. Doing so, the deck becomes \[2 4 3 6 5 1\].

The first line of the input contains one integer $n$ ( $1\le n\le 52$ ) — the number of cards in the deck.

The second line contains $n$ integers $c_1, c_2, \dots, c_n$ — the cards in the deck. The first card is $c_1$ , the second is $c_2$ and so on.

It is guaranteed that for all $i=1,\dots,n$ there is exactly one $j\in\{1,\dots,n\}$ such that $c_j = i$ .

On the first line, print the number $q$ of operations you perform (it must hold $0\le q\le n$ ).

Then, print $q$ lines, each describing one operation.

To describe an operation, print on a single line the number $k$ of parts you are going to split the deck in, followed by the size of the $k$ parts: $|D_1|, |D_2|, \dots , |D_k|$ .

It must hold $2\le k\le n$ , and $|D_i|\ge 1$ for all $i=1,\dots,k$ , and $|D_1|+|D_2|+\cdots + |D_k| = n$ .

It can be proven that it is always possible to sort the deck performing at most $n$ operations. If there are several ways to sort the deck you can output any of them.