Playlist for Polycarp (hard version)
Contest: Codeforces Round 568 (Div. 2)
Difficulty rating: 2600
Tags: combinatorics, dp

The only difference between easy and hard versions is constraints.

Polycarp loves to listen to music, so he never leaves the player, even on the way home from the university. Polycarp overcomes the distance from the university to the house in exactly $$$T$$$ minutes.

In the player, Polycarp stores $$$n$$$ songs, each of which is characterized by two parameters: $$$t_i$$$ and $$$g_i$$$, where $$$t_i$$$ is the length of the song in minutes ($$$1 \le t_i \le 50$$$), $$$g_i$$$ is its genre ($$$1 \le g_i \le 3$$$).

Polycarp wants to create such a playlist so that he can listen to music all the time on the way from the university to his home, and at the time of his arrival home, the playlist is over. Polycarp never interrupts songs and always listens to them from beginning to end. Thus, if he started listening to the $$$i$$$-th song, he would spend exactly $$$t_i$$$ minutes on its listening. Polycarp also does not like when two songs of the same genre play in a row (i.e. successively/adjacently) or when the songs in his playlist are repeated.

Help Polycarpus count the number of different sequences of songs (their order matters), the total duration is exactly $$$T$$$, such that there are no two consecutive songs of the same genre in them and all the songs in the playlist are different.

Input format:
The first line of the input contains two integers $$$n$$$ and $$$T$$$ ($$$1 \le n \le 50, 1 \le T \le 2500$$$) — the number of songs in the player and the required total duration, respectively.

Next, the $$$n$$$ lines contain descriptions of songs: the $$$i$$$-th line contains two integers $$$t_i$$$ and $$$g_i$$$ ($$$1 \le t_i \le 50, 1 \le g_i \le 3$$$) — the duration of the $$$i$$$-th song and its genre, respectively.

Output format:
Output one integer — the number of different sequences of songs, the total length of exactly $$$T$$$, such that there are no two consecutive songs of the same genre in them and all the songs in the playlist are different. Since the answer may be huge, output it modulo $$$10^9 + 7$$$ (that is, the remainder when dividing the quantity by $$$10^9 + 7$$$).

Constraints:
Time limit: 5.0 s
Memory limit: 256.0 MB

Note:
In the first example, Polycarp can make any of the $$$6$$$ possible playlist by rearranging the available songs: $$$[1, 2, 3]$$$, $$$[1, 3, 2]$$$, $$$[2, 1, 3]$$$, $$$[2, 3, 1]$$$, $$$[3, 1, 2]$$$ and $$$[3, 2, 1]$$$ (indices of the songs are given).

In the second example, the first and second songs cannot go in succession (since they have the same genre). Thus, Polycarp can create a playlist in one of $$$2$$$ possible ways: $$$[1, 3, 2]$$$ and $$$[2, 3, 1]$$$ (indices of the songs are given).

In the third example, Polycarp can make the following playlists: $$$[1, 2, 3]$$$, $$$[1, 3, 2]$$$, $$$[2, 1, 3]$$$, $$$[2, 3, 1]$$$, $$$[3, 1, 2]$$$, $$$[3, 2, 1]$$$, $$$[1, 4]$$$, $$$[4, 1]$$$, $$$[2, 3, 4]$$$ and $$$[4, 3, 2]$$$ (indices of the songs are given).

Example input:
3 3
1 1
1 2
1 3

Example output:
6

Example input:
3 3
1 1
1 1
1 3

Example output:
2

Editorial:
Consider all genres are from $$$0$$$ to $$$2$$$ instead of from $$$1$$$ to $$$3$$$. It will be easier to deal with 0-based indices.
Let's find two arrays:
• $$$a[i][s]$$$ = number of ways to choose a subset of exactly $$$i$$$ songs of the genre $$$0$$$ with total duration of exactly $$$s$$$;
• $$$bc[i][j][s]$$$ = number of ways to choose a subset of exactly $$$i$$$ songs of the genre $$$1$$$ and $$$j$$$ songs of the genre $$$2$$$ with total duration of exactly $$$s$$$.
You can easily do it with DP. Here is the sample code to do it while reading the input:
This part works in $$$O(n^3 \cdot T)$$$ with a really small constant factor.
Let's calculate the array $$$ways[i][j][k][lst]$$$ = number of ways to put $$$i$$$ zeroes, $$$j$$$ ones and $$$k$$$ twos in a row (its length is $$$i+j+k$$$) in such a way that no two equal values go consecutive and the row ends on $$$lst$$$. You can also use DP to find the values:
This part works in $$$O(n^3)$$$ with a really small constant factor.
And now if the final part of the solution. Let's iterate over all possible ways how songs with the genre $$$0$$$ can be in a playlist. It means that we try all possible counts of such songs ($$$c[0]$$$ in the code below) and their total duration ($$$durs0$$$ in the code below). Now we know the total time for genres $$$1$$$ and $$$2$$$, it equals to $$$T-durs0$$$. Let's iterate over all possible counts of songs with genre=$$$1$$$ and songs with genre=$$$2$$$ (say, $$$c[1]$$$ and $$$c[2]$$$). For fixed $$$c[0]$$$, $$$durs0$$$, $$$c[1]$$$ and $$$c[2]$$$ there are $$$x$$$ ways to choose songs, where $$$x=a[c[0]][durs0] \cdot b[c[1]][c[2]][T-durs]$$$. To count all possible orders of songs, let's multiply $$$x$$$ on:
• $$$c[0]!$$$ — number of permutations of songs with genre=$$$0$$$;
• $$$c[1]!$$$ — number of permutations of songs with genre=$$$1$$$;
• $$$c[2]!$$$ — number of permutations of songs with genre=$$$2$$$;
• $$$ways[c[0]][c[1]][c[2]][0]+ways[c[0]][c[1]][c[2]][1]+ways[c[0]][c[1]][c[2]][2]$$$ - number ways to order them in the required way.
After all the multiplications add the result to the answer.
The following code illustrates this part of the solution:
This part works in $$$O(n^3 \cdot T)$$$ with a really small constant factor.
So the total complexity is $$$O(n^3 \cdot T)$$$ (the constant factor is significantly less than $$$1$$$).