200582: [AtCoder]ARC058 E - Iroha and Haiku

Memory Limit：512 MB Time Limit：4 S
Judge Style：Text Compare Creator：
Submit：0 Solved：0

Description

Score : $700$ points

Problem Statement

Haiku is a short form of Japanese poetry. A Haiku consists of three phrases with 5, 7 and 5 syllables, in this order.

Iroha is looking for $X,Y,Z$-Haiku (defined below) in integer sequences.

Consider all integer sequences of length $N$ whose elements are between $1$ and $10$, inclusive. Out of those $10^N$ sequences, how many contain an $X,Y,Z$-Haiku?

Here, an integer sequence $a_0, a_1, ..., a_{N-1}$ is said to contain an $X,Y,Z$-Haiku if and only if there exist four indices $x, y, z, w (0 ≦ x < y < z < w ≦ N)$ such that all of the following are satisfied:

  • $a_x + a_{x+1} + ... + a_{y-1} = X$
  • $a_y + a_{y+1} + ... + a_{z-1} = Y$
  • $a_z + a_{z+1} + ... + a_{w-1} = Z$

Since the answer can be extremely large, print the number modulo $10^9+7$.

Constraints

  • $3 ≦ N ≦ 40$
  • $1 ≦ X ≦ 5$
  • $1 ≦ Y ≦ 7$
  • $1 ≦ Z ≦ 5$

Input

The input is given from Standard Input in the following format:

$N$ $X$ $Y$ $Z$

Output

Print the number of the sequences that contain an $X,Y,Z$-Haiku, modulo $10^9+7$.

Sample Input 1

3 5 7 5

Sample Output 1

1

Here, the only sequence that contains a $5,7,5$-Haiku is $[5, 7, 5]$.

Sample Input 2

4 5 7 5

Sample Output 2

34

Sample Input 3

37 4 2 3

Sample Output 3

863912418

Sample Input 4

40 5 7 5

Sample Output 4

562805100

Input

题意翻译

- 若 $a=\{a_1,a_2,\cdots a_n\}$ 存在 $1\le x<y<z<w\le n+1$ 满足 $\sum\limits_{i=x}^{y-1}a_i=X,\sum\limits_{i=y}^{z-1}a_i=Y,\sum\limits_{i=z}^{w-1}a_i=Z$ 时，则称数列 $a$ 是**好的**。 - 求在所有长度为 $n$ 且 $a_i\in\mathbb{N}^{+}\cap[1,10]$ 的 $10^n$ 个序列 $a$ 中，有多少个序列是**好的**，答案对 $10^9+7$ 取模。 - $3\le n\le40$，$1\le X\le5$，$1\le Y\le7$，$1\le Z\le5$。

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