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feat: add solutions to lc problem: No.2338
No.2338.Count the Number of Ideal Arrays
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‎solution/2300-2399/2338.Count the Number of Ideal Arrays/README.md

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@@ -59,8 +59,20 @@
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<!-- 这里可写通用的实现逻辑 -->
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我们注意到，$maxValue$ 的最大值不超过 $10^4$，如果数组不含重复元素，那么数组最多只会有不超过 $16$ 个元素。
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**方法一：记忆化搜索 + 组合计数**
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**方法二：动态规划**
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设 $dp[i][j]$ 表示以 $i$ 结尾，且由 $j$ 个不同元素构成的序列的方案数。初始值 $dp[i][1]=1$。
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考虑 $n$ 个小球，最终划分为 $j$ 份，那么可以用“隔板法”，即在 $n-1$ 个位置上插入 $j-1$ 个隔板，那么组合数为 $C_{n-1}^{j-1}$ 。
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我们可以预处理组合数 $C[i][j]$，根据递推公式 $C[i][j]=C[i-1][j]+C[i-1][j-1]$ 求得，特别地，当 $j=0$ 时，$C[i][j]=1$。
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最终的答案为 $\sum\limits_{i=1}^{k}\sum\limits_{j=1}^{\log_2 k + 1}dp[i][j] \times C_{n-1}^{j-1}$ 。其中 $k$ 表示数组的最大值，即 $maxValue$。
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<!-- tabs:start -->
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### **Python3**
@@ -84,14 +96,37 @@ class Solution:
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mod = 10**9 + 7
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for i in range(n):
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for j in range(min(16, i + 1)):
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c[i][j] = 1 if j == 0 else (
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c[i - 1][j] + c[i - 1][j - 1]) % mod
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c[i][j] = 1 if j == 0 else (c[i - 1][j] + c[i - 1][j - 1]) % mod
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ans = 0
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for i in range(1, maxValue + 1):
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ans = (ans + dfs(i, 1)) % mod
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return ans
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```
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```python
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class Solution:
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def idealArrays(self, n: int, maxValue: int) -> int:
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c = [[0] * 16 for _ in range(n)]
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mod = 10**9 + 7
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for i in range(n):
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for j in range(min(16, i + 1)):
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c[i][j] = 1 if j == 0 else (c[i - 1][j] + c[i - 1][j - 1]) % mod
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dp = [[0] * 16 for _ in range(maxValue + 1)]
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for i in range(1, maxValue + 1):
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dp[i][1] = 1
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for j in range(1, 15):
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for i in range(1, maxValue + 1):
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k = 2
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while k * i <= maxValue:
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dp[k * i][j + 1] = (dp[k * i][j + 1] + dp[i][j]) % mod
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k += 1
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ans = 0
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for i in range(1, maxValue + 1):
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for j in range(1, 16):
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ans = (ans + dp[i][j] * c[-1][j - 1]) % mod
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return ans
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```
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### **Java**
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<!-- 这里可写当前语言的特殊实现逻辑 -->
@@ -141,6 +176,40 @@ class Solution {
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}
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```
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```java
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class Solution {
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private static final int MOD = (int) 1e9 + 7;
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public int idealArrays(int n, int maxValue) {
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int[][] c = new int[n][16];
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for (int i = 0; i < n; ++i) {
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for (int j = 0; j <= i && j < 16; ++j) {
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c[i][j] = j == 0 ? 1 : (c[i - 1][j] + c[i - 1][j - 1]) % MOD;
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}
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}
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long[][] dp = new long[maxValue + 1][16];
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for (int i = 1; i <= maxValue; ++i) {
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dp[i][1] = 1;
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}
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for (int j = 1; j < 15; ++j) {
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for (int i = 1; i <= maxValue; ++i) {
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int k = 2;
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for (; k * i <= maxValue; ++k) {
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dp[k * i][j + 1] = (dp[k * i][j + 1] + dp[i][j]) % MOD;
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}
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}
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}
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long ans = 0;
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for (int i = 1; i <= maxValue; ++i) {
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for (int j = 1; j < 16; ++j) {
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ans = (ans + dp[i][j] * c[n - 1][j - 1]) % MOD;
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}
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}
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return (int) ans;
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}
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}
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```
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### **C++**
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```cpp
@@ -176,6 +245,37 @@ public:
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};
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```
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```cpp
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using ll = long long;
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class Solution {
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public:
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const int mod = 1e9 + 7;
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int idealArrays(int n, int maxValue) {
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vector<vector<int>> c(n, vector<int>(16));
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for (int i = 0; i < n; ++i)
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for (int j = 0; j <= i && j < 16; ++j)
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c[i][j] = j == 0 ? 1 : (c[i - 1][j] + c[i - 1][j - 1]) % mod;
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vector<vector<ll>> dp(maxValue + 1, vector<ll>(16));
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for (int i = 1; i <= maxValue; ++i) dp[i][1] = 1;
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for (int j = 1; j < 15; ++j)
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{
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for (int i = 1; i <= maxValue; ++i)
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{
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int k = 2;
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for (; k * i <= maxValue; ++k) dp[k * i][j + 1] = (dp[k * i][j + 1] + dp[i][j]) % mod;
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}
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}
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ll ans = 0;
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for (int i = 1; i <= maxValue; ++i)
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for (int j = 1; j < 16; ++j)
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ans = (ans + dp[i][j] * c[n - 1][j - 1]) % mod;
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return (int) ans;
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}
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};
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```
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### **Go**
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```go
@@ -224,6 +324,45 @@ func idealArrays(n int, maxValue int) int {
224324
}
225325
```
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```go
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func idealArrays(n int, maxValue int) int {
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mod := int(1e9) + 7
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c := make([][]int, n)
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for i := range c {
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c[i] = make([]int, 16)
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}
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for i := 0; i < n; i++ {
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for j := 0; j <= i && j < 16; j++ {
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if j == 0 {
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c[i][j] = 1
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} else {
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c[i][j] = (c[i-1][j] + c[i-1][j-1]) % mod
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}
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}
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}
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dp := make([][]int, maxValue+1)
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for i := range dp {
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dp[i] = make([]int, 16)
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dp[i][1] = 1
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}
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for j := 1; j < 15; j++ {
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for i := 1; i <= maxValue; i++ {
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k := 2
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for ; k*i <= maxValue; k++ {
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dp[k*i][j+1] = (dp[k*i][j+1] + dp[i][j]) % mod
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}
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}
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}
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ans := 0
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for i := 1; i <= maxValue; i++ {
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for j := 1; j < 16; j++ {
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ans = (ans + dp[i][j]*c[n-1][j-1]) % mod
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}
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}
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return ans
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}
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```
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227366
### **TypeScript**
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229368
```ts

‎solution/2300-2399/2338.Count the Number of Ideal Arrays/README_EN.md

Lines changed: 129 additions & 2 deletions
Original file line numberDiff line numberDiff line change
@@ -76,14 +76,37 @@ class Solution:
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mod = 10**9 + 7
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for i in range(n):
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for j in range(min(16, i + 1)):
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c[i][j] = 1 if j == 0 else (
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c[i - 1][j] + c[i - 1][j - 1]) % mod
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c[i][j] = 1 if j == 0 else (c[i - 1][j] + c[i - 1][j - 1]) % mod
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ans = 0
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for i in range(1, maxValue + 1):
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ans = (ans + dfs(i, 1)) % mod
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return ans
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```
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```python
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class Solution:
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def idealArrays(self, n: int, maxValue: int) -> int:
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c = [[0] * 16 for _ in range(n)]
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mod = 10**9 + 7
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for i in range(n):
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for j in range(min(16, i + 1)):
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c[i][j] = 1 if j == 0 else (c[i - 1][j] + c[i - 1][j - 1]) % mod
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dp = [[0] * 16 for _ in range(maxValue + 1)]
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for i in range(1, maxValue + 1):
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dp[i][1] = 1
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for j in range(1, 15):
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for i in range(1, maxValue + 1):
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k = 2
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while k * i <= maxValue:
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dp[k * i][j + 1] = (dp[k * i][j + 1] + dp[i][j]) % mod
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k += 1
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ans = 0
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for i in range(1, maxValue + 1):
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for j in range(1, 16):
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ans = (ans + dp[i][j] * c[-1][j - 1]) % mod
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return ans
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```
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### **Java**
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89112
```java
@@ -131,6 +154,40 @@ class Solution {
131154
}
132155
```
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```java
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class Solution {
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private static final int MOD = (int) 1e9 + 7;
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public int idealArrays(int n, int maxValue) {
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int[][] c = new int[n][16];
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for (int i = 0; i < n; ++i) {
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for (int j = 0; j <= i && j < 16; ++j) {
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c[i][j] = j == 0 ? 1 : (c[i - 1][j] + c[i - 1][j - 1]) % MOD;
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}
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}
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long[][] dp = new long[maxValue + 1][16];
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for (int i = 1; i <= maxValue; ++i) {
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dp[i][1] = 1;
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}
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for (int j = 1; j < 15; ++j) {
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for (int i = 1; i <= maxValue; ++i) {
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int k = 2;
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for (; k * i <= maxValue; ++k) {
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dp[k * i][j + 1] = (dp[k * i][j + 1] + dp[i][j]) % MOD;
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}
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}
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}
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long ans = 0;
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for (int i = 1; i <= maxValue; ++i) {
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for (int j = 1; j < 16; ++j) {
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ans = (ans + dp[i][j] * c[n - 1][j - 1]) % MOD;
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}
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}
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return (int) ans;
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}
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}
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```
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### **C++**
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```cpp
@@ -166,6 +223,37 @@ public:
166223
};
167224
```
168225
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```cpp
227+
using ll = long long;
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class Solution {
230+
public:
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const int mod = 1e9 + 7;
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int idealArrays(int n, int maxValue) {
234+
vector<vector<int>> c(n, vector<int>(16));
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for (int i = 0; i < n; ++i)
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for (int j = 0; j <= i && j < 16; ++j)
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c[i][j] = j == 0 ? 1 : (c[i - 1][j] + c[i - 1][j - 1]) % mod;
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vector<vector<ll>> dp(maxValue + 1, vector<ll>(16));
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for (int i = 1; i <= maxValue; ++i) dp[i][1] = 1;
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for (int j = 1; j < 15; ++j)
241+
{
242+
for (int i = 1; i <= maxValue; ++i)
243+
{
244+
int k = 2;
245+
for (; k * i <= maxValue; ++k) dp[k * i][j + 1] = (dp[k * i][j + 1] + dp[i][j]) % mod;
246+
}
247+
}
248+
ll ans = 0;
249+
for (int i = 1; i <= maxValue; ++i)
250+
for (int j = 1; j < 16; ++j)
251+
ans = (ans + dp[i][j] * c[n - 1][j - 1]) % mod;
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return (int) ans;
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}
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};
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```
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### **Go**
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171259
```go
@@ -214,6 +302,45 @@ func idealArrays(n int, maxValue int) int {
214302
}
215303
```
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```go
306+
func idealArrays(n int, maxValue int) int {
307+
mod := int(1e9) + 7
308+
c := make([][]int, n)
309+
for i := range c {
310+
c[i] = make([]int, 16)
311+
}
312+
for i := 0; i < n; i++ {
313+
for j := 0; j <= i && j < 16; j++ {
314+
if j == 0 {
315+
c[i][j] = 1
316+
} else {
317+
c[i][j] = (c[i-1][j] + c[i-1][j-1]) % mod
318+
}
319+
}
320+
}
321+
dp := make([][]int, maxValue+1)
322+
for i := range dp {
323+
dp[i] = make([]int, 16)
324+
dp[i][1] = 1
325+
}
326+
for j := 1; j < 15; j++ {
327+
for i := 1; i <= maxValue; i++ {
328+
k := 2
329+
for ; k*i <= maxValue; k++ {
330+
dp[k*i][j+1] = (dp[k*i][j+1] + dp[i][j]) % mod
331+
}
332+
}
333+
}
334+
ans := 0
335+
for i := 1; i <= maxValue; i++ {
336+
for j := 1; j < 16; j++ {
337+
ans = (ans + dp[i][j]*c[n-1][j-1]) % mod
338+
}
339+
}
340+
return ans
341+
}
342+
```
343+
217344
### **TypeScript**
218345

219346
```ts

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