feat: add solutions to lc problem: No.1866 (doocs#1981)

No.1866.Number of Ways to Rearrange Sticks With K Sticks Visible

Author: yanglbme

solution/1800-1899/1866.Number of Ways to Rearrange Sticks With K Sticks Visible/README.md

@@ -210,6 +210,39 @@ func rearrangeSticks(n int, k int) int {
 }
 ```
 
+### **TypeScript**
+
+```ts
+function rearrangeSticks(n: number, k: number): number {
+    const mod = 10 ** 9 + 7;
+    const f: number[][] = Array.from({ length: n + 1 }, () =>
+        Array.from({ length: k + 1 }, () => 0),
+    );
+    f[0][0] = 1;
+    for (let i = 1; i <= n; ++i) {
+        for (let j = 1; j <= k; ++j) {
+            f[i][j] = (f[i - 1][j - 1] + (i - 1) * f[i - 1][j]) % mod;
+        }
+    }
+    return f[n][k];
+}
+```
+
+```ts
+function rearrangeSticks(n: number, k: number): number {
+    const mod = 10 ** 9 + 7;
+    const f: number[] = Array(n + 1).fill(0);
+    f[0] = 1;
+    for (let i = 1; i <= n; ++i) {
+        for (let j = k; j; --j) {
+            f[j] = (f[j] * (i - 1) + f[j - 1]) % mod;
+        }
+        f[0] = 0;
+    }
+    return f[k];
+}
+```
+
 ### **...**
 
 ```

solution/1800-1899/1866.Number of Ways to Rearrange Sticks With K Sticks Visible/README_EN.md

@@ -49,6 +49,24 @@ The visible sticks are underlined.
 
 ## Solutions
 
+**Solution 1: Dynamic Programming**
+
+We define $f[i][j]$ to represent the number of permutations of length $i$ in which exactly $j$ sticks can be seen. Initially, $f[0][0]=1$ and the rest $f[i][j]=0$. The answer is $f[n][k]$.
+
+Consider whether the last stick can be seen. If it can be seen, it must be the longest. Then there are $i - 1$ sticks in front of it, and exactly $j - 1$ sticks can be seen, which is $f[i - 1][j - 1]$. If the last stick cannot be seen, it can be any one except the longest stick. Then there are $i - 1$ sticks in front of it, and exactly $j$ sticks can be seen, which is $f[i - 1][j] \times (i - 1)$.
+
+Therefore, the state transition equation is:
+
+$$
+f[i][j] = f[i - 1][j - 1] + f[i - 1][j] \times (i - 1)
+$$
+
+The final answer is $f[n][k]$.
+
+We notice that $f[i][j]$ is only related to $f[i - 1][j - 1]$ and $f[i - 1][j]$, so we can use a one-dimensional array to optimize the space complexity.
+
+The time complexity is $O(n \times k)$, and the space complexity is $O(k)$. Here, $n$ and $k$ are the two integers given in the problem.
+
 <!-- tabs:start -->
 
 ### **Python3**
@@ -185,6 +203,39 @@ func rearrangeSticks(n int, k int) int {
 }
 ```
 
+### **TypeScript**
+
+```ts
+function rearrangeSticks(n: number, k: number): number {
+    const mod = 10 ** 9 + 7;
+    const f: number[][] = Array.from({ length: n + 1 }, () =>
+        Array.from({ length: k + 1 }, () => 0),
+    );
+    f[0][0] = 1;
+    for (let i = 1; i <= n; ++i) {
+        for (let j = 1; j <= k; ++j) {
+            f[i][j] = (f[i - 1][j - 1] + (i - 1) * f[i - 1][j]) % mod;
+        }
+    }
+    return f[n][k];
+}
+```
+
+```ts
+function rearrangeSticks(n: number, k: number): number {
+    const mod = 10 ** 9 + 7;
+    const f: number[] = Array(n + 1).fill(0);
+    f[0] = 1;
+    for (let i = 1; i <= n; ++i) {
+        for (let j = k; j; --j) {
+            f[j] = (f[j] * (i - 1) + f[j - 1]) % mod;
+        }
+        f[0] = 0;
+    }
+    return f[k];
+}
+```
+
 ### **...**
 
 ```

solution/1800-1899/1866.Number of Ways to Rearrange Sticks With K Sticks Visible/Solution.ts

@@ -0,0 +1,12 @@
+function rearrangeSticks(n: number, k: number): number {
+    const mod = 10 ** 9 + 7;
+    const f: number[] = Array(n + 1).fill(0);
+    f[0] = 1;
+    for (let i = 1; i <= n; ++i) {
+        for (let j = k; j; --j) {
+            f[j] = (f[j] * (i - 1) + f[j - 1]) % mod;
+        }
+        f[0] = 0;
+    }
+    return f[k];
+}