[USACO23JAN] Moo Route G

Luogu

IDLGP9018

Time2000ms

Memory256MB

DifficultyP5

USACO2023组合数学排列组合

Farmer Nhoj dropped Bessie in the middle of nowhere! At time $t=0$, Bessie is located at $x=0$ on an infinite number line. She frantically searches for an exit by moving left or right by $1$ unit each second. However, there actually is no exit and after $T$ seconds, Bessie is back at $x=0$, tired and resigned. Farmer Nhoj tries to track Bessie but only knows how many times Bessie crosses $x=0.5,1.5,2.5,\cdots,(N−1).5$ , given by an array $A_0,A_1, \cdots ,A_{N−1} (1 \le N \le 10^5, 1 \le A_i \le 10^6)$. Bessie never reaches $x>N$ nor $x<0$. In particular, Bessie's route can be represented by a string of $T=\sum\limits_{i=0}^{N-1}A_i$ Ls and Rs where the ith character represents the direction Bessie moves in during the ith second. The number of direction changes is defined as the number of occurrences of $LR$s plus the number of occurrences of $RL$s. Please help Farmer Nhoj count the number of routes Bessie could have taken that are consistent with $A$ and minimize the number of direction changes. It is guaranteed that there is at least one valid route. ## Input The first line contains $N$. The second line contains $A_0,A_1, \cdots ,A_{N−1}$. ## Output The number of routes Bessie could have taken, modulo $10^9+7$. [samples] ## Note ### Explanation for Sample 1 Bessie must change direction at least 5 times. There are two routes corresponding to Bessie changing direction exactly 5 times: $\texttt{RRLRLLRRLL}$ $\texttt{RRLLRRLRLL}$ ### Scoring - Inputs $2-4$: $N \le 2$ and $\max(A_i) \le 10^3$ - Inputs $5-7$: $N \le 2$ - Inputs $8-11$: $\max(A_i) \le 10^3$ - Inputs $12-21$: No additional constraints.

Samples

Input #1

2
4 6

Output #1

API Response (JSON)

{
  "problem": {
    "name": "[USACO23JAN] Moo Route G",
    "description": {
      "content": "Farmer Nhoj dropped Bessie in the middle of nowhere! At time $t=0$, Bessie is located at $x=0$ on an infinite number line. She frantically searches for an exit by moving left or right by $1$ unit each",
      "description_type": "Markdown"
    },
    "platform": "Luogu",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": {
      "LuoguStyle": "P5"
    },
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "LGP9018"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Farmer Nhoj dropped Bessie in the middle of nowhere! At time $t=0$, Bessie is located at $x=0$ on an infinite number line. She frantically searches for an exit by moving left or right by $1$ unit each...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments