Colorful

AtCoder

IDfps_24_t

Time8000ms

Memory256MB

Difficulty—

You are given a sequence of positive integers $A = (A_1, A_2, \dots, A_N)$ of length $N$, and a positive integer $T$. There are $\sum_{i=1}^N A_i$ distinct locations. Each location is painted with a color represented by an integer, and there are exactly $A_i$ locations painted with color $i$. Initially, you choose one of the locations painted with color $1$, move to that location, and mark it. Then, you repeat the following operation exactly $T$ times: * Choose any location with a different color from your current one, and move there. Count the number of ways such that after all $T$ operations, you are back at the location you initially marked. Output the result modulo $998244353$. ## Constraints * $1 \leq N \leq 10^5$ * $1 \leq T \leq 10^{18}$ * $1 \leq A_i \leq 10^9$ * All input values are integers ## Input The input is given from standard input in the following format: $N$ $T$ $A_1$ $A_2$ $\dots$ $A_N$ ## Partial Score This problem has partial scoring: * If you solve all datasets with $N \leq 3 \times 10^3$, you will earn $5$ points. [samples]

Samples

Input #1

3 3
2 1 2

Output #1

4

Let us label the locations as follows:

*   The initially marked location: location $1$
*   The other location painted with color $1$: location $2$
*   The location painted with color $2$: location $3$
*   The two locations painted with color $3$: locations $4$ and $5$

Then, there are $4$ valid sequences of moves:

*   $1 \to 3 \to 4 \to 1$
*   $1 \to 3 \to 5 \to 1$
*   $1 \to 4 \to 3 \to 1$
*   $1 \to 5 \to 3 \to 1$

Input #2

10 31415926535897932
766294630 440423914 59187620 725560241 585990757 965580536 623321126 550925214 122410709 549392045

Output #2

66487687

API Response (JSON)

{
  "problem": {
    "name": "Colorful",
    "description": {
      "content": "You are given a sequence of positive integers $A = (A_1, A_2, \\dots, A_N)$ of length $N$, and a positive integer $T$. There are $\\sum_{i=1}^N A_i$ distinct locations. Each location is painted with a c",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 8000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "fps_24_t"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a sequence of positive integers $A = (A_1, A_2, \\dots, A_N)$ of length $N$, and a positive integer $T$.\nThere are $\\sum_{i=1}^N A_i$ distinct locations. Each location is painted with a c...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments