Redistribution of Piles

AtCoder

IDabc313_g

Time2000ms

Memory256MB

Difficulty—

There are $N$ plates numbered $1$ through $N$. Dish $i$ has $a_i$ stones on it. There is also an empty bag. You can perform the following two kinds of operations any number of times (possibly zero) in any order. * Remove one stone from each plate with one or more stones. Put the removed stones into the bag. * Take $N$ stones out of the bag, and put one stone to each plate. This operation can be performed only when the bag has $N$ or more stones. Let $b_i$ be the number of stones on plate $i$ after you finished the operations. Print the number, modulo $998244353$, of sequences of integers $(b_1, b_2, \dots, b_N)$ of length $N$ that can result from the operations. ## Constraints * $1 \leq N \leq 2 \times 10^5$ * $0 \leq a_i \leq 10^9$ ## Input The input is given from Standard Input in the following format: $N$ $a_1$ $a_2$ $\dots$ $a_N$ [samples]

Samples

Input #1

3
3 1 3

Output #1

7

For example, $b$ becomes $(2, 1, 2)$ by the following procedure.

*   Perform the first operation. $b$ becomes $(2, 0, 2)$.
*   Perform the first operation. $b$ becomes $(1, 0, 1)$.
*   Perform the second operation. $b$ becomes $(2, 1, 2)$.

The following seven sequences can be the resulting $b$.

*   $(0, 0, 0)$
*   $(1, 0, 1)$
*   $(1, 1, 1)$
*   $(2, 0, 2)$
*   $(2, 1, 2)$
*   $(2, 2, 2)$
*   $(3, 1, 3)$

Input #2

1
0

Output #2

1

There are one sequence, $(0)$, that can be the resulting $b$.

Input #3

5
1 3 5 7 9

Output #3

Input #4

10
766294629 440423913 59187619 725560240 585990756 965580535 623321125 550925213 122410708 549392044

Output #4

666174028

API Response (JSON)

{
  "problem": {
    "name": "Redistribution of Piles",
    "description": {
      "content": "There are $N$ plates numbered $1$ through $N$. Dish $i$ has $a_i$ stones on it. There is also an empty bag.   You can perform the following two kinds of operations any number of times (possibly zero) ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc313_g"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ plates numbered $1$ through $N$. Dish $i$ has $a_i$ stones on it. There is also an empty bag.  \nYou can perform the following two kinds of operations any number of times (possibly zero) ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments