Count Strictly Increasing Sequences

AtCoder

IDabc292_g

Time2000ms

Memory256MB

Difficulty—

You are given a sequence $S_1,\ldots,S_N$ of length-$M$ strings consisting of digits (`0123456789`) and `?`. There are $10^q$ ways to replace the occurrences of `?` with digits independently, where $q$ is the total number of `?` in $S_1,\ldots,S_N$. Among them, how many satisfy $S_1\lt S_2 \lt \ldots \lt S_N$ when the resulting strings are seen as integers? Find this count modulo $998244353$. The resulting strings may have leading zeros. For instance, `0000000292` is seen as $292$. ## Constraints * $2 \leq N \leq 40$ * $1 \leq M \leq 40$ * $N$ and $M$ are integers. * $S_i$ is a string of length $M$ consisting of digits and `?`. ## Input The input is given from Standard Input in the following format: $N$ $M$ $S_1$ $\vdots$ $S_N$ [samples]

Samples

Input #1

3 2
?0
??
05

Output #1

4

Here are the four desired replacements.

*   Replace the first character of $S_1$ with `0`, and the first and second characters of $S_2$ with `0` and `1`, respectively.
*   Replace the first character of $S_1$ with `0`, and the first and second characters of $S_2$ with `0` and `2`, respectively.
*   Replace the first character of $S_1$ with `0`, and the first and second characters of $S_2$ with `0` and `3`, respectively.
*   Replace the first character of $S_1$ with `0`, and the first and second characters of $S_2$ with `0` and `4`, respectively.

Input #2

2 1
0
0

Output #2

Input #3

10 10
1?22??37?4
1??8?0??49
3?02??8044
51?4?8?7??
5?9?20???2
68?7?6?800
?3??2???23
?442312158
??2??921?8
????5?96??

Output #3

137811792

Find the count modulo $998244353$.

API Response (JSON)

{
  "problem": {
    "name": "Count Strictly Increasing Sequences",
    "description": {
      "content": "You are given a sequence $S_1,\\ldots,S_N$ of length-$M$ strings consisting of digits (`0123456789`) and `?`. There are $10^q$ ways to replace the occurrences of `?` with digits independently, where $q",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc292_g"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a sequence $S_1,\\ldots,S_N$ of length-$M$ strings consisting of digits (`0123456789`) and `?`.\nThere are $10^q$ ways to replace the occurrences of `?` with digits independently, where $q...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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