Mex on Blackboard

AtCoder

IDarc156_b

Time2000ms

Memory256MB

Difficulty—

For a finite multiset $S$ of non-negative integers, let us define $\mathrm{mex}(S)$ as the smallest non-negative integer not in $S$. For instance, $\mathrm{mex}(\lbrace 0,0, 1,3\rbrace ) = 2, \mathrm{mex}(\lbrace 1 \rbrace) = 0, \mathrm{mex}(\lbrace \rbrace) = 0$. There are $N$ non-negative integers on a blackboard. The $i$\-th integer is $A_i$. You will perform the following operation exactly $K$ times. * Choose zero or more integers on the blackboard. Let $S$ be the multiset of chosen integers, and write $\mathrm{mex}(S)$ on the blackboard once. How many multisets can be the multiset of integers on the final blackboard? Find this count modulo $998244353$. ## Constraints * $1 \leq N,K \leq 2\times 10^5$ * $0\leq A_i\leq 2\times 10^5$ * All numbers in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $K$ $A_1$ $A_2$ $\ldots$ $A_N$ [samples]

Samples

Input #1

3 1
0 1 3

Output #1

3

The following three multisets can be obtained by the operations.

*   $\lbrace 0,0,1,3 \rbrace$
*   $\lbrace 0,1,1,3\rbrace$
*   $\lbrace 0,1,2,3 \rbrace$

For instance, you can get $\lbrace 0,1,1,3\rbrace$ by choosing the $0$ on the blackboard to let $S=\lbrace 0\rbrace$ in the operation.

Input #2

2 1
0 0

Output #2

2

The following two multisets can be obtained by the operations.

*   $\lbrace 0,0,0 \rbrace$
*   $\lbrace 0,0,1\rbrace$

Note that you may choose zero integers in the operation.

Input #3

5 10
3 1 4 1 5

Output #3

API Response (JSON)

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    "name": "Mex on Blackboard",
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      "content": "For a finite multiset $S$ of non-negative integers, let us define $\\mathrm{mex}(S)$ as the smallest non-negative integer not in $S$. For instance, $\\mathrm{mex}(\\lbrace 0,0, 1,3\\rbrace ) = 2, \\mathrm{",
      "description_type": "Markdown"
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    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
    "is_remote": true,
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    "sign": "arc156_b"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "For a finite multiset $S$ of non-negative integers, let us define $\\mathrm{mex}(S)$ as the smallest non-negative integer not in $S$. For instance, $\\mathrm{mex}(\\lbrace 0,0, 1,3\\rbrace ) = 2, \\mathrm{...",
      "is_translate": false,
      "language": "English"
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}

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