Prefix Mex Sequence

AtCoder

IDarc170_c

Time2000ms

Memory256MB

Difficulty—

For a sequence $X$ composed of a finite number of non-negative integers, we define $\mathrm{mex}(X)$ as the smallest non-negative integer not in $X$. For example, $\mathrm{mex}((0,0, 1,3)) = 2, \mathrm{mex}((1)) = 0, \mathrm{mex}(()) = 0$. You are given a sequence $S=(S_1,\ldots,S_N)$ of length $N$ where each element is $0$ or $1$. Find the number, modulo $998244353$, of sequences $A=(A_1,A_2,\ldots,A_N)$ of length $N$ consisting of integers between $0$ and $M$, inclusive, that satisfy the following condition: * For each $i (1\leq i\leq N)$, $A_i = \mathrm{mex}((A_1,A_2,\ldots,A_{i-1}))$ if $S_i=1$, and $A_i \neq \mathrm{mex}((A_1,A_2,\ldots,A_{i-1}))$ if $S_i=0$. ## Constraints * $1 \leq N \leq 5000$ * $0 \leq M \leq 10^9$ * $S_i$ is $0$ or $1$. * All input numbers are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $S_1$ $\ldots$ $S_N$ [samples]

Samples

Input #1

4 2
1 0 0 1

Output #1

4

The following four sequences satisfy the conditions:

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

Input #2

10 1000000000
0 0 1 0 0 0 1 0 1 0

Output #2

587954969

Be sure to find the count modulo $998244353$.

API Response (JSON)

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      "content": "For a sequence $X$ composed of a finite number of non-negative integers, we define $\\mathrm{mex}(X)$ as the smallest non-negative integer not in $X$. For example, $\\mathrm{mex}((0,0, 1,3)) = 2, \\mathr",
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      "statement_type": "Markdown",
      "content": "For a sequence $X$ composed of a finite number of non-negative integers, we define $\\mathrm{mex}(X)$ as the smallest non-negative integer not in $X$. For example, $\\mathrm{mex}((0,0, 1,3)) = 2, \\mathr...",
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