Simple Solitaire

AtCoder

IDarc146_f

Time10000ms

Memory256MB

Difficulty—

The following process is carried out on a permutation $P$ of $(1,2,\dots,N)$. > We have $N$ cards, numbered $1$ to $N$. Card $i$ has the integer $P_i$ written on it. > There are an integer $X=1$ and a boy called PCT, who initially has nothing. PCT does the following procedure for each $i=1,2,\dots,N$ in this order. > > * Get Card $i$. Then, repeat the following action as long as he has a card with $X$ written on it: > > * eat the card with $X$ written on it, and then add $1$ to $X$. > > * If PCT currently has $M$ or more cards, throw away all cards he has and terminate the process, without performing any more procedures. Here, let us define the score of the permutation $P$ as follows: * if the process is terminated by throwing away cards, the score of $P$ is $0$; * if the process is carried out through the end without throwing away cards, the score of $P$ is $\prod_{i=1}^{N-1}$ $($ the number of cards PCT has at the end of the $i$\-th procedure $)$. There are $N!$ permutations $P$ of $(1,2,\dots,N)$. Find the sum of the scores of all those permutations, modulo $998244353$. ## Constraints * $2 \le N \le 2 \times 10^5$ * $2 \le M \le N$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

3 2

Output #1

1

For $P=(3,1,2)$, the process goes as follows.

*   The first procedure:
    *   PCT gets Card $1$.
    *   PCT currently has $1$ card, so he goes on.
*   The second procedure:
    *   PCT gets Card $2$.
    *   PCT eats Card $2$ and make $X = 2$.
    *   PCT currently has $1$ card, so he goes on.
*   The third procedure:
    *   PCT gets Card $3$.
    *   PCT eats Cards $1,3$ and make $X = 4$.
    *   PCT currently has $0$ cards, so he goes on.

The process is carried out through the end, so the score of $(3,1,2)$ is $1 \times 1 = 1$.
Other than $(3,1,2)$, there is no permutation with a score of $1$ or greater, so the answer is $1$.

Input #2

3 3

Output #2

Input #3

146146 146

Output #3

103537573

API Response (JSON)

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    "name": "Simple Solitaire",
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      "content": "The following process is carried out on a permutation $P$ of $(1,2,\\dots,N)$. > We have $N$ cards, numbered $1$ to $N$. Card $i$ has the integer $P_i$ written on it. > There are an integer $X=1$ and ",
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      "memory_limit": 262144
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    "difficulty": "None",
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "The following process is carried out on a permutation $P$ of $(1,2,\\dots,N)$.\n\n> We have $N$ cards, numbered $1$ to $N$. Card $i$ has the integer $P_i$ written on it.\n> There are an integer $X=1$ and ...",
      "is_translate": false,
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