Ex - Colorfulness

AtCoder

IDabc260_h

Time8000ms

Memory256MB

Difficulty—

There are $N$ balls numbered $1$ through $N$. Ball $i$ is painted in Color $a_i$. For a permutation $P = (P_1, P_2, \dots, P_N)$ of $(1, 2, \dots, N)$, let us define $C(P)$ as follows: * The number of pairs of adjacent balls with different colors when Balls $P_1, P_2, \dots, P_N$ are lined up in a row in this order. Let $S_N$ be the set of all permutations of $(1, 2, \dots, N)$. Also, let us define $F(k)$ by: $\displaystyle F(k) = \left(\sum_{P \in S_N}C(P)^k \right) \bmod 998244353$. Enumerate $F(1), F(2), \dots, F(M)$. ## Constraints * $2 \leq N \leq 2.5 \times 10^5$ * $1 \leq M \leq 2.5 \times 10^5$ * $1 \leq a_i \leq N$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $a_1$ $a_2$ $\dots$ $a_N$ [samples]

Samples

Input #1

3 4
1 1 2

Output #1

8 12 20 36

Here is the list of all possible pairs of $(P, C(P))$.

*   If $P=(1,2,3)$, then $C(P) = 1$.
*   If $P=(1,3,2)$, then $C(P) = 2$.
*   If $P=(2,1,3)$, then $C(P) = 1$.
*   If $P=(2,3,1)$, then $C(P) = 2$.
*   If $P=(3,1,2)$, then $C(P) = 1$.
*   If $P=(3,2,1)$, then $C(P) = 1$.

We may obtain the answers by assigning these values into $F(k)$. For instance, $F(1) = 1^1 + 2^1 + 1^1 + 2^1 + 1^1 + 1^1 = 8$.

Input #2

2 1
1 1

Output #2

Input #3

10 5
3 1 4 1 5 9 2 6 5 3

Output #3

30481920 257886720 199419134 838462446 196874334

API Response (JSON)

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  "problem": {
    "name": "Ex - Colorfulness",
    "description": {
      "content": "There are $N$ balls numbered $1$ through $N$. Ball $i$ is painted in Color $a_i$. For a permutation $P = (P_1, P_2, \\dots, P_N)$ of $(1, 2, \\dots, N)$, let us define $C(P)$ as follows: *   The number",
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ balls numbered $1$ through $N$. Ball $i$ is painted in Color $a_i$.\nFor a permutation $P = (P_1, P_2, \\dots, P_N)$ of $(1, 2, \\dots, N)$, let us define $C(P)$ as follows:\n\n*   The number...",
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