Ex - Colorfulness
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Ex - Colorfulness
Score : $600$ points
Problem Statement
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:
Output
Print the answers in the following format:
``` $F(1)$ $F(2)$ $\dots$ $F(M)$ ```3 4
1 1 2
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, $F(1) = 1^1 + 2^1 + 1^1 + 2^1 + 1^1 + 1^1 = 8$.
2 1
1 1
0
10 5
3 1 4 1 5 9 2 6 5 3
30481920 257886720 199419134 838462446 196874334