Doping
Description
We call an array $a$ of length $n$ fancy if for each $1 < i \le n$ it holds that $a_i = a_{i-1} + 1$.
Let's call $f(p)$ applied to a permutation$^\dagger$ of length $n$ as the minimum number of subarrays it can be partitioned such that each one of them is fancy. For example $f([1,2,3]) = 1$, while $f([3,1,2]) = 2$ and $f([3,2,1]) = 3$.
Given $n$ and a permutation $p$ of length $n$, we define a permutation $p'$ of length $n$ to be $k$-special if and only if:
- $p'$ is lexicographically smaller$^\ddagger$ than $p$, and
- $f(p') = k$.
Your task is to count for each $1 \le k \le n$ the number of $k$-special permutations modulo $m$.
$^\dagger$ A permutation is an array consisting of $n$ distinct integers from $1$ to $n$ in arbitrary order. For example, $[2,3,1,5,4]$ is a permutation, but $[1,2,2]$ is not a permutation ($2$ appears twice in the array) and $[1,3,4]$ is also not a permutation ($n=3$ but there is $4$ in the array).
$^\ddagger$ A permutation $a$ of length $n$ is lexicographically smaller than a permutation $b$ of length $n$ if and only if the following holds: in the first position where $a$ and $b$ differ, the permutation $a$ has a smaller element than the corresponding element in $b$.
The first line contains two integers $n$ and $m$ ($1 \le n \le 2000$, $10 \le m \le 10^9$) — the length of the permutation and the required modulo.
The second line contains $n$ distinct integers $p_1, p_2, \ldots, p_n$ ($1 \le p_i \le n$) — the permutation $p$.
Print $n$ integers, where the $k$-th integer is the number of $k$-special permutations modulo $m$.
Input
The first line contains two integers $n$ and $m$ ($1 \le n \le 2000$, $10 \le m \le 10^9$) — the length of the permutation and the required modulo.
The second line contains $n$ distinct integers $p_1, p_2, \ldots, p_n$ ($1 \le p_i \le n$) — the permutation $p$.
Output
Print $n$ integers, where the $k$-th integer is the number of $k$-special permutations modulo $m$.
4 666012
1 3 4 2
3 10
3 2 1
7 1000000000
7 2 1 3 5 4 6
10 11
10 9 8 7 6 5 4 3 2 1
1 0 1 1
1 2 2
1 6 40 201 705 1635 1854
1 9 9 0 1 5 5 0 1 0
Note
In the first example, the permutations that are lexicographically smaller than $[1,3,4,2]$ are:
- $[1,2,3,4]$, $f([1,2,3,4])=1$;
- $[1,2,4,3]$, $f([1,2,4,3])=3$;
- $[1,3,2,4]$, $f([1,3,2,4])=4$.
Thus our answer is $[1,0,1,1]$.
In the second example, the permutations that are lexicographically smaller than $[3,2,1]$ are:
- $[1,2,3]$, $f([1,2,3])=1$;
- $[1,3,2]$, $f([1,3,2])=3$;
- $[2,1,3]$, $f([2,1,3])=3$;
- $[2,3,1]$, $f([2,3,1])=2$;
- $[3,1,2]$, $f([3,1,2])=2$.
Thus our answer is $[1,2,2]$.