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E - At Least One

Score : $500$ points

Problem Statement

You are given an integer $M$ and $N$ pairs of integers $(A_1, B_1), (A_2, B_2), \dots, (A_N, B_N)$.
For all $i$, it holds that $1 \leq A_i \lt B_i \leq M$.

A sequence $S$ is said to be a good sequence if the following conditions are satisfied:

• $S$ is a contiguous subsequence of the sequence $(1,2,3,..., M)$.
• For all $i$, $S$ contains at least one of $A_i$ and $B_i$.

Let $f(k)$ be the number of possible good sequences of length $k$.
Enumerate $f(1), f(2), \dots, f(M)$.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $2 \leq M \leq 2 \times 10^5$
• $1 \leq A_i \lt B_i \leq M$
• All values in input are integers.

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Input

Input is given from Standard Input in the following format:

$N$ $M$

$A_1$ $B_1$

$A_2$ $B_2$

$\vdots$

$A_N$ $B_N$

Output

Print the answers in the following format:

``` $f(1)$ $f(2)$ $\dots$ $f(M)$ ```

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3 5
1 3
1 4
2 5

0 1 3 2 1

Here is the list of all possible good sequences.

• $(1,2)$
• $(1,2,3)$
• $(2,3,4)$
• $(3,4,5)$
• $(1,2,3,4)$
• $(2,3,4,5)$
• $(1,2,3,4,5)$

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1 2
1 2

2 1

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5 9
1 5
1 7
5 6
5 8
2 6

0 0 1 2 4 4 3 2 1