Longest Substring

For a string $S$ of length $n ≥ 1$ and a positive integer $k$ ($1 ≤ k ≤ n$), a non-empty substring of $S$ is called a $k$-substring if the substring appears exactly $k$ times. Such $k$ occurrences are not necessarily disjoint, i.e., are possibly overlapping. For example, if $S =$ "ababa", the $k$-substrings of $S$ for every $k = 1, \dots , 5$ are as follows:

• There are four $1$-substrings in $S$, "abab", "ababa", "bab", and "baba" because these substrings appear exactly once in $S$. Note that "aba" is not a $1$-substring because it appears twice.
• There are four $2$-substrings in $S$, "ab", "aba", "b", and "ba". The substring "ab" appears exactly twice without overlapping. Two occurrences of the substring "aba" are overlapped at a common character "a", but it does not appear three times or more.
• There is only one $3$-substring in $S$, "a".
• Neither $4$-substrings nor $5$-substrings exist in $S$.

For a $k$-substring $T$ of $S$, let $d(T)$ be the maximum number of the disjoint occurrences of $T$ in $S$. For example, a $2$-substring $T =$ "ab" can be selected twice without overlapping, that is, the maximum number of the disjoint occurrences is two, so $d(T) = 2$. For a $2$-substring $T =$ "aba", it cannot be selected twice without overlapping, so $d(T) = 1$. For a $3$-substring $T =$ "a", it can be selected three times without overlapping, which is the maximum, so $d(T) = 3$.

Let $f(k)$ be the length of the longest one among all $k$-substring $T$ with the largest $d(T)$ for $1 ≤ k ≤ n$. For example, $f(k)$ for $S =$ "ababa" and $k = 1, \dots , 5$ is as follows:

• For $k = 1$, all $1$-substrings $T$ can be selected only once without overlapping, so $d(T) = 1$. Thus, the longest one among all $1$-substrings with $d(T) = 1$ is "ababa", so $f(1) = 5$.
• For $k = 2$, $d(T) = 1$ for $T =$ "aba", but $d(T) = 2$ for the other $2$-substrings $T =$ "ab", "b", "ba". Among $2$-substrings with $d(T) = 2$, "ab" and "ba" are the longest ones, so $f(2) = 2$.
• For $k = 3$, $f(3) = 1$ because there is only one $3$-substring "a".
• For $k = 4, 5$, there are no $k$-substrings, so $f(4) = 0$ and $f(5) = 0$.

Given a string $S$ of length $n$, write a program to output $n$ values of $f(k)$ from $k = 1$ to $k = n$. For the above example, the output should be $5$ $2$ $1$ $0$ $0$.

Your program is to read from standard input. The input starts with a line containing the string $S$ consisting of $n$ ($1 ≤ n ≤ 50\\,000$) lowercase alphabets.

Your program is to write to standard output. Print exactly one line. The line should contain exactly $n$ nonnegative integers, separated by a space, that represent $f(k)$ from $k = 1$ to $k = n$ in order, that is, $f(1)$ $f(2)$ $\dots$ $f(n)$. Note that $f(k)$ should be zero if there is no $k$-substring for some $k$.