RoAlgo PreOJI 2024 XI-XII | Dominant

Task

A sequence $s_1, s_2, \ldots, s_k$ of natural numbers is called $xy$-dominant for given values $x$ and $y$ if:

1. There are at least $x$ distinct numbers in the sequence.
2. We can choose $x$ distinct numbers from the sequence such that the sum of their frequencies is greater than or equal to $y$.

For example, for $x=2$ and $y=4$:

• The sequence $[1,2,1,2,3]$ is $xy$-dominant (we can choose the numbers $1$ and $2$, which appear a total of $2+2=4$ times in the sequence).
• The sequence $[1,1,1,1]$ is not $xy$-dominant (there are not two distinct values in the sequence).
• The sequence $[1,2,1,3]$ is not $xy$-dominant (there are not two distinct values in the sequence that appear a total of at least $4$ times).

Given $n, x, y$, and a sequence of natural numbers $a_1, a_2, \ldots, a_n$,

Find the number of $xy$-dominant subarrays of the sequence $a$.

Input data

The first line of the file dominant.in contains the numbers $n$, $x$, and $y$.

The second line contains the sequence $a_1, a_2, \ldots, a_n$.

Output data

The first line of the file dominant.out will display the number of $xy$-dominant subarrays of the sequence $a$.

Constraints and clarifications

• $1 \leq n \leq 3 \cdot 10^5$
• $1 \leq x \leq$ number of distinct values in the sequence $a$
• $1 \leq y, a_i \leq n$
• To obtain points for a specific subtask, at least one source sent must pass all tests in that subtask.

Example 1

dominant.in

4 3 1
1 2 3 4

dominant.out

3

Explanation

The $xy$-dominant subarrays are $[1,2,3]$, $[2,3,4]$, and $[1,2,3,4]$.

Example 2

dominant.in

10 2 4
1 1 1 2 1 2 2 2 1 2

dominant.out

28

Explanation

There are a total of KaTeX parse error: Can't use function '$' in math mode at position 3: 28$̲ xy$*-dominant* subarrays, including$[1,1,1,2]$,$[2,1,2,2]$, and$[1,1,1,2,1,2,2,2,1,2]$.

Example 3

dominant.in

14 3 8
5 3 5 1 4 5 1 4 2 2 4 5 4 4

dominant.out

15

Explanation

There are a total of KaTeX parse error: Can't use function '$' in math mode at position 3: 15$̲ xy$*-dominant* subarrays, such as$[5,3,5,1,4,5,1,4,2,2,4]$.