A13073 | New Year and Ascent Sequence

A sequence $a = [a_1, a_2, \ldots, a_l]$ of length $l$ has an ascent if there exists a pair of indices $(i, j)$ such that $1 \le i < j \le l$ and $a_i < a_j$ . For example, the sequence $[0, 2, 0, 2, 0]$ has an ascent because of the pair $(1, 4)$ , but the sequence $[4, 3, 3, 3, 1]$ doesn't have an ascent.

Let's call a concatenation of sequences $p$ and $q$ the sequence that is obtained by writing down sequences $p$ and $q$ one right after another without changing the order. For example, the concatenation of the $[0, 2, 0, 2, 0]$ and $[4, 3, 3, 3, 1]$ is the sequence $[0, 2, 0, 2, 0, 4, 3, 3, 3, 1]$ . The concatenation of sequences $p$ and $q$ is denoted as $p+q$ .

Gyeonggeun thinks that sequences with ascents bring luck. Therefore, he wants to make many such sequences for the new year. Gyeonggeun has $n$ sequences $s_1, s_2, \ldots, s_n$ which may have different lengths.

Gyeonggeun will consider all $n^2$ pairs of sequences $s_x$ and $s_y$ ( $1 \le x, y \le n$ ), and will check if its concatenation $s_x + s_y$ has an ascent. Note that he may select the same sequence twice, and the order of selection matters.

Please count the number of pairs ( $x, y$ ) of sequences $s_1, s_2, \ldots, s_n$ whose concatenation $s_x + s_y$ contains an ascent.

The first line contains the number $n$ ( $1 \le n \le 100\,000$ ) denoting the number of sequences.

The next $n$ lines contain the number $l_i$ ( $1 \le l_i$ ) denoting the length of $s_i$ , followed by $l_i$ integers $s_{i, 1}, s_{i, 2}, \ldots, s_{i, l_i}$ ( $0 \le s_{i, j} \le 10^6$ ) denoting the sequence $s_i$ .

It is guaranteed that the sum of all $l_i$ does not exceed $100\,000$ .

Print a single integer, the number of pairs of sequences whose concatenation has an ascent.