Div. 1 500 from here is the same. You can see the editorial in the comment.

The naive dp is something like this: dp[i][j]= answer for the first i elements given that the last chosen element has value j. The problem here is that j can be too large. However note that we can partition the integers into certain partitions, and only the partition to which j belongs should matter: to be precise, let t₁ < t₂ < ... < t_N be the set {L₁, ... , L_n, R₁, ... , R_n} in sorted order, then it only matters to which of the intervals (t_i, t_i - 1) j belongs to. So the dp will now look like this: dp[i][j]= answer for the first i elements given that the last element belongs to the j'th interval. The transitions will be like this: iterate over the k < i for which the number is in another interval, multiply the number of increasing subsequences of length k - i contained in the j'th interval (O(1) formula is easy to derive) by and add it to result. There are a few minor details to work out here (which I unfortuntely I couldn't do in contest time :( ).