Fruit Sequences

给定一个 01 串，设 $ f(l,r) $ 为从 $ l $ 到 $ r $ 的子串中，最长的全部为 $ 1 $ 的子串的长度。 求 $ \sum_{l=1}^n\sum_{r=l}^n f(l,r) $

Zookeeper is buying a carton of fruit to feed his pet wabbit. The fruits are a sequence of apples and oranges, which is represented by a binary string $ s_1s_2\ldots s_n $ of length $ n $ . $ 1 $ represents an apple and $ 0 $ represents an orange. Since wabbit is allergic to eating oranges, Zookeeper would like to find the longest contiguous sequence of apples. Let $ f(l,r) $ be the longest contiguous sequence of apples in the substring $ s_{l}s_{l+1}\ldots s_{r} $ . Help Zookeeper find $ \sum_{l=1}^{n} \sum_{r=l}^{n} f(l,r) $ , or the sum of $ f $ across all substrings.

The first line contains a single integer $ n $ $ (1 \leq n \leq 5 \cdot 10^5) $ . The next line contains a binary string $ s $ of length $ n $ $ (s_i \in \{0,1\}) $

Print a single integer: $ \sum_{l=1}^{n} \sum_{r=l}^{n} f(l,r) $ .

4
0110

12

7
1101001

30

12
011100011100

156

In the first test, there are ten substrings. The list of them (we let $ [l,r] $ be the substring $ s_l s_{l+1} \ldots s_r $ ): - $ [1,1] $ : 0 - $ [1,2] $ : 01 - $ [1,3] $ : 011 - $ [1,4] $ : 0110 - $ [2,2] $ : 1 - $ [2,3] $ : 11 - $ [2,4] $ : 110 - $ [3,3] $ : 1 - $ [3,4] $ : 10 - $ [4,4] $ : 0 The lengths of the longest contiguous sequence of ones in each of these ten substrings are $ 0,1,2,2,1,2,2,1,1,0 $ respectively. Hence, the answer is $ 0+1+2+2+1+2+2+1+1+0 = 12 $ .