I. Mind Bloom
dp
This is the way it always was. This is the way it always will be. All will be forgotten again soon...Jellyfish is playing a one-player card game called ""Slay the Spire"". There are \(n\) cards in total numbered from \(1\) to \(n\). The \(i\)-th card has power \(c_i\).There is a binary string \(s\) of length \(n\). If \(s_i = \texttt{0}\), the \(i\)-th card is initially in the draw pile. If \(s_i = \texttt{1}\), the \(i\)-th card is initially in Jellyfish's hand.Jellyfish will repeat the following process until either her hand or the draw pile is empty. Let \(x\) be the power of the card with the largest power in her hand. Place a single card with power \(x\) back into the draw pile. Randomly draw \(x\) cards from the draw pile. All subsets of \(x\) cards from the draw pile have an equal chance of being drawn. If there are fewer than \(x\) cards in the draw pile, Jellyfish will draw all cards. At the end of this process, find the probability that Jellyfish can empty the draw pile, modulo \(1\,000\,000\,007\).Formally, let \(M=1\,000\,000\,007\). It can be shown that the answer can be expressed as an irreducible fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \not \equiv 0 \pmod{M}\). Output the integer equal to \(p \cdot q^{-1} \bmod M\). In other words, output such an integer \(x\) that \(0 \le x < M\) and \(x \cdot q \equiv p \pmod{M}\).
Each test contains multiple test cases. The first line contains the number of test cases \(t\) (\(1\leq t\leq 100\)). The description of the test cases follows.The first line of each test case contains a single integer \(n\) (\(1 \leq n \leq 120\)) — the number of cards.The second line of each test case contains \(n\) integers \(c_1,c_2,\ldots,c_n\) (\(0 \leq c_i \leq n\)) — the powers of the cards. It is guaranteed that \(c_1 \leq c_2 \leq \ldots \leq c_n\).The third line of each test case contains a binary string \(s\) of length \(n\). If \(s_i = \texttt{0}\), the \(i\)-th card is initially in the draw pile. If \(s_i = \texttt{1}\), the \(i\)-th card is initially in Jellyfish's hand.It is guaranteed that the sum of \(n^2\) over all test cases does not exceed \(120^2\).
For each test case, output the probability that Jellyfish can empty the draw pile modulo \(1\,000\,000\,007\).
In the first test case, Jellyfish will keep playing cards with power \(1\) until Jellyfish draws a card with power \(0\) or power \(2\). If Jellyfish draws a card with power \(0\), she will eventually empty her hand. If Jellyfish draws a card with power \(2\), she will eventually empty the draw pile. Since there is an equal chance of drawing \(0\) or \(2\), the answer is \(\frac{1}{2}\), and \(2 \cdot 500\,000\,004 \equiv 1 \pmod {10^9+7}\)