BZOJ #2863. Angry Führer

/ 某岛

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set up f_{i,j} It means that there are exactly j schemes whose in-degree is 0, there are:

(1)   \begin{align*} f_{i,j} &= \sum_{j=1}^{i} \binom{i}{j} \sum_{k=1}^{i-j} f_{i-j,k} (2^j-1)^k 2^{j(i-j-k)} \end{align*}

The complexity is O(n3), we can use the prefix and + redundancy to optimize the enumeration in the inner layer, there are:

(2)   \begin{align*} f_i &= \sum_{j=1}^{i} (-1)^{j-1} \binom{i}{j} \sum_{k=1}^{i-j} f_{i-j} 2^{j(i-j)} \end{align*}

Here, Newton’s binomial expansion is used, (1+x)^n, and x = -1 is substituted.

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