I believe everyone will FWT and FMT.

If you don't , I recommend you to have a look at VFK Of 2015 National training team .

Let the complete set be $$U=\{1,2,\ldots,n\}$$, Suppose we care about $$f_S$$ The set in $$S$$ yes $$U$$ Subset .

Here you are. $$c_i,d_i$$, Make
$b_i=(1+c_ix^{d_i})$

seek
$g=\prod_{i}b_i$
The product of two set power series is set merging convolution （or）/ Set symmetric difference convolution （xor） One of the .

Might as well set $$d_S$$ Different from each other （ Otherwise you can use DP/ Combination number or something ）.

Make
\begin{align} a_S&=\sum_{d_i=S}c_i\\ f_S&=1+a_Sx^S \end{align}

Violence

Do it again for each set power series FMT/FWT, And then we'll just ride them together , Switch back .

Time complexity ：$$O(n4^n)$$

It's too slow , Because it doesn't use the special condition of this problem .

Set or convolution

For a set power series $$f$$, Definition $$f$$ The Mobius transformation of is a set power series $$\hat f$$, among
\begin{align} \hat f_S=\sum_{T\subseteq S}f_T \end{align}
In turn, , Definition $$\hat f$$ The Mobius inversion of is $$f$$, From the inclusion exclusion principle, we can get
$f_S=\sum_{T\subseteq S}{(-1)}^{|S|-|T|}\hat f_T$
I believe everyone is familiar with the above contents .

Back to the formula we asked for ：
\begin{align} \hat g_T&=\prod_S \hat{f_{S}}_T\\ &=\prod_S \sum_{K\subseteq T}f_{S,K}\\ &=\prod_{S\subseteq T}{(1+a_S)} \end{align}
It's similar to the ordinary Mobius transformation ？

Put all the $$a_S$$ add $$1$$, Change the addition of Mobius transformation to multiplication , You can get $$\hat g_T$$ 了 .

Time complexity ：$$O(n2^n)$$

Set symmetric difference convolution

We still need to use those formulas .
\begin{align} \hat g_T&=\prod_S\hat {f_S}_T\\ &=\prod_S\sum_{K}f_{S,K}{(-1)}^{|K\cap T|}\\ &=\prod_S(1+a_S{(-1)}^{|S\cap T|}) \end{align}
This is also very similar to Walsh transformation , however $${(-1)}^{|S\cap T|}$$ Only in $$a_S$$ above , So we can't put $$a_S$$ Add $$1$$ After doing the variant Walsh transformation .

But we can maintain another $$\hat h_T=\prod_S(1-a_S{(-1)}^{|S\cap T|})$$, Take the Walsh transformation of $$-\hat g_T$$ Replacing the $$\hat h_T$$, You can do it .

Time complexity ：$$O(n2^n)$$

First pit

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