A13694 | Bitwise Magic

You are given a positive integer $k$ and an array $a_1, a_2, \ldots, a_n$ of non-negative distinct integers not smaller than $k$ and not greater than $2^c-1$ .

In each of the next $k$ seconds, one element is chosen randomly equiprobably out of all $n$ elements and decreased by $1$ .

For each integer $x$ , $0 \leq x \leq 2^c - 1$ , you need to find the probability that in the end the bitwise XOR of all elements of the array is equal to $x$ .

Each of these values can be represented as an irreducible fraction $\frac{p}{q}$ , and you need to find the value of $p \cdot q^{-1}$ modulo $998\,244\,353$ .

The first line of input contains three integers $n, k, c$ ( $1 \leq n \leq (2^c - k)$ , $1 \leq k \leq 16$ , $1 \leq c \leq 16$ ).

The second line contains $n$ distinct integers $a_1, a_2, \ldots, a_n$ ( $k \leq a_i \leq 2^c-1$ ).

Print $2^c$ integers: the probability that the bitwise XOR is equal to $x$ in the end for $x$ in $\{0, 1, \ldots, 2^c-1\}$ modulo $998\,244\,353$ .