Codeforces Round 754 (Div. 2) |
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Finished |
An integer array $$$a$$$ of length $$$n$$$ is said to be a PalindORme if ($$$a_{1}$$$ $$$|$$$ $$$a_{2} $$$ $$$|$$$ $$$ \ldots $$$ $$$|$$$ $$$ a_{i}) = (a_{{n - i + 1}} $$$ $$$|$$$ $$$ \ldots $$$ $$$|$$$ $$$ a_{{n - 1}} $$$ $$$|$$$ $$$ a_{n}) $$$ for all $$$ 1 \leq i \leq n$$$, where $$$|$$$ denotes the bitwise OR operation.
An integer array $$$a$$$ of length $$$n$$$ is considered to be good if its elements can be rearranged to form a PalindORme. Formally, array $$$a$$$ is good if there exists a permutation $$$p_1, p_2, \ldots p_n$$$ (an array where each integer from $$$1$$$ to $$$n$$$ appears exactly once) for which $$$a_{p_1}, a_{p_2}, \ldots a_{p_n}$$$ is a PalindORme.
Find the number of good arrays of length $$$n$$$, consisting only of integers in the range $$$[0, 2^{k} - 1]$$$, and print it modulo some prime $$$m$$$.
Two arrays $$$a_1, a_2, \ldots, a_n$$$ and $$$b_1, b_2, \ldots, b_n$$$ are considered to be different if there exists any $$$i$$$ $$$(1 \leq i \leq n)$$$ such that $$$a_i \ne b_i$$$.
The first and only line of the input contains three integers $$$n$$$, $$$k$$$ and $$$m$$$ ($$$1 \leq n,k \leq 80$$$, $$$10^8 \lt m \lt 10^9$$$). It is guaranteed that $$$m$$$ is prime.
Print a single integer — the number of good arrays modulo $$$m$$$.
1 1 998244353
2
3 2 999999733
40
7 3 796735397
1871528
2 46 606559127
177013
In the first sample, both the possible arrays $$$[0]$$$ and $$$[1]$$$ are good.
In the second sample, some examples of good arrays are:
Note that $$$[1, 1, 0]$$$, $$$[1, 0, 1]$$$ and $$$[0, 1, 1]$$$ are all good arrays and are considered to be different according to the definition in the statement.
In the third sample, an example of a good array is $$$[1, 0, 1, 4, 2, 5, 4]$$$. It can be rearranged to an array $$$b = [1, 5, 0, 2, 4, 4, 1]$$$ which is a PalindORme because:
Here $$$\mathrm{OR}(l, r)$$$ denotes $$$b_{l}$$$ $$$|$$$ $$$b_{l+1} $$$ $$$|$$$ $$$ \ldots $$$ $$$|$$$ $$$ b_{r}$$$
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