






# Algorithm for the pigeonhole sorting




def __A    (      __lowerCAmelCase	)->  List[str]:





						"""simple docstring"""



						_UpperCAmelCase									=			min(__lowerCAmelCase	)  # min() finds the minimum value
						_UpperCAmelCase									=			max(__lowerCAmelCase	)  # max() finds the maximum value

						_UpperCAmelCase									=			max_val - min_val + 1  # size is difference of max and min values plus one

						# list of pigeonholes of size equal to the variable size
						_UpperCAmelCase									=			[0] * size

						# Populate the pigeonholes.
						for x in a:
												assert isinstance(__lowerCAmelCase ,		__lowerCAmelCase	), "integers only please"
												holes[x - min_val] += 1

						# Putting the elements back into the array in an order.
						_UpperCAmelCase									=			0
						for count in range(__lowerCAmelCase	):
												while holes[count] > 0:
																		holes[count] -= 1
																		_UpperCAmelCase									=			count + min_val
																		i += 1


def __A    (      )->  Dict:





						"""simple docstring"""



						_UpperCAmelCase									=			[8, 3, 2, 7, 4, 6, 8]
						pigeonhole_sort(__lowerCAmelCase	)
						print('Sorted order is:' ,		' '.join(__lowerCAmelCase	)	)


if __name__ == "__main__":
		main()


"""simple docstring"""




from __future__ import annotations

from functools import lru_cache
from math import ceil

_A			:       Optional[Any]   		=       1_00

_A			:       Optional[int]   		=       set(range(3, NUM_PRIMES, 2))
primes.add(2)
_A			:       int

for prime in range(3, ceil(NUM_PRIMES**0.5), 2):
   if prime not in primes:
      continue
   primes.difference_update(set(range(prime * prime, NUM_PRIMES, prime)))


@lru_cache(maxsize=100  )
def   __magic_name__	(      __snake_case				:       int  ) ->				set[int]:
       if number_to_partition < 0:
              return set()
       elif number_to_partition == 0:
              return {1}

       lowercase    :				set[int]       		=							set()
       lowercase    :				int
       lowercase    :				int

       for prime in primes:
              if prime > number_to_partition:
                     continue
              for sub in partition(number_to_partition - prime  ):
                     ret.add(sub * prime  )

       return ret


def   __magic_name__	(      __snake_case				:       int = 5000  ) ->				int | None:
       for number_to_partition in range(1     , __snake_case  ):
              if len(partition(__snake_case  )  ) > number_unique_partitions:
                     return number_to_partition
       return None


if __name__ == "__main__":
   print(F"{solution() = }")


