from collections import defaultdict
from collections.abc import Generator, Iterator
from itertools import chain, combinations, combinations_with_replacement, permutations, product
from typing import Any, Literal, NoReturn
from typing_extensions import Never

from sympy import Basic, Symbol

cartes = product

def is_palindromic(s, i=..., j=...) -> bool: ...
def flatten(iterable, levels=..., cls=...) -> list[Any]: ...
def unflatten(iter, n=...) -> list[Any]: ...
def reshape(seq, how) -> Any: ...
def group(seq, multiple=...) -> list[list[Any]] | list[tuple[Any, int]]: ...
def iproduct(*iterables) -> Generator[tuple[()] | tuple[Any] | tuple[Any, Any] | Any, Any, None]: ...
def multiset(seq) -> dict[Any, int]: ...
def ibin(
    n, bits=..., str=...
) -> (
    list[int]
    | Iterator[Never]
    | permutations[Any]
    | Iterator[tuple[()]]
    | product[tuple[Any, ...]]
    | str
    | Generator[str, None, None]
): ...
def variations(
    seq, n, repetition=...
) -> Iterator[Never] | permutations[Any] | Iterator[tuple[()]] | product[tuple[Any, ...]]: ...
def subsets(seq, k=..., repetition=...) -> chain[tuple[Any, ...]] | combinations[Any] | combinations_with_replacement[Any]: ...
def filter_symbols(iterator, exclude) -> Generator[Any, Any, None]: ...
def numbered_symbols(
    prefix=..., cls=..., start=..., exclude=..., *args, **assumptions
) -> Generator[Symbol | Any, Any, NoReturn]: ...
def capture(func) -> str: ...
def sift(seq, keyfunc, binary=...) -> defaultdict[Any, list[Any]] | tuple[list[Any], list[Any]]: ...
def take(iter, n) -> list[Any]: ...
def dict_merge(*dicts) -> dict[Any, Any]: ...
def common_prefix(*seqs) -> list[Any]: ...
def common_suffix(*seqs) -> list[Any]: ...
def prefixes(seq) -> Generator[Any, Any, None]: ...
def postfixes(seq) -> Generator[Any, Any, None]: ...
def topological_sort(graph, key=...) -> list[Any]: ...
def strongly_connected_components(G) -> list[Any]: ...
def connected_components(G) -> list[Any]: ...
def rotate_left(x, y) -> list[Any]: ...
def rotate_right(x, y) -> list[Any]: ...
def least_rotation(x, key=...) -> int: ...
def multiset_combinations(m, n, g=...) -> Generator[Any | list[Any], Any, None]: ...
def multiset_permutations(m, size=..., g=...) -> Generator[list[Any | int] | Any, Any, None]: ...
def multiset_partitions(
    multiset, m=...
) -> Generator[list[list[int]] | list[list[Any]] | list[list[str] | Any] | list[Any], Any, None]: ...
def partitions(
    n, m=..., k=..., size=...
) -> Generator[tuple[Literal[0], dict[Any, Any]] | tuple[int, dict[int, int]] | dict[int, int], Any, None]: ...
def ordered_partitions(n, m=..., sort=...) -> Generator[Any | list[Any], Any, None]: ...
def binary_partitions(n) -> Generator[Any | list[Any], Any, None]: ...
def has_dups(seq) -> bool: ...
def has_variety(seq) -> bool: ...
def uniq(seq, result=...) -> Generator[Any, Any, None]: ...
def generate_bell(
    n,
) -> Generator[
    tuple[Literal[0]] | tuple[Literal[0], Literal[1]] | tuple[Literal[1], Literal[0]] | Any | tuple[int, ...], Any, None
]: ...
def generate_involutions(n) -> Generator[tuple[int, ...], Any, None]: ...
def multiset_derangements(s) -> Generator[list[Any] | list[None] | Any, Any, None]: ...
def random_derangement(t, choice=..., strict=...) -> str | list[None] | list[Any]: ...
def generate_derangements(s) -> Generator[list[Any] | list[None], Any, None]: ...
def necklaces(n, k, free=...) -> Generator[tuple[Any, ...], Any, None]: ...
def bracelets(n, k) -> Generator[tuple[Any, ...], Any, None]: ...
def generate_oriented_forest(n) -> Generator[list[int], Any, None]: ...
def minlex(seq, directed=..., key=...) -> tuple[Any, ...] | list[Any]: ...
def runs(seq, op=...) -> list[Any]: ...
def sequence_partitions(l, n, /) -> Generator[list[Any] | Any, Any, None]: ...
def sequence_partitions_empty(l, n, /) -> Generator[list[Any] | Any, Any, None]: ...
def kbins(
    l, k, ordered=...
) -> Generator[list[Any] | Any | list[list[int]] | list[list[Any]] | list[list[str] | Any], Any, None]: ...
def permute_signs(t) -> Generator[Any, Any, None]: ...
def signed_permutations(t) -> Generator[Any, None, None]: ...
def rotations(s, dir=...) -> Generator[list[Any] | Any, Any, None]: ...
def roundrobin(*iterables) -> Generator[Any, Any, None]: ...

class NotIterable: ...

def iterable(i, exclude=...) -> bool: ...
def is_sequence(i, include=...) -> bool: ...
def postorder_traversal(node, keys=...) -> Generator[Any | Basic, Any, None]: ...
def interactive_traversal(expr) -> Basic: ...
def default_sort_key(
    *args, **kwargs
) -> (
    tuple[tuple[Literal[5], Literal[0], str], tuple[int, tuple[Any, ...]], Any, Any]
    | tuple[tuple[Literal[10, 0], Literal[0], str | Any], tuple[int, tuple[Any, ...]] | tuple[Literal[1], tuple[str]], Any, Any]
): ...
def ordered(*args, **kwargs) -> Generator[Any, Any, None]: ...