itertools — Functions creating iterators for efficient looping

이 모듈은 APL, Haskell 및 SML의 구성물들에서 영감을 얻은 여러 이터레이터 빌딩 블록을 구현합니다. 각각을 파이썬에 적합한 형태로 개선했습니다.

이 모듈은 자체적으로 혹은 조합하여 유용한 빠르고 메모리 효율적인 도구의 핵심 집합을 표준화합니다. 함께 모여, 순수 파이썬에서 간결하고 효율적으로 특수화된 도구를 구성할 수 있도록 하는 “이터레이터 대수(iterator algebra)”를 형성합니다.

예를 들어, SML은 테이블 화 도구를 제공합니다: 시퀀스 f(0), f(1), ...를 생성하는 tabulate(f). map()count()를 결합하여 map(f, count())를 형성해서 파이썬에서도 같은 효과를 얻을 수 있습니다.

These tools and their built-in counterparts also work well with the high-speed functions in the operator module. For example, the multiplication operator can be mapped across two vectors to form an efficient dot-product: sum(starmap(operator.mul, zip(vec1, vec2, strict=True))).

무한 이터레이터:





[start[, step]]

start, start+step, start+2*step, …

count(10) 10 11 12 13 14 ...



p0, p1, … plast, p0, p1, …

cycle('ABCD') A B C D A B C D ...


elem [,n]

elem, elem, elem, … 끝없이 또는 최대 n 번

repeat(10, 3) 10 10 10

가장 짧은 입력 시퀀스에서 종료되는 이터레이터:





p [,func]

p0, p0+p1, p0+p1+p2, …

accumulate([1,2,3,4,5]) 1 3 6 10 15


p, n

(p0, p1, …, p_n-1), …

batched('ABCDEFG', n=3) ABC DEF G


p, q, …

p0, p1, … plast, q0, q1, …

chain('ABC', 'DEF') A B C D E F



p0, p1, … plast, q0, q1, …

chain.from_iterable(['ABC', 'DEF']) A B C D E F


data, selectors

(d[0] if s[0]), (d[1] if s[1]), …

compress('ABCDEF', [1,0,1,0,1,1]) A C E F


predicate, seq

seq[n], seq[n+1], starting when predicate fails

dropwhile(lambda x: x<5, [1,4,6,3,8]) 6 3 8


predicate, seq

elements of seq where predicate(elem) fails

filterfalse(lambda x: x<5, [1,4,6,3,8]) 6 8


iterable[, key]

key(v)의 값으로 그룹화된 서브 이터레이터들


seq, [start,] stop [, step]

seq[start:stop:step]의 요소들

islice('ABCDEFG', 2, None) C D E F G



(p[0], p[1]), (p[1], p[2])



func, seq

func(*seq[0]), func(*seq[1]), …

starmap(pow, [(2,5), (3,2), (10,3)]) 32 9 1000


predicate, seq

seq[0], seq[1], until predicate fails

takewhile(lambda x: x<5, [1,4,6,3,8]) 1 4


it, n

it1, it2, … itn 하나의 이터레이터를 n개의 이터레이터로 나눕니다


p, q, …

(p[0], q[0]), (p[1], q[1]), …

zip_longest('ABCD', 'xy', fillvalue='-') Ax By C- D-

조합형 이터레이터:





p, q, … [repeat=1]

데카르트 곱(cartesian product), 중첩된 for 루프와 동등합니다


p[, r]

r-길이 튜플들, 모든 가능한 순서, 반복되는 요소 없음


p, r

r-길이 튜플들, 정렬된 순서, 반복되는 요소 없음


p, r

r-길이 튜플들, 정렬된 순서, 반복되는 요소 있음


product('ABCD', repeat=2)


permutations('ABCD', 2)


combinations('ABCD', 2)


combinations_with_replacement('ABCD', 2)


Itertool Functions

다음 모듈 함수는 모두 이터레이터를 생성하고 반환합니다. 일부는 길이가 무한한 스트림을 제공해서, 스트림을 자르는 함수나 루프로만 액세스해야 합니다.

itertools.accumulate(iterable[, function, *, initial=None])

Make an iterator that returns accumulated sums or accumulated results from other binary functions.

The function defaults to addition. The function should accept two arguments, an accumulated total and a value from the iterable.

If an initial value is provided, the accumulation will start with that value and the output will have one more element than the input iterable.

대략 다음과 동등합니다:

def accumulate(iterable, function=operator.add, *, initial=None):
    'Return running totals'
    # accumulate([1,2,3,4,5]) → 1 3 6 10 15
    # accumulate([1,2,3,4,5], initial=100) → 100 101 103 106 110 115
    # accumulate([1,2,3,4,5], operator.mul) → 1 2 6 24 120

    iterator = iter(iterable)
    total = initial
    if initial is None:
            total = next(iterator)
        except StopIteration:

    yield total
    for element in iterator:
        total = function(total, element)
        yield total

The function argument can be set to min() for a running minimum, max() for a running maximum, or operator.mul() for a running product. Amortization tables can be built by accumulating interest and applying payments:

>>> data = [3, 4, 6, 2, 1, 9, 0, 7, 5, 8]
>>> list(accumulate(data, max))              # running maximum
[3, 4, 6, 6, 6, 9, 9, 9, 9, 9]
>>> list(accumulate(data, operator.mul))     # running product
[3, 12, 72, 144, 144, 1296, 0, 0, 0, 0]

# Amortize a 5% loan of 1000 with 10 annual payments of 90
>>> update = lambda balance, payment: round(balance * 1.05) - payment
>>> list(accumulate(repeat(90, 10), update, initial=1_000))
[1000, 960, 918, 874, 828, 779, 728, 674, 618, 559, 497]

최종 누적값만 반환하는 유사한 함수에 대해서는 functools.reduce()를 참조하십시오.

Added in version 3.2.

버전 3.3에서 변경: Added the optional function parameter.

버전 3.8에서 변경: 선택적 initial 매개 변수를 추가했습니다.

itertools.batched(iterable, n)

Batch data from the iterable into tuples of length n. The last batch may be shorter than n.

Loops over the input iterable and accumulates data into tuples up to size n. The input is consumed lazily, just enough to fill a batch. The result is yielded as soon as the batch is full or when the input iterable is exhausted:

>>> flattened_data = ['roses', 'red', 'violets', 'blue', 'sugar', 'sweet']
>>> unflattened = list(batched(flattened_data, 2))
>>> unflattened
[('roses', 'red'), ('violets', 'blue'), ('sugar', 'sweet')]

대략 다음과 동등합니다:

def batched(iterable, n):
    # batched('ABCDEFG', 3) → ABC DEF G
    if n < 1:
        raise ValueError('n must be at least one')
    iterator = iter(iterable)
    while batch := tuple(islice(iterator, n)):
        yield batch

Added in version 3.12.


첫 번째 이터러블에서 소진될 때까지 요소를 반환한 다음 이터러블로 넘어가고, 이런 식으로 iterables의 모든 이터러블이 소진될 때까지 진행하는 이터레이터를 만듭니다. 여러 시퀀스를 단일 시퀀스처럼 처리하는 데 사용됩니다. 대략 다음과 동등합니다:

def chain(*iterables):
    # chain('ABC', 'DEF') → A B C D E F
    for iterable in iterables:
        yield from iterable
classmethod chain.from_iterable(iterable)

chain()의 대체 생성자. 게으르게 평가되는 단일 이터러블 인자에서 연쇄 입력을 가져옵니다. 대략 다음과 동등합니다:

def from_iterable(iterables):
    # chain.from_iterable(['ABC', 'DEF']) → A B C D E F
    for iterable in iterables:
        yield from iterable
itertools.combinations(iterable, r)

입력 iterable에서 요소의 길이 r 서브 시퀀스들을 반환합니다.

The output is a subsequence of product() keeping only entries that are subsequences of the iterable. The length of the output is given by math.comb() which computes n! / r! / (n - r)! when 0 r n or zero when r > n.

The combination tuples are emitted in lexicographic order according to the order of the input iterable. If the input iterable is sorted, the output tuples will be produced in sorted order.

Elements are treated as unique based on their position, not on their value. If the input elements are unique, there will be no repeated values within each combination.

대략 다음과 동등합니다:

def combinations(iterable, r):
    # combinations('ABCD', 2) → AB AC AD BC BD CD
    # combinations(range(4), 3) → 012 013 023 123

    pool = tuple(iterable)
    n = len(pool)
    if r > n:
    indices = list(range(r))

    yield tuple(pool[i] for i in indices)
    while True:
        for i in reversed(range(r)):
            if indices[i] != i + n - r:
        indices[i] += 1
        for j in range(i+1, r):
            indices[j] = indices[j-1] + 1
        yield tuple(pool[i] for i in indices)
itertools.combinations_with_replacement(iterable, r)

입력 iterable에서 요소의 길이 r 서브 시퀀스들을 반환하는데, 개별 요소를 두 번 이상 반복할 수 있습니다.

The output is a subsequence of product() that keeps only entries that are subsequences (with possible repeated elements) of the iterable. The number of subsequence returned is (n + r - 1)! / r! / (n - 1)! when n > 0.

The combination tuples are emitted in lexicographic order according to the order of the input iterable. if the input iterable is sorted, the output tuples will be produced in sorted order.

Elements are treated as unique based on their position, not on their value. If the input elements are unique, the generated combinations will also be unique.

대략 다음과 동등합니다:

def combinations_with_replacement(iterable, r):
    # combinations_with_replacement('ABC', 2) → AA AB AC BB BC CC

    pool = tuple(iterable)
    n = len(pool)
    if not n and r:
    indices = [0] * r

    yield tuple(pool[i] for i in indices)
    while True:
        for i in reversed(range(r)):
            if indices[i] != n - 1:
        indices[i:] = [indices[i] + 1] * (r - i)
        yield tuple(pool[i] for i in indices)

Added in version 3.1.

itertools.compress(data, selectors)

Make an iterator that returns elements from data where the corresponding element in selectors is true. Stops when either the data or selectors iterables have been exhausted. Roughly equivalent to:

def compress(data, selectors):
    # compress('ABCDEF', [1,0,1,0,1,1]) → A C E F
    return (datum for datum, selector in zip(data, selectors) if selector)

Added in version 3.1.

itertools.count(start=0, step=1)

Make an iterator that returns evenly spaced values beginning with start. Can be used with map() to generate consecutive data points or with zip() to add sequence numbers. Roughly equivalent to:

def count(start=0, step=1):
    # count(10) → 10 11 12 13 14 ...
    # count(2.5, 0.5) → 2.5 3.0 3.5 ...
    n = start
    while True:
        yield n
        n += step

부동 소수점 숫자로 count 할 때, (start + step * i for i in count())와 같은 곱셈 코드를 대체하여 때로 더 나은 정확도를 얻을 수 있습니다.

버전 3.1에서 변경: step 인자를 추가하고 정수가 아닌 인자를 허용했습니다.


Make an iterator returning elements from the iterable and saving a copy of each. When the iterable is exhausted, return elements from the saved copy. Repeats indefinitely. Roughly equivalent to:

def cycle(iterable):
    # cycle('ABCD') → A B C D A B C D A B C D ...
    saved = []
    for element in iterable:
        yield element
    while saved:
        for element in saved:
            yield element

This itertool may require significant auxiliary storage (depending on the length of the iterable).

itertools.dropwhile(predicate, iterable)

Make an iterator that drops elements from the iterable while the predicate is true and afterwards returns every element. Roughly equivalent to:

def dropwhile(predicate, iterable):
    # dropwhile(lambda x: x<5, [1,4,6,3,8]) → 6 3 8

    iterator = iter(iterable)
    for x in iterator:
        if not predicate(x):
            yield x

    for x in iterator:
        yield x

Note this does not produce any output until the predicate first becomes false, so this itertool may have a lengthy start-up time.

itertools.filterfalse(predicate, iterable)

Make an iterator that filters elements from the iterable returning only those for which the predicate returns a false value. If predicate is None, returns the items that are false. Roughly equivalent to:

def filterfalse(predicate, iterable):
    # filterfalse(lambda x: x<5, [1,4,6,3,8]) → 6 8
    if predicate is None:
        predicate = bool
    for x in iterable:
        if not predicate(x):
            yield x
itertools.groupby(iterable, key=None)

iterable에서 연속적인 키와 그룹을 반환하는 이터레이터를 만듭니다. key는 각 요소의 키값을 계산하는 함수입니다. 지정되지 않거나 None이면, key의 기본값은 항등함수(identity function)이고 요소를 변경하지 않고 반환합니다. 일반적으로, iterable은 같은 키 함수로 이미 정렬되어 있어야 합니다.

groupby()의 작동은 유닉스의 uniq 필터와 유사합니다. 키 함수의 값이 변경될 때마다 중단(break)이나 새 그룹을 생성합니다 (이것이 일반적으로 같은 키 함수를 사용하여 데이터를 정렬해야 하는 이유입니다). 이 동작은 입력 순서와 관계없이 공통 요소를 집계하는 SQL의 GROUP BY와 다릅니다.

반환되는 그룹 자체는 groupby()와 하부 이터러블(iterable)을 공유하는 이터레이터입니다. 소스가 공유되므로, groupby() 객체가 진행하면, 이전 그룹은 이 더는 보이지 않게 됩니다. 따라서, 나중에 데이터가 필요하면, 리스트로 저장해야 합니다:

groups = []
uniquekeys = []
data = sorted(data, key=keyfunc)
for k, g in groupby(data, keyfunc):
    groups.append(list(g))      # Store group iterator as a list

groupby()는 대략 다음과 동등합니다:

def groupby(iterable, key=None):
    # [k for k, g in groupby('AAAABBBCCDAABBB')] → A B C D A B
    # [list(g) for k, g in groupby('AAAABBBCCD')] → AAAA BBB CC D

    keyfunc = (lambda x: x) if key is None else key
    iterator = iter(iterable)
    exhausted = False

    def _grouper(target_key):
        nonlocal curr_value, curr_key, exhausted
        yield curr_value
        for curr_value in iterator:
            curr_key = keyfunc(curr_value)
            if curr_key != target_key:
            yield curr_value
        exhausted = True

        curr_value = next(iterator)
    except StopIteration:
    curr_key = keyfunc(curr_value)

    while not exhausted:
        target_key = curr_key
        curr_group = _grouper(target_key)
        yield curr_key, curr_group
        if curr_key == target_key:
            for _ in curr_group:
itertools.islice(iterable, stop)
itertools.islice(iterable, start, stop[, step])

Make an iterator that returns selected elements from the iterable. Works like sequence slicing but does not support negative values for start, stop, or step.

If start is zero or None, iteration starts at zero. Otherwise, elements from the iterable are skipped until start is reached.

If stop is None, iteration continues until the iterator is exhausted, if at all. Otherwise, it stops at the specified position.

If step is None, the step defaults to one. Elements are returned consecutively unless step is set higher than one which results in items being skipped.

대략 다음과 동등합니다:

def islice(iterable, *args):
    # islice('ABCDEFG', 2) → A B
    # islice('ABCDEFG', 2, 4) → C D
    # islice('ABCDEFG', 2, None) → C D E F G
    # islice('ABCDEFG', 0, None, 2) → A C E G

    s = slice(*args)
    start = 0 if s.start is None else s.start
    stop = s.stop
    step = 1 if s.step is None else s.step
    if start < 0 or (stop is not None and stop < 0) or step <= 0:
        raise ValueError

    indices = count() if stop is None else range(max(start, stop))
    next_i = start
    for i, element in zip(indices, iterable):
        if i == next_i:
            yield element
            next_i += step

Return successive overlapping pairs taken from the input iterable.

The number of 2-tuples in the output iterator will be one fewer than the number of inputs. It will be empty if the input iterable has fewer than two values.

대략 다음과 동등합니다:

def pairwise(iterable):
    # pairwise('ABCDEFG') → AB BC CD DE EF FG
    iterator = iter(iterable)
    a = next(iterator, None)
    for b in iterator:
        yield a, b
        a = b

Added in version 3.10.

itertools.permutations(iterable, r=None)

Return successive r length permutations of elements from the iterable.

r이 지정되지 않았거나 None이면, r의 기본값은 iterable의 길이이며 가능한 모든 최대 길이 순열이 생성됩니다.

The output is a subsequence of product() where entries with repeated elements have been filtered out. The length of the output is given by math.perm() which computes n! / (n - r)! when 0 r n or zero when r > n.

The permutation tuples are emitted in lexicographic order according to the order of the input iterable. If the input iterable is sorted, the output tuples will be produced in sorted order.

Elements are treated as unique based on their position, not on their value. If the input elements are unique, there will be no repeated values within a permutation.

대략 다음과 동등합니다:

def permutations(iterable, r=None):
    # permutations('ABCD', 2) → AB AC AD BA BC BD CA CB CD DA DB DC
    # permutations(range(3)) → 012 021 102 120 201 210

    pool = tuple(iterable)
    n = len(pool)
    r = n if r is None else r
    if r > n:

    indices = list(range(n))
    cycles = list(range(n, n-r, -1))
    yield tuple(pool[i] for i in indices[:r])

    while n:
        for i in reversed(range(r)):
            cycles[i] -= 1
            if cycles[i] == 0:
                indices[i:] = indices[i+1:] + indices[i:i+1]
                cycles[i] = n - i
                j = cycles[i]
                indices[i], indices[-j] = indices[-j], indices[i]
                yield tuple(pool[i] for i in indices[:r])
itertools.product(*iterables, repeat=1)

입력 이터러블들(iterables)의 데카르트 곱.

대략 제너레이터 표현식에서의 중첩된 for-루프와 동등합니다. 예를 들어, product(A, B)((x,y) for x in A for y in B)와 같은 것을 반환합니다.

중첩된 루프는 매 이터레이션마다 가장 오른쪽 요소가 진행되는 주행 거리계처럼 순환합니다. 이 패턴은 사전식 순서를 만들어서 입력의 이터러블들이 정렬되어 있다면, 곱(product) 튜플이 정렬된 순서로 방출됩니다.

이터러블의 자신과의 곱을 계산하려면, 선택적 repeat 키워드 인자를 사용하여 반복 횟수를 지정하십시오. 예를 들어, product(A, repeat=4)product(A, A, A, A)와 같은 것을 뜻합니다.

이 함수는 실제 구현이 메모리에 중간 결과를 쌓지 않는다는 점을 제외하고 다음 코드와 대략 동등합니다:

def product(*iterables, repeat=1):
    # product('ABCD', 'xy') → Ax Ay Bx By Cx Cy Dx Dy
    # product(range(2), repeat=3) → 000 001 010 011 100 101 110 111

    pools = [tuple(pool) for pool in iterables] * repeat

    result = [[]]
    for pool in pools:
        result = [x+[y] for x in result for y in pool]

    for prod in result:
        yield tuple(prod)

product()가 실행되기 전에, 입력 이터러블을 완전히 소비하여, 곱을 생성하기 위해 값의 풀(pool)을 메모리에 유지합니다. 따라서, 유한 입력에만 유용합니다.

itertools.repeat(object[, times])

Make an iterator that returns object over and over again. Runs indefinitely unless the times argument is specified.

대략 다음과 동등합니다:

def repeat(object, times=None):
    # repeat(10, 3) → 10 10 10
    if times is None:
        while True:
            yield object
        for i in range(times):
            yield object

A common use for repeat is to supply a stream of constant values to map or zip:

>>> list(map(pow, range(10), repeat(2)))
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81]
itertools.starmap(function, iterable)

Make an iterator that computes the function using arguments obtained from the iterable. Used instead of map() when argument parameters have already been “pre-zipped” into tuples.

The difference between map() and starmap() parallels the distinction between function(a,b) and function(*c). Roughly equivalent to:

def starmap(function, iterable):
    # starmap(pow, [(2,5), (3,2), (10,3)]) → 32 9 1000
    for args in iterable:
        yield function(*args)
itertools.takewhile(predicate, iterable)

Make an iterator that returns elements from the iterable as long as the predicate is true. Roughly equivalent to:

def takewhile(predicate, iterable):
    # takewhile(lambda x: x<5, [1,4,6,3,8]) → 1 4
    for x in iterable:
        if not predicate(x):
        yield x

Note, the element that first fails the predicate condition is consumed from the input iterator and there is no way to access it. This could be an issue if an application wants to further consume the input iterator after takewhile has been run to exhaustion. To work around this problem, consider using more-iterools before_and_after() instead.

itertools.tee(iterable, n=2)

단일 iterable에서 n 개의 독립 이터레이터를 반환합니다.

대략 다음과 동등합니다:

def tee(iterable, n=2):
    iterator = iter(iterable)
    shared_link = [None, None]
    return tuple(_tee(iterator, shared_link) for _ in range(n))

def _tee(iterator, link):
        while True:
            if link[1] is None:
                link[0] = next(iterator)
                link[1] = [None, None]
            value, link = link
            yield value
    except StopIteration:

Once a tee() has been created, the original iterable should not be used anywhere else; otherwise, the iterable could get advanced without the tee objects being informed.

tee iterators are not threadsafe. A RuntimeError may be raised when simultaneously using iterators returned by the same tee() call, even if the original iterable is threadsafe.

이 이터레이터 도구에는 상당한 보조 기억 장치가 필요할 수 있습니다 (일시적으로 저장해야 하는 데이터양에 따라 다릅니다). 일반적으로, 다른 이터레이터가 시작하기 전에 하나의 이터레이터가 대부분이나 모든 데이터를 사용하면, tee() 대신 list()를 사용하는 것이 더 빠릅니다.

itertools.zip_longest(*iterables, fillvalue=None)

Make an iterator that aggregates elements from each of the iterables.

If the iterables are of uneven length, missing values are filled-in with fillvalue. If not specified, fillvalue defaults to None.

Iteration continues until the longest iterable is exhausted.

대략 다음과 동등합니다:

def zip_longest(*iterables, fillvalue=None):
    # zip_longest('ABCD', 'xy', fillvalue='-') → Ax By C- D-

    iterators = list(map(iter, iterables))
    num_active = len(iterators)
    if not num_active:

    while True:
        values = []
        for i, iterator in enumerate(iterators):
                value = next(iterator)
            except StopIteration:
                num_active -= 1
                if not num_active:
                iterators[i] = repeat(fillvalue)
                value = fillvalue
        yield tuple(values)

If one of the iterables is potentially infinite, then the zip_longest() function should be wrapped with something that limits the number of calls (for example islice() or takewhile()).

Itertools 조리법

이 섹션에서는 기존 itertools를 빌딩 블록으로 사용하여 확장 도구 집합을 만드는 방법을 보여줍니다.

The primary purpose of the itertools recipes is educational. The recipes show various ways of thinking about individual tools — for example, that chain.from_iterable is related to the concept of flattening. The recipes also give ideas about ways that the tools can be combined — for example, how starmap() and repeat() can work together. The recipes also show patterns for using itertools with the operator and collections modules as well as with the built-in itertools such as map(), filter(), reversed(), and enumerate().

A secondary purpose of the recipes is to serve as an incubator. The accumulate(), compress(), and pairwise() itertools started out as recipes. Currently, the sliding_window(), iter_index(), and sieve() recipes are being tested to see whether they prove their worth.

Substantially all of these recipes and many, many others can be installed from the more-itertools project found on the Python Package Index:

python -m pip install more-itertools

Many of the recipes offer the same high performance as the underlying toolset. Superior memory performance is kept by processing elements one at a time rather than bringing the whole iterable into memory all at once. Code volume is kept small by linking the tools together in a functional style. High speed is retained by preferring “vectorized” building blocks over the use of for-loops and generators which incur interpreter overhead.

import collections
import contextlib
import functools
import math
import operator
import random

def take(n, iterable):
    "Return first n items of the iterable as a list."
    return list(islice(iterable, n))

def prepend(value, iterable):
    "Prepend a single value in front of an iterable."
    # prepend(1, [2, 3, 4]) → 1 2 3 4
    return chain([value], iterable)

def tabulate(function, start=0):
    "Return function(0), function(1), ..."
    return map(function, count(start))

def repeatfunc(func, times=None, *args):
    "Repeat calls to func with specified arguments."
    if times is None:
        return starmap(func, repeat(args))
    return starmap(func, repeat(args, times))

def flatten(list_of_lists):
    "Flatten one level of nesting."
    return chain.from_iterable(list_of_lists)

def ncycles(iterable, n):
    "Returns the sequence elements n times."
    return chain.from_iterable(repeat(tuple(iterable), n))

def tail(n, iterable):
    "Return an iterator over the last n items."
    # tail(3, 'ABCDEFG') → E F G
    return iter(collections.deque(iterable, maxlen=n))

def consume(iterator, n=None):
    "Advance the iterator n-steps ahead. If n is None, consume entirely."
    # Use functions that consume iterators at C speed.
    if n is None:
        collections.deque(iterator, maxlen=0)
        next(islice(iterator, n, n), None)

def nth(iterable, n, default=None):
    "Returns the nth item or a default value."
    return next(islice(iterable, n, None), default)

def quantify(iterable, predicate=bool):
    "Given a predicate that returns True or False, count the True results."
    return sum(map(predicate, iterable))

def first_true(iterable, default=False, predicate=None):
    "Returns the first true value or the *default* if there is no true value."
    # first_true([a,b,c], x) → a or b or c or x
    # first_true([a,b], x, f) → a if f(a) else b if f(b) else x
    return next(filter(predicate, iterable), default)

def all_equal(iterable, key=None):
    "Returns True if all the elements are equal to each other."
    # all_equal('4٤௪౪໔', key=int) → True
    return len(take(2, groupby(iterable, key))) <= 1

def unique_justseen(iterable, key=None):
    "Yield unique elements, preserving order. Remember only the element just seen."
    # unique_justseen('AAAABBBCCDAABBB') → A B C D A B
    # unique_justseen('ABBcCAD', str.casefold) → A B c A D
    if key is None:
        return map(operator.itemgetter(0), groupby(iterable))
    return map(next, map(operator.itemgetter(1), groupby(iterable, key)))

def unique_everseen(iterable, key=None):
    "Yield unique elements, preserving order. Remember all elements ever seen."
    # unique_everseen('AAAABBBCCDAABBB') → A B C D
    # unique_everseen('ABBcCAD', str.casefold) → A B c D
    seen = set()
    if key is None:
        for element in filterfalse(seen.__contains__, iterable):
            yield element
        for element in iterable:
            k = key(element)
            if k not in seen:
                yield element

def unique(iterable, key=None, reverse=False):
   "Yield unique elements in sorted order. Supports unhashable inputs."
   # unique([[1, 2], [3, 4], [1, 2]]) → [1, 2] [3, 4]
   return unique_justseen(sorted(iterable, key=key, reverse=reverse), key=key)

def sliding_window(iterable, n):
    "Collect data into overlapping fixed-length chunks or blocks."
    # sliding_window('ABCDEFG', 4) → ABCD BCDE CDEF DEFG
    iterator = iter(iterable)
    window = collections.deque(islice(iterator, n - 1), maxlen=n)
    for x in iterator:
        yield tuple(window)

def grouper(iterable, n, *, incomplete='fill', fillvalue=None):
    "Collect data into non-overlapping fixed-length chunks or blocks."
    # grouper('ABCDEFG', 3, fillvalue='x') → ABC DEF Gxx
    # grouper('ABCDEFG', 3, incomplete='strict') → ABC DEF ValueError
    # grouper('ABCDEFG', 3, incomplete='ignore') → ABC DEF
    iterators = [iter(iterable)] * n
    match incomplete:
        case 'fill':
            return zip_longest(*iterators, fillvalue=fillvalue)
        case 'strict':
            return zip(*iterators, strict=True)
        case 'ignore':
            return zip(*iterators)
        case _:
            raise ValueError('Expected fill, strict, or ignore')

def roundrobin(*iterables):
    "Visit input iterables in a cycle until each is exhausted."
    # roundrobin('ABC', 'D', 'EF') → A D E B F C
    # Algorithm credited to George Sakkis
    iterators = map(iter, iterables)
    for num_active in range(len(iterables), 0, -1):
        iterators = cycle(islice(iterators, num_active))
        yield from map(next, iterators)

def partition(predicate, iterable):
    """Partition entries into false entries and true entries.

    If *predicate* is slow, consider wrapping it with functools.lru_cache().
    # partition(is_odd, range(10)) → 0 2 4 6 8   and  1 3 5 7 9
    t1, t2 = tee(iterable)
    return filterfalse(predicate, t1), filter(predicate, t2)

def subslices(seq):
    "Return all contiguous non-empty subslices of a sequence."
    # subslices('ABCD') → A AB ABC ABCD B BC BCD C CD D
    slices = starmap(slice, combinations(range(len(seq) + 1), 2))
    return map(operator.getitem, repeat(seq), slices)

def iter_index(iterable, value, start=0, stop=None):
    "Return indices where a value occurs in a sequence or iterable."
    # iter_index('AABCADEAF', 'A') → 0 1 4 7
    seq_index = getattr(iterable, 'index', None)
    if seq_index is None:
        iterator = islice(iterable, start, stop)
        for i, element in enumerate(iterator, start):
            if element is value or element == value:
                yield i
        stop = len(iterable) if stop is None else stop
        i = start
        with contextlib.suppress(ValueError):
            while True:
                yield (i := seq_index(value, i, stop))
                i += 1

def iter_except(func, exception, first=None):
    "Convert a call-until-exception interface to an iterator interface."
    # iter_except(d.popitem, KeyError) → non-blocking dictionary iterator
    with contextlib.suppress(exception):
        if first is not None:
            yield first()
        while True:
            yield func()

The following recipes have a more mathematical flavor:

def powerset(iterable):
    "powerset([1,2,3]) → () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)"
    s = list(iterable)
    return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))

def sum_of_squares(iterable):
    "Add up the squares of the input values."
    # sum_of_squares([10, 20, 30]) → 1400
    return math.sumprod(*tee(iterable))

def reshape(matrix, cols):
    "Reshape a 2-D matrix to have a given number of columns."
    # reshape([(0, 1), (2, 3), (4, 5)], 3) →  (0, 1, 2), (3, 4, 5)
    return batched(chain.from_iterable(matrix), cols)

def transpose(matrix):
    "Swap the rows and columns of a 2-D matrix."
    # transpose([(1, 2, 3), (11, 22, 33)]) → (1, 11) (2, 22) (3, 33)
    return zip(*matrix, strict=True)

def matmul(m1, m2):
    "Multiply two matrices."
    # matmul([(7, 5), (3, 5)], [(2, 5), (7, 9)]) → (49, 80), (41, 60)
    n = len(m2[0])
    return batched(starmap(math.sumprod, product(m1, transpose(m2))), n)

def convolve(signal, kernel):
    """Discrete linear convolution of two iterables.
    Equivalent to polynomial multiplication.

    Convolutions are mathematically commutative; however, the inputs are
    evaluated differently.  The signal is consumed lazily and can be
    infinite. The kernel is fully consumed before the calculations begin.

    # convolve([1, -1, -20], [1, -3]) → 1 -4 -17 60
    # convolve(data, [0.25, 0.25, 0.25, 0.25]) → Moving average (blur)
    # convolve(data, [1/2, 0, -1/2]) → 1st derivative estimate
    # convolve(data, [1, -2, 1]) → 2nd derivative estimate
    kernel = tuple(kernel)[::-1]
    n = len(kernel)
    padded_signal = chain(repeat(0, n-1), signal, repeat(0, n-1))
    windowed_signal = sliding_window(padded_signal, n)
    return map(math.sumprod, repeat(kernel), windowed_signal)

def polynomial_from_roots(roots):
    """Compute a polynomial's coefficients from its roots.

       (x - 5) (x + 4) (x - 3)  expands to:   x³ -4x² -17x + 60
    # polynomial_from_roots([5, -4, 3]) → [1, -4, -17, 60]
    factors = zip(repeat(1), map(operator.neg, roots))
    return list(functools.reduce(convolve, factors, [1]))

def polynomial_eval(coefficients, x):
    """Evaluate a polynomial at a specific value.

    Computes with better numeric stability than Horner's method.
    # Evaluate x³ -4x² -17x + 60 at x = 5
    # polynomial_eval([1, -4, -17, 60], x=5) → 0
    n = len(coefficients)
    if not n:
        return type(x)(0)
    powers = map(pow, repeat(x), reversed(range(n)))
    return math.sumprod(coefficients, powers)

def polynomial_derivative(coefficients):
    """Compute the first derivative of a polynomial.

       f(x)  =  x³ -4x² -17x + 60
       f'(x) = 3x² -8x  -17
    # polynomial_derivative([1, -4, -17, 60]) → [3, -8, -17]
    n = len(coefficients)
    powers = reversed(range(1, n))
    return list(map(operator.mul, coefficients, powers))

def sieve(n):
    "Primes less than n."
    # sieve(30) → 2 3 5 7 11 13 17 19 23 29
    if n > 2:
        yield 2
    data = bytearray((0, 1)) * (n // 2)
    for p in iter_index(data, 1, start=3, stop=math.isqrt(n) + 1):
        data[p*p : n : p+p] = bytes(len(range(p*p, n, p+p)))
    yield from iter_index(data, 1, start=3)

def factor(n):
    "Prime factors of n."
    # factor(99) → 3 3 11
    # factor(1_000_000_000_000_007) → 47 59 360620266859
    # factor(1_000_000_000_000_403) → 1000000000000403
    for prime in sieve(math.isqrt(n) + 1):
        while not n % prime:
            yield prime
            n //= prime
            if n == 1:
    if n > 1:
        yield n

def totient(n):
    "Count of natural numbers up to n that are coprime to n."
    # totient(12) → 4 because len([1, 5, 7, 11]) == 4
    for prime in set(factor(n)):
        n -= n // prime
    return n