itertools — Functions creating iterators for efficient looping

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Este módulo implementa un número de piezas básicas iterator inspiradas en constructs de APL, Haskell y SML. Cada pieza ha sido reconvertida a una forma apropiada para Python.

El módulo estandariza un conjunto base de herramientas rápidas y eficientes en memoria, útiles por sí mismas o en combinación con otras. Juntas, forman un «álgebra de iteradores», haciendo posible la construcción de herramientas especializadas, sucintas y eficientes, en Python puro.

Por ejemplo, SML provee una herramienta de tabulación tabulate(f), que produce una secuencia f(0), f(1), .... En Python, se puede lograr el mismo efecto al combinar map() y count() para formar map(f, count()).

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))).

Iteradores infinitos:

Iterador	Argumentos	Resultados	Ejemplo
count()	[start[, step]]	start, start+step, start+2*step, …	count(10) → 10 11 12 13 14 ...
cycle()	p	p0, p1, … plast, p0, p1, …	cycle('ABCD') → A B C D A B C D ...
repeat()	elem [,n]	elem, elem, elem, … indefinidamente o hasta n veces	repeat(10, 3) → 10 10 10

Iteradores que terminan en la secuencia de entrada más corta:

Iterador	Argumentos	Resultados	Ejemplo
accumulate()	p [,func]	p0, p0+p1, p0+p1+p2, …	accumulate([1,2,3,4,5]) → 1 3 6 10 15
batched()	p, n	(p0, p1, …, p_n-1), …	batched('ABCDEFG', n=3) → ABC DEF G
chain()	p, q, …	p0, p1, … plast, q0, q1, …	chain('ABC', 'DEF') → A B C D E F
chain.from_iterable()	iterable	p0, p1, … plast, q0, q1, …	chain.from_iterable(['ABC', 'DEF']) → A B C D E F
compress()	data, selectors	(d[0] if s[0]), (d[1] if s[1]), …	compress('ABCDEF', [1,0,1,0,1,1]) → A C E F
dropwhile()	predicate, seq	seq[n], seq[n+1], starting when predicate fails	dropwhile(lambda x: x<5, [1,4,6,3,8]) → 6 3 8
filterfalse()	predicate, seq	elements of seq where predicate(elem) fails	filterfalse(lambda x: x<5, [1,4,6,3,8]) → 6 8
groupby()	iterable[, key]	sub-iteradores agrupados según el valor de key(v)
islice()	seq, [start,] stop [, step]	elementos de seq[start:stop:step]	islice('ABCDEFG', 2, None) → C D E F G
pairwise()	iterable	(p[0], p[1]), (p[1], p[2])	pairwise('ABCDEFG') → AB BC CD DE EF FG
starmap()	func, seq	func(*seq[0]), func(*seq[1]), …	starmap(pow, [(2,5), (3,2), (10,3)]) → 32 9 1000
takewhile()	predicate, seq	seq[0], seq[1], until predicate fails	takewhile(lambda x: x<5, [1,4,6,3,8]) → 1 4
tee()	it, n	it1, it2, … itn divide un iterador en n
zip_longest()	p, q, …	(p[0], q[0]), (p[1], q[1]), …	zip_longest('ABCD', 'xy', fillvalue='-') → Ax By C- D-

Iteradores combinatorios:

Iterador	Argumentos	Resultados
product()	p, q, … [repeat=1]	producto cartesiano, equivalente a un bucle for anidado
permutations()	p[, r]	tuplas de longitud r, en todas los órdenes posibles, sin elementos repetidos
combinations()	p, r	tuplas de longitud r, ordenadas, sin elementos repetidos
combinations_with_replacement()	p, r	tuplas de longitud r, ordenadas, con elementos repetidos

Ejemplos	Resultados
product('ABCD', repeat=2)	AA AB AC AD BA BB BC BD CA CB CC CD DA DB DC DD
permutations('ABCD', 2)	AB AC AD BA BC BD CA CB CD DA DB DC
combinations('ABCD', 2)	AB AC AD BC BD CD
combinations_with_replacement('ABCD', 2)	AA AB AC AD BB BC BD CC CD DD

Itertool Functions

Todas las funciones del siguiente módulo construyen y retornan iteradores. Algunas proveen flujos infinitos, por lo que deberían ser sólo manipuladas por funciones o bucles que cortan el flujo.

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.

Aproximadamente equivalente a:

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:
        try:
            total = next(iterator)
        except StopIteration:
            return

    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]

Para una función similar que retorne únicamente el valor final acumulado, revisa functools.reduce().

Added in version 3.2.

Distinto en la versión 3.3: Added the optional function parameter.

Distinto en la versión 3.8: Adicionó el argumento opcional initial.

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')]

Aproximadamente equivalente a:

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.

itertools.chain(*iterables)

Crea un iterador que retorna elementos del primer iterable hasta que es consumido, para luego proceder con el siguiente iterable, hasta que todos los iterables son consumidos. Se utiliza para tratar secuencias consecutivas como unas sola secuencia. Aproximadamente equivalente a:

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

classmethod chain.from_iterable(iterable)

Constructor alternativo para chain(). Obtiene entradas enlazadas de un mismo argumento que se evalúa perezosamente. Aproximadamente equivalente a:

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)

Retorna subsecuencias de longitud r con elementos del iterable de entrada.

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.

Aproximadamente equivalente a:

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:
        return
    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:
                break
        else:
            return
        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)

Retorna subsecuencias, de longitud r, con elementos del iterable de entrada, permitiendo que haya elementos individuales repetidos más de una vez.

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.

Aproximadamente equivalente a:

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:
        return
    indices = [0] * r

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

When counting with floating-point numbers, better accuracy can sometimes be achieved by substituting multiplicative code such as: (start + step * i for i in count()).

Distinto en la versión 3.1: Se adicionó el argumento step y se permitieron argumentos diferentes a enteros.

itertools.cycle(iterable)

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
        saved.append(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
            break

    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)

Crea un iterador que retorna claves consecutivas y grupos del iterable. key es una función que calcula un valor clave para cada elemento. Si no se especifica o es None, key es una función de identidad por defecto y retorna el elemento sin cambios. Generalmente, el iterable necesita estar ordenado con la misma función key.

El funcionamiento de groupby() es similar al del filtro uniq en Unix. Genera un salto o un nuevo grupo cada vez que el valor de la función clave cambia (por lo que usualmente es necesario ordenar los datos usando la misma función clave). Ese comportamiento difiere del de GROUP BY de SQL, el cual agrega elementos comunes sin importar el orden de entrada.

El grupo retornado es un iterador mismo que comparte el iterable subyacente con groupby(). Al compartir la fuente, cuando el objeto groupby() se avanza, el grupo previo deja de ser visible. En ese caso, si los datos se necesitan posteriormente, se deberían almacenar como lista:

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

groupby() es aproximadamente equivalente a:

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:
                return
            yield curr_value
        exhausted = True

    try:
        curr_value = next(iterator)
    except StopIteration:
        return
    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:
                pass

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.

Aproximadamente equivalente a:

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

itertools.pairwise(iterable)

Retorna los sucesivos pares superpuestos tomados de la entrada iterable.

El número de tuplas de 2 elementos en el iterador de salida será uno menos que el número de entradas. Estará vacío si la entrada iterable tiene menos de dos valores.

Aproximadamente equivalente a:

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.

Si r no es especificado o si es None, entonces por defecto r será igual a la longitud de iterable y todas las permutaciones de máxima longitud serán generadas.

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.

Aproximadamente equivalente a:

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:
        return

    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
            else:
                j = cycles[i]
                indices[i], indices[-j] = indices[-j], indices[i]
                yield tuple(pool[i] for i in indices[:r])
                break
        else:
            return

itertools.product(*iterables, repeat=1)

Producto cartesiano de los iterables de entrada.

Aproximadamente equivalente a tener bucles for anidados en un generador. Por ejemplo, product(A, B) es equivalente a ((x,y) for x in A for y in B).

Los bucles anidados hacen ciclos como un cuentapasos o taxímetro, con el elemento más hacia la derecha avanzando en cada iteración. Este patrón crea un orden lexicográfico en el que, si los iterables de entrada están ordenados, las tuplas producidas son emitidas de manera ordenada.

Para calcular el producto de un iterable consigo mismo, especifica el número de repeticiones con el argumento opcional repeat. Por ejemplo, product(A, repeat=4) es equivalente a product(A, A, A, A).

Esta función es aproximadamente equivalente al código siguiente, exceptuando que la implementación real no acumula resultados intermedios en memoria:

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)

Antes de que product() se ejecute, consume completamente los iterables de entrada, manteniendo grupos de valores en la memoria para generar los productos. En consecuencia, solo es útil con entradas finitas.

itertools.repeat(object[, times])

Crea un iterador que retorna object una y otra vez. Se ejecuta indefinidamente a menos que se especifique el argumento times.

Aproximadamente equivalente a:

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

Un uso común de repeat es proporcionar un flujo de valores constantes a map o 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.

La diferencia entre map() y starmap() es paralela a la distinción entre function(a,b) y function(*c). Aproximadamente equivalente a:

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):
            break
        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)

Retorna n iteradores independientes de un mismo iterador.

Aproximadamente equivalente a:

def tee(iterable, n=2):
    if n < 0:
        raise ValueError
    if n == 0:
        return ()
    iterator = _tee(iterable)
    result = [iterator]
    for _ in range(n - 1):
        result.append(_tee(iterator))
    return tuple(result)

class _tee:

    def __init__(self, iterable):
        it = iter(iterable)
        if isinstance(it, _tee):
            self.iterator = it.iterator
            self.link = it.link
        else:
            self.iterator = it
            self.link = [None, None]

    def __iter__(self):
        return self

    def __next__(self):
        link = self.link
        if link[1] is None:
            link[0] = next(self.iterator)
            link[1] = [None, None]
        value, self.link = link
        return value

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.

Esta herramienta de iteración puede requerir almacenamiento auxiliar significativo (dependiendo de qué tantos datos necesitan ser almacenados). En general, si un iterador utiliza todos o la mayoría de los datos antes que otro iterador comience, es más rápido utilizar list() en vez de tee().

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.

Aproximadamente equivalente a:

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:
        return

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

Fórmulas con itertools

Esta sección muestra fórmulas para crear un conjunto de herramientas extendido usando las herramientas de itertools como piezas básicas.

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)
    else:
        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):
            seen.add(element)
            yield element
    else:
        for element in iterable:
            k = key(element)
            if k not in seen:
                seen.add(k)
                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:
        window.append(x)
        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
    else:
        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.

    Article:  https://betterexplained.com/articles/intuitive-convolution/
    Video:    https://www.youtube.com/watch?v=KuXjwB4LzSA
    """
    # 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:
                return
    if n > 1:
        yield n

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