Tuples

The k-tuples over $\{1, \dots, n\}$: all $n^k$ ordered selections with repetition.

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Tuples(n, k)the $n^k$ length-$k$ tuples

Tuples(list, k)the $\lvert list \rvert^k$ length-$k$ tuples of the list's own elements, as a $List$ (up to a size cap)

Typed signatures

Type	Package	Applies
((integer<0..>, integer<0..>) -> indexed_collection<list<integer>>) & ((list<any>, integer<0..>) -> list<list<any>>)	combinatorics	always

Domain: Combinatorial collections

Details

• Count is $n^k$
• Elements are ordered as mixed-radix (base $n$) counting

Definition

Epsil, as the implementation runs it.

count(_n, _k)

1 if _k == 0 else _n ^ _k

unrank(_n, _k, _r)

Map(
     Function((_r - _r % _n ^ (_k - j)) / _n ^ (_k - j) % _n + 1, j),
     Range(1, _k, 1),
   )

rank(_n, _k, _x)

Fold(Function(acc * _n + (_x[j] - 1), acc, j), 0, Range(1, _k, 1))

valid(_n, _k, _x)

Length(_x) == _k && Fold(Function(
                                   ok_j && 1 <= _x[j] && _x[j] <= _n + 1 - 1,
                                   ok_j,
                                   j,
                                 ), True, Range(1, Length(_x), 1))

Examples

Basic #

$2^3$

live ↗

In$\mathrm{Count}(\mathrm{Tuples}(2, 3))$Out$8$

live ↗

In$\mathrm{Tuples}(2, 3)[1]$Out$\bigl\lbrack1, 1, 1\bigr\rbrack$

Applications #

live ↗

In$\mathrm{Thread}(\mathrm{Rule}(\mathrm{Tuples}(\bigl\lbrack1, 0\bigr\rbrack, 3), \mathrm{IntegerDigits}(30, 2, 8)))$Out$\bigl\lbrack\mathrm{Rule}(\bigl\lbrack1, 1, 1\bigr\rbrack, 0), \mathrm{Rule}(\bigl\lbrack1, 1, 0\bigr\rbrack, 0), \mathrm{Rule}(\bigl\lbrack1, 0, 1\bigr\rbrack, 0), \mathrm{Rule}(\bigl\lbrack1, 0, 0\bigr\rbrack, 1), \mathrm{Rule}(\bigl\lbrack0, 1, 1\bigr\rbrack, 1), \mathrm{Rule}(\bigl\lbrack0, 1, 0\bigr\rbrack, 1), \mathrm{Rule}(\bigl\lbrack0, 0, 1\bigr\rbrack, 1), \mathrm{Rule}(\bigl\lbrack0, 0, 0\bigr\rbrack, 0)\bigr\rbrack$

About

Examples	3
Checked against	Wolfram Language 3 agree
Crosswalk	WikipediaMathWorldWikidataOEISWolfram Language

See also: Subsets, Multisets