257 lines
10 KiB
Markdown
257 lines
10 KiB
Markdown
---
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title: Permutations
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TARGET DECK: Obsidian::STEM
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FILE TAGS: combinatorics set
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tags:
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- combinatorics
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- set
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---
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## Overview
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A **permutation** of some $n$ objects is a (possible) rearrangement of those $n$ objects. The number of permutations is $n!$ since there are $n$ possible ways to pick the first object, $(n - 1)$ possible ways to pick the second, and so on.
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```c
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void permutations_aux(
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const size_t n,
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int A[static n],
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int res[static n],
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uint64_t choices
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) {
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if (!choices) {
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for (size_t i = 0; i < n; ++i) {
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printf("%d ", A[res[i]]);
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}
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printf("\n");
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return;
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}
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unsigned int weight = n - bit_weight(choices);
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for (unsigned int i = 0; i < 64; ++i) {
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uint64_t next = 1L << i;
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if (choices & next) {
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res[weight] = i;
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permutations_aux(n, A, res, choices & ~next);
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}
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}
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}
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void permutations(const size_t n, int A[static n]) {
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int *res = malloc(sizeof(int) * n);
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permutations_aux(n, A, res, (1L << n) - 1);
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free(res);
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}
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```
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The above approach prints out all permutations of a given array, provided the array contains at most `64` digits. It relies on `bit_weight` as defined in [[binary/index|binary]].
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%%ANKI
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Basic
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What *is* a permutation?
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Back: An ordered arrangement of some collection of objects.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788567-->
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END%%
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%%ANKI
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Basic
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How many permutations are there of $n$ objects?
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Back: $n!$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788573-->
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END%%
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%%ANKI
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Basic
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How is $n!$ written recursively?
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Back: As $n(n - 1)!$.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708451749781-->
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END%%
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%%ANKI
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Basic
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How is $n!$ permutations of $n$ objects derived?
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Back: There are $n$ choices for the first position, $n - 1$ choices for the second, etc.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788576-->
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END%%
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%%ANKI
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Basic
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What combinatorial concept explains $n!$ permutations of $n$ objects?
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Back: The multiplicative principle.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788580-->
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END%%
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%%ANKI
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Basic
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What combinatorial concept is often associated with the factorial?
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Back: Permutations.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788583-->
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END%%
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%%ANKI
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Basic
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How does sorting relate to the concept of permutations?
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Back: Sorting aims to efficiently find a specific permutation.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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Tags: algorithm
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<!--ID: 1708366788587-->
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END%%
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%%ANKI
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Basic
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What symbol denotes "$n$ factorial"?
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Back: $n!$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788590-->
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END%%
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%%ANKI
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Basic
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$n!$ is shorthand for what other closed formula?
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Back: $\Pi_{k=1}^n k$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788594-->
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END%%
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%%ANKI
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Basic
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What is the identity element of $\cdot$ (multiplication)?
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Back: $1$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1708366788597-->
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END%%
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%%ANKI
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Basic
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What does $0!$ (factorial) evaluate to?
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Back: $1$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788600-->
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END%%
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%%ANKI
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Basic
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*Why* might $0! = 1$ (barring convention)?
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Back: Because the empty product is $1$, the multiplication identity.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788603-->
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END%%
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%%ANKI
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Basic
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What combinatorial concept explains the number of bijective functions between two finite sets?
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Back: Permutations (factorials).
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788606-->
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END%%
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%%ANKI
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Basic
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How many bijective functions exist between $\{1, 2, 3\}$ and $\{a, b, c\}$?
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Back: $3!$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788610-->
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END%%
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%%ANKI
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Basic
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How many bijective functions exist between finite sets $A$ and $B$ where $|A| = |B| = n$?
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Back: $n!$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788613-->
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END%%
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## Falling Factorials
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If we generalize to choosing $k \leq n$ elements of $k$ objects, we can calculate the $k$-permutation of $n$. This is denoted as $(n)_k$, sometimes called the **falling factorial**. $$(n)_k = \frac{n!}{(n - k)!}$$
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The derivation works by noting that we have $n - 0$ possible ways to pick the first object, $n - 1$ ways to pick the second, up until $n - (k - 1)$ ways to pick the last object.
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%%ANKI
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Basic
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What *is* a $k$-permutation?
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Back: An ordered arrangement, containing $k$ elements, of some collection of objects.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788616-->
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END%%
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%%ANKI
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Basic
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What is the closed formula for falling factorial $(n)_k$?
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Back: $$(n)_k = \frac{n!}{(n - k)!}$$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788619-->
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END%%
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%%ANKI
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Basic
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How is the number of $k$-permutations of $n$ objects denoted?
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Back: $(n)_k$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788622-->
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END%%
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%%ANKI
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Basic
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How is $n!$ written equivalently as a falling factorial?
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Back: $(n)_n$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708781334241-->
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END%%
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%%ANKI
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Basic
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How is $(n)_k$ $k$-permutations of $n$ objects derived?
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Back: There are $n$ choices for the first position, $n - 1$ choices for the second, etc. up until $n - (k - 1)$ choices for the last position.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788625-->
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END%%
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%%ANKI
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Basic
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What does $(n)_n$ evaluate to?
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Back: $n!$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788628-->
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END%%
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%%ANKI
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Basic
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What does $(n)_0$ evaluate to?
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Back: $1$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788631-->
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END%%
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%%ANKI
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Cloze
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In a $k$-permutation of $n$ objects, there are $n - 0$ choices for first object and {$n - (k - 1)$} choices for the last object.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788634-->
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END%%
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%%ANKI
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Basic
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What combinatorial concept explains the number of injective functions between two finite sets?
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Back: $k$-permutations (falling factorials).
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788638-->
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END%%
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%%ANKI
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Basic
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How many injective functions exist between $\{1, 2, 3\}$ and $\{a, b, c, d, e\}$?
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Back: $(5)_3$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708366788641-->
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END%%
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## References
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* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf). |