Prove that there exists a set of ${}_ktext{P}_n$ permutations of $(1, 2, …, k)$ where no two permutations...
up vote
1
down vote
favorite
Let $k$ and $n$ be positive integers with $k ge n$. Then I want to prove that we can find ${}_ktext{P}_n$ permutations of $(1, 2, ..., k)$ of length k such that no two permutations have n elements matching.
For example, if $k = 4$ and $n = 2$, we find ${}_4text{P}_2 = 12$ permutations of $(1, 2, 3, 4)$ with the property:
$1234, 1342, 1423, 2143, 2314, 2431, 3124, 3241, 3412, 4132, 4213, 4321$
If we pick any two of these permutations, they will not have more than one number in the same spot in both permutations. The first two digits use each of the 12 permutations of two distinct elements from ${1, 2, 3, 4}$ exactly once, as do the last two digits, the first and last digit, etc.
I cannot find an explicit construction through induction or backwards induction (or anything else) nor can I find a graph theory proof of this.
combinatorics
add a comment |
up vote
1
down vote
favorite
Let $k$ and $n$ be positive integers with $k ge n$. Then I want to prove that we can find ${}_ktext{P}_n$ permutations of $(1, 2, ..., k)$ of length k such that no two permutations have n elements matching.
For example, if $k = 4$ and $n = 2$, we find ${}_4text{P}_2 = 12$ permutations of $(1, 2, 3, 4)$ with the property:
$1234, 1342, 1423, 2143, 2314, 2431, 3124, 3241, 3412, 4132, 4213, 4321$
If we pick any two of these permutations, they will not have more than one number in the same spot in both permutations. The first two digits use each of the 12 permutations of two distinct elements from ${1, 2, 3, 4}$ exactly once, as do the last two digits, the first and last digit, etc.
I cannot find an explicit construction through induction or backwards induction (or anything else) nor can I find a graph theory proof of this.
combinatorics
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $k$ and $n$ be positive integers with $k ge n$. Then I want to prove that we can find ${}_ktext{P}_n$ permutations of $(1, 2, ..., k)$ of length k such that no two permutations have n elements matching.
For example, if $k = 4$ and $n = 2$, we find ${}_4text{P}_2 = 12$ permutations of $(1, 2, 3, 4)$ with the property:
$1234, 1342, 1423, 2143, 2314, 2431, 3124, 3241, 3412, 4132, 4213, 4321$
If we pick any two of these permutations, they will not have more than one number in the same spot in both permutations. The first two digits use each of the 12 permutations of two distinct elements from ${1, 2, 3, 4}$ exactly once, as do the last two digits, the first and last digit, etc.
I cannot find an explicit construction through induction or backwards induction (or anything else) nor can I find a graph theory proof of this.
combinatorics
Let $k$ and $n$ be positive integers with $k ge n$. Then I want to prove that we can find ${}_ktext{P}_n$ permutations of $(1, 2, ..., k)$ of length k such that no two permutations have n elements matching.
For example, if $k = 4$ and $n = 2$, we find ${}_4text{P}_2 = 12$ permutations of $(1, 2, 3, 4)$ with the property:
$1234, 1342, 1423, 2143, 2314, 2431, 3124, 3241, 3412, 4132, 4213, 4321$
If we pick any two of these permutations, they will not have more than one number in the same spot in both permutations. The first two digits use each of the 12 permutations of two distinct elements from ${1, 2, 3, 4}$ exactly once, as do the last two digits, the first and last digit, etc.
I cannot find an explicit construction through induction or backwards induction (or anything else) nor can I find a graph theory proof of this.
combinatorics
combinatorics
asked Nov 12 at 3:07
Perry Ainsworth
495
495
add a comment |
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2994780%2fprove-that-there-exists-a-set-of-k-textp-n-permutations-of-1-2-k%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown