Can all of the first and second differences of a Costas array be at least 3 in magnitude?
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A Costas array $pi = d_1 d_2 ldots d_n$ (one-line form) of order $n$ is a permutation $pi in S_n$ such that the $n-r$ differences
$d_{r+1} - d_{1} , d_{r+2} - d_{2} , ldots , d_n - d_{n-r}$
are distinct for each $r$ , $1 leq r leq n-1$.
Find an example of a Costas array $pi$ for some $n$ such that the $2n-3$ inequalities
$|d_2 - d_1| , |d_3 - d_2| , ldots , |d_n - d_{n-1}| geq 3$
and
$|d_3 - d_1| , |d_4 - d_2| , ldots , |d_n - d_{n-2}| geq 3$
hold or else prove that no such permutation exists.
Remarks: No such Costas arrays have been found using Beard's database of Costas arrays out to order $1030$ (the database is available at https://ieee-dataport.org/open-access/costas-arrays-and-enumeration-order-1030 and is exhaustive out to order $29$.) Costas arrays are known to exist for infinitely many (but not all) orders by constructions involving the infinitude of primes.
combinatorics discrete-mathematics permutations
asked Nov 26 at 16:58
Bill
294
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up vote
0
down vote
favorite
A Costas array $pi = d_1 d_2 ldots d_n$ (one-line form) of order $n$ is a permutation $pi in S_n$ such that the $n-r$ differences
$d_{r+1} - d_{1} , d_{r+2} - d_{2} , ldots , d_n - d_{n-r}$
are distinct for each $r$ , $1 leq r leq n-1$.
Find an example of a Costas array $pi$ for some $n$ such that the $2n-3$ inequalities
$|d_2 - d_1| , |d_3 - d_2| , ldots , |d_n - d_{n-1}| geq 3$
and
$|d_3 - d_1| , |d_4 - d_2| , ldots , |d_n - d_{n-2}| geq 3$
hold or else prove that no such permutation exists.
Remarks: No such Costas arrays have been found using Beard's database of Costas arrays out to order $1030$ (the database is available at https://ieee-dataport.org/open-access/costas-arrays-and-enumeration-order-1030 and is exhaustive out to order $29$.) Costas arrays are known to exist for infinitely many (but not all) orders by constructions involving the infinitude of primes.
combinatorics discrete-mathematics permutations
asked Nov 26 at 16:58
Bill
294
You said $n=11$ doesn't work. What am I missing here: 1 4 7 10 2 5 8 11 3 6 9?
– Todor Markov
Nov 26 at 17:15
1
Ach! I found a flaw in my verification code. Todor Markov your example does work. I am also interested in the same question for the restricted class of permutations known as Costas arrays. I will make the appropriate edits. Thank you.
– Bill
Nov 26 at 18:01
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
A Costas array $pi = d_1 d_2 ldots d_n$ (one-line form) of order $n$ is a permutation $pi in S_n$ such that the $n-r$ differences
$d_{r+1} - d_{1} , d_{r+2} - d_{2} , ldots , d_n - d_{n-r}$
are distinct for each $r$ , $1 leq r leq n-1$.
Find an example of a Costas array $pi$ for some $n$ such that the $2n-3$ inequalities
$|d_2 - d_1| , |d_3 - d_2| , ldots , |d_n - d_{n-1}| geq 3$
and
$|d_3 - d_1| , |d_4 - d_2| , ldots , |d_n - d_{n-2}| geq 3$
hold or else prove that no such permutation exists.
Remarks: No such Costas arrays have been found using Beard's database of Costas arrays out to order $1030$ (the database is available at https://ieee-dataport.org/open-access/costas-arrays-and-enumeration-order-1030 and is exhaustive out to order $29$.) Costas arrays are known to exist for infinitely many (but not all) orders by constructions involving the infinitude of primes.
combinatorics discrete-mathematics permutations
asked Nov 26 at 16:58
Bill
294
A Costas array $pi = d_1 d_2 ldots d_n$ (one-line form) of order $n$ is a permutation $pi in S_n$ such that the $n-r$ differences
$d_{r+1} - d_{1} , d_{r+2} - d_{2} , ldots , d_n - d_{n-r}$
are distinct for each $r$ , $1 leq r leq n-1$.
Find an example of a Costas array $pi$ for some $n$ such that the $2n-3$ inequalities
$|d_2 - d_1| , |d_3 - d_2| , ldots , |d_n - d_{n-1}| geq 3$
and
$|d_3 - d_1| , |d_4 - d_2| , ldots , |d_n - d_{n-2}| geq 3$
hold or else prove that no such permutation exists.
Remarks: No such Costas arrays have been found using Beard's database of Costas arrays out to order $1030$ (the database is available at https://ieee-dataport.org/open-access/costas-arrays-and-enumeration-order-1030 and is exhaustive out to order $29$.) Costas arrays are known to exist for infinitely many (but not all) orders by constructions involving the infinitude of primes.
combinatorics discrete-mathematics permutations
combinatorics discrete-mathematics permutations
asked Nov 26 at 16:58
Bill
294
asked Nov 26 at 16:58
Bill
294
edited Nov 26 at 18:29
asked Nov 26 at 16:58
Bill
294
asked Nov 26 at 16:58
Bill
294
asked Nov 26 at 16:58
Bill
294
294
You said $n=11$ doesn't work. What am I missing here: 1 4 7 10 2 5 8 11 3 6 9?
– Todor Markov
Nov 26 at 17:15
1
Ach! I found a flaw in my verification code. Todor Markov your example does work. I am also interested in the same question for the restricted class of permutations known as Costas arrays. I will make the appropriate edits. Thank you.
– Bill
Nov 26 at 18:01
add a comment |
You said $n=11$ doesn't work. What am I missing here: 1 4 7 10 2 5 8 11 3 6 9?
– Todor Markov
Nov 26 at 17:15
1
Ach! I found a flaw in my verification code. Todor Markov your example does work. I am also interested in the same question for the restricted class of permutations known as Costas arrays. I will make the appropriate edits. Thank you.
– Bill
Nov 26 at 18:01
You said $n=11$ doesn't work. What am I missing here: 1 4 7 10 2 5 8 11 3 6 9?
– Todor Markov
Nov 26 at 17:15
You said $n=11$ doesn't work. What am I missing here: 1 4 7 10 2 5 8 11 3 6 9?
– Todor Markov
Nov 26 at 17:15
1
1
Ach! I found a flaw in my verification code. Todor Markov your example does work. I am also interested in the same question for the restricted class of permutations known as Costas arrays. I will make the appropriate edits. Thank you.
– Bill
Nov 26 at 18:01
Ach! I found a flaw in my verification code. Todor Markov your example does work. I am also interested in the same question for the restricted class of permutations known as Costas arrays. I will make the appropriate edits. Thank you.
– Bill
Nov 26 at 18:01
add a comment |
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You said $n=11$ doesn't work. What am I missing here: 1 4 7 10 2 5 8 11 3 6 9?
– Todor Markov
Nov 26 at 17:15
1
Ach! I found a flaw in my verification code. Todor Markov your example does work. I am also interested in the same question for the restricted class of permutations known as Costas arrays. I will make the appropriate edits. Thank you.
– Bill
Nov 26 at 18:01