Examples of strictly monotone functions mapping all positive integer sequences with finite support to...
up vote
0
down vote
favorite
Are there interesting examples of strictly monotone total functions between the positive integer sequences with finite support and the positive integers themselves, aside from the prime exponent multiplication (canonical representation evaluation)? The function should be minimal in the sense that it is not produced by composing another function of this kind with a strictly monotone map between integer sequences.
If there are multitude such functions, are there ones with easier or harder to compute preimage (in the asymptotic complexity sense, e.g. when compared to prime number factorization)?
Edit: The ordering of the integer sequences in the function's domain is assumed to be lexicographic product ordering.
(SW engineer, CS undergraduate background. Thanks.)
number-theory prime-factorization
edited Nov 23 at 11:57
amWhy
191k27223439
asked Feb 21 at 13:00
simeonz
285
|
show 1 more comment
up vote
0
down vote
favorite
Are there interesting examples of strictly monotone total functions between the positive integer sequences with finite support and the positive integers themselves, aside from the prime exponent multiplication (canonical representation evaluation)? The function should be minimal in the sense that it is not produced by composing another function of this kind with a strictly monotone map between integer sequences.
If there are multitude such functions, are there ones with easier or harder to compute preimage (in the asymptotic complexity sense, e.g. when compared to prime number factorization)?
Edit: The ordering of the integer sequences in the function's domain is assumed to be lexicographic product ordering.
(SW engineer, CS undergraduate background. Thanks.)
number-theory prime-factorization
edited Nov 23 at 11:57
amWhy
191k27223439
asked Feb 21 at 13:00
simeonz
285
What's the ordering on the set of sequences?
– 5xum
Feb 21 at 13:34
@5xum Yes, I noticed the omission, and made an edit after the original post. I am improvising, so hopefully the requirements are not so weak as to the point of being trivial. I tried to eliminate trivial compositions, such as with monotone functions and permutations, but something else may slip through the cracks. Or I may be confused about the entire issue altogether.
– simeonz
Feb 21 at 13:44
@5xum: After some consideration, my edit was erroneous. Lexicographic ordering was useful for banning trivial compositions, but does not apply to canonical representations, so i'll correct this.
– simeonz
Feb 21 at 14:21
The product ordering only applies to sequences of equal length, doesn't it? What's the ordering on sequences of different lengths?
– Rahul
Nov 23 at 12:00
My idea was that the product ordering should include the zeros to the extent of the longer support, acting on them as infinitary Cartesian product (which they are, albeit with finite support). The truth is that I still have hard time formulating the question properly, partly due to lack of mathematical maturity on my side, and also because the question is difficult in itself. I just want to find out whether there are other bijective monotonic representations of numbers as sequences with finite support, similar to the prime factorization, but not trivially correspondent to it through remapping.
– simeonz
Nov 29 at 22:40
|
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Are there interesting examples of strictly monotone total functions between the positive integer sequences with finite support and the positive integers themselves, aside from the prime exponent multiplication (canonical representation evaluation)? The function should be minimal in the sense that it is not produced by composing another function of this kind with a strictly monotone map between integer sequences.
If there are multitude such functions, are there ones with easier or harder to compute preimage (in the asymptotic complexity sense, e.g. when compared to prime number factorization)?
Edit: The ordering of the integer sequences in the function's domain is assumed to be lexicographic product ordering.
(SW engineer, CS undergraduate background. Thanks.)
number-theory prime-factorization
edited Nov 23 at 11:57
amWhy
191k27223439
asked Feb 21 at 13:00
simeonz
285
Are there interesting examples of strictly monotone total functions between the positive integer sequences with finite support and the positive integers themselves, aside from the prime exponent multiplication (canonical representation evaluation)? The function should be minimal in the sense that it is not produced by composing another function of this kind with a strictly monotone map between integer sequences.
If there are multitude such functions, are there ones with easier or harder to compute preimage (in the asymptotic complexity sense, e.g. when compared to prime number factorization)?
Edit: The ordering of the integer sequences in the function's domain is assumed to be lexicographic product ordering.
(SW engineer, CS undergraduate background. Thanks.)
number-theory prime-factorization
number-theory prime-factorization
edited Nov 23 at 11:57
amWhy
191k27223439
asked Feb 21 at 13:00
simeonz
285
edited Nov 23 at 11:57
amWhy
191k27223439
asked Feb 21 at 13:00
simeonz
285
edited Nov 23 at 11:57
amWhy
191k27223439
edited Nov 23 at 11:57
amWhy
191k27223439
edited Nov 23 at 11:57
amWhy
191k27223439
191k27223439
asked Feb 21 at 13:00
simeonz
285
asked Feb 21 at 13:00
simeonz
285
asked Feb 21 at 13:00
simeonz
285
285
What's the ordering on the set of sequences?
– 5xum
Feb 21 at 13:34
@5xum Yes, I noticed the omission, and made an edit after the original post. I am improvising, so hopefully the requirements are not so weak as to the point of being trivial. I tried to eliminate trivial compositions, such as with monotone functions and permutations, but something else may slip through the cracks. Or I may be confused about the entire issue altogether.
– simeonz
Feb 21 at 13:44
@5xum: After some consideration, my edit was erroneous. Lexicographic ordering was useful for banning trivial compositions, but does not apply to canonical representations, so i'll correct this.
– simeonz
Feb 21 at 14:21
The product ordering only applies to sequences of equal length, doesn't it? What's the ordering on sequences of different lengths?
– Rahul
Nov 23 at 12:00
My idea was that the product ordering should include the zeros to the extent of the longer support, acting on them as infinitary Cartesian product (which they are, albeit with finite support). The truth is that I still have hard time formulating the question properly, partly due to lack of mathematical maturity on my side, and also because the question is difficult in itself. I just want to find out whether there are other bijective monotonic representations of numbers as sequences with finite support, similar to the prime factorization, but not trivially correspondent to it through remapping.
– simeonz
Nov 29 at 22:40
|
show 1 more comment
What's the ordering on the set of sequences?
– 5xum
Feb 21 at 13:34
@5xum Yes, I noticed the omission, and made an edit after the original post. I am improvising, so hopefully the requirements are not so weak as to the point of being trivial. I tried to eliminate trivial compositions, such as with monotone functions and permutations, but something else may slip through the cracks. Or I may be confused about the entire issue altogether.
– simeonz
Feb 21 at 13:44
@5xum: After some consideration, my edit was erroneous. Lexicographic ordering was useful for banning trivial compositions, but does not apply to canonical representations, so i'll correct this.
– simeonz
Feb 21 at 14:21
The product ordering only applies to sequences of equal length, doesn't it? What's the ordering on sequences of different lengths?
– Rahul
Nov 23 at 12:00
My idea was that the product ordering should include the zeros to the extent of the longer support, acting on them as infinitary Cartesian product (which they are, albeit with finite support). The truth is that I still have hard time formulating the question properly, partly due to lack of mathematical maturity on my side, and also because the question is difficult in itself. I just want to find out whether there are other bijective monotonic representations of numbers as sequences with finite support, similar to the prime factorization, but not trivially correspondent to it through remapping.
– simeonz
Nov 29 at 22:40
What's the ordering on the set of sequences?
– 5xum
Feb 21 at 13:34
What's the ordering on the set of sequences?
– 5xum
Feb 21 at 13:34
@5xum Yes, I noticed the omission, and made an edit after the original post. I am improvising, so hopefully the requirements are not so weak as to the point of being trivial. I tried to eliminate trivial compositions, such as with monotone functions and permutations, but something else may slip through the cracks. Or I may be confused about the entire issue altogether.
– simeonz
Feb 21 at 13:44
@5xum Yes, I noticed the omission, and made an edit after the original post. I am improvising, so hopefully the requirements are not so weak as to the point of being trivial. I tried to eliminate trivial compositions, such as with monotone functions and permutations, but something else may slip through the cracks. Or I may be confused about the entire issue altogether.
– simeonz
Feb 21 at 13:44
@5xum: After some consideration, my edit was erroneous. Lexicographic ordering was useful for banning trivial compositions, but does not apply to canonical representations, so i'll correct this.
– simeonz
Feb 21 at 14:21
@5xum: After some consideration, my edit was erroneous. Lexicographic ordering was useful for banning trivial compositions, but does not apply to canonical representations, so i'll correct this.
– simeonz
Feb 21 at 14:21
The product ordering only applies to sequences of equal length, doesn't it? What's the ordering on sequences of different lengths?
– Rahul
Nov 23 at 12:00
The product ordering only applies to sequences of equal length, doesn't it? What's the ordering on sequences of different lengths?
– Rahul
Nov 23 at 12:00
My idea was that the product ordering should include the zeros to the extent of the longer support, acting on them as infinitary Cartesian product (which they are, albeit with finite support). The truth is that I still have hard time formulating the question properly, partly due to lack of mathematical maturity on my side, and also because the question is difficult in itself. I just want to find out whether there are other bijective monotonic representations of numbers as sequences with finite support, similar to the prime factorization, but not trivially correspondent to it through remapping.
– simeonz
Nov 29 at 22:40
My idea was that the product ordering should include the zeros to the extent of the longer support, acting on them as infinitary Cartesian product (which they are, albeit with finite support). The truth is that I still have hard time formulating the question properly, partly due to lack of mathematical maturity on my side, and also because the question is difficult in itself. I just want to find out whether there are other bijective monotonic representations of numbers as sequences with finite support, similar to the prime factorization, but not trivially correspondent to it through remapping.
– simeonz
Nov 29 at 22:40
|
show 1 more comment
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
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%2f2660215%2fexamples-of-strictly-monotone-functions-mapping-all-positive-integer-sequences-w%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
What's the ordering on the set of sequences?
– 5xum
Feb 21 at 13:34
@5xum Yes, I noticed the omission, and made an edit after the original post. I am improvising, so hopefully the requirements are not so weak as to the point of being trivial. I tried to eliminate trivial compositions, such as with monotone functions and permutations, but something else may slip through the cracks. Or I may be confused about the entire issue altogether.
– simeonz
Feb 21 at 13:44
@5xum: After some consideration, my edit was erroneous. Lexicographic ordering was useful for banning trivial compositions, but does not apply to canonical representations, so i'll correct this.
– simeonz
Feb 21 at 14:21
The product ordering only applies to sequences of equal length, doesn't it? What's the ordering on sequences of different lengths?
– Rahul
Nov 23 at 12:00
My idea was that the product ordering should include the zeros to the extent of the longer support, acting on them as infinitary Cartesian product (which they are, albeit with finite support). The truth is that I still have hard time formulating the question properly, partly due to lack of mathematical maturity on my side, and also because the question is difficult in itself. I just want to find out whether there are other bijective monotonic representations of numbers as sequences with finite support, similar to the prime factorization, but not trivially correspondent to it through remapping.
– simeonz
Nov 29 at 22:40