Calculating the number of all the possible decreasing sequences made from a set of points?
$begingroup$
Forgive my english, not native, but here it goes:
We are given several values: {a1, a2, a3... ai}
We can create the point Ai only if the sum of its non-negative integer coordinates xi+yi = ai
We also define a decreasing sequence if for each 2 points - Ai(xi, yi) and A(i+1) (x(i+1), y(i+1)), we have xi ≤ x(i+1) and yi ≥ y(i+1). (see image below)
Sequence example
Example:
a1 = 4
a2 = 5
a3 = 3
This should give us 10 possible decreasing sequences.
How could I calculate this? Thank you.
sequences-and-series combinatorics recurrence-relations
asked Jan 17 at 16:27
SciencephileSciencephile
404
$endgroup$
add a comment |
$begingroup$
Forgive my english, not native, but here it goes:
We are given several values: {a1, a2, a3... ai}
We can create the point Ai only if the sum of its non-negative integer coordinates xi+yi = ai
We also define a decreasing sequence if for each 2 points - Ai(xi, yi) and A(i+1) (x(i+1), y(i+1)), we have xi ≤ x(i+1) and yi ≥ y(i+1). (see image below)
Sequence example
Example:
a1 = 4
a2 = 5
a3 = 3
This should give us 10 possible decreasing sequences.
How could I calculate this? Thank you.
sequences-and-series combinatorics recurrence-relations
asked Jan 17 at 16:27
SciencephileSciencephile
404
$endgroup$
$begingroup$
To clarify, the coordinates must be nonnegative integers? That is, choosing $a_1=(-1,5)$ is not allowed?
$endgroup$
– Mike Earnest
Jan 17 at 16:36
$begingroup$
Yes, this is correct, my bad for forgetting.
$endgroup$
– Sciencephile
Jan 17 at 16:38
add a comment |
$begingroup$
Forgive my english, not native, but here it goes:
We are given several values: {a1, a2, a3... ai}
We can create the point Ai only if the sum of its non-negative integer coordinates xi+yi = ai
We also define a decreasing sequence if for each 2 points - Ai(xi, yi) and A(i+1) (x(i+1), y(i+1)), we have xi ≤ x(i+1) and yi ≥ y(i+1). (see image below)
Sequence example
Example:
a1 = 4
a2 = 5
a3 = 3
This should give us 10 possible decreasing sequences.
How could I calculate this? Thank you.
sequences-and-series combinatorics recurrence-relations
asked Jan 17 at 16:27
SciencephileSciencephile
404
$endgroup$
Forgive my english, not native, but here it goes:
We are given several values: {a1, a2, a3... ai}
We can create the point Ai only if the sum of its non-negative integer coordinates xi+yi = ai
We also define a decreasing sequence if for each 2 points - Ai(xi, yi) and A(i+1) (x(i+1), y(i+1)), we have xi ≤ x(i+1) and yi ≥ y(i+1). (see image below)
Sequence example
Example:
a1 = 4
a2 = 5
a3 = 3
This should give us 10 possible decreasing sequences.
How could I calculate this? Thank you.
sequences-and-series combinatorics recurrence-relations
sequences-and-series combinatorics recurrence-relations
asked Jan 17 at 16:27
SciencephileSciencephile
404
asked Jan 17 at 16:27
SciencephileSciencephile
404
edited Jan 17 at 16:38
Sciencephile
asked Jan 17 at 16:27
SciencephileSciencephile
404
asked Jan 17 at 16:27
SciencephileSciencephile
404
asked Jan 17 at 16:27
SciencephileSciencephile
404
404
$begingroup$
To clarify, the coordinates must be nonnegative integers? That is, choosing $a_1=(-1,5)$ is not allowed?
$endgroup$
– Mike Earnest
Jan 17 at 16:36
$begingroup$
Yes, this is correct, my bad for forgetting.
$endgroup$
– Sciencephile
Jan 17 at 16:38
add a comment |
$begingroup$
To clarify, the coordinates must be nonnegative integers? That is, choosing $a_1=(-1,5)$ is not allowed?
$endgroup$
– Mike Earnest
Jan 17 at 16:36
$begingroup$
Yes, this is correct, my bad for forgetting.
$endgroup$
– Sciencephile
Jan 17 at 16:38
$begingroup$
To clarify, the coordinates must be nonnegative integers? That is, choosing $a_1=(-1,5)$ is not allowed?
$endgroup$
– Mike Earnest
Jan 17 at 16:36
$begingroup$
To clarify, the coordinates must be nonnegative integers? That is, choosing $a_1=(-1,5)$ is not allowed?
$endgroup$
– Mike Earnest
Jan 17 at 16:36
$begingroup$
Yes, this is correct, my bad for forgetting.
$endgroup$
– Sciencephile
Jan 17 at 16:38
$begingroup$
Yes, this is correct, my bad for forgetting.
$endgroup$
– Sciencephile
Jan 17 at 16:38
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let $f(k,x)$ be the number of integer sequences $x_1,x_2,ldots, x_k$ with
- $0le x_1le x_2leldotsle x_k=x$
- $a_1-x_1ge a_2-x_2geldots ge a_k-x_kge 0$
You want to calculate $sum_{x=0}^{a_i}f(i,x)$.
You may achieve this by using the recursion
$$f(k+1,x)=begin{cases}0&text{if }x>a_{k+1}\sum_{j=0}^{x} f(k,j)&text{if} xle a_{k+1}le a_k\
sum_{j=0}^{x+a_k-a_{k+1}} f(k,j)&text{otherwise}
end{cases} $$
answered Jan 17 at 16:52
Hagen von EitzenHagen von Eitzen
283k23273508
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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%2f3077198%2fcalculating-the-number-of-all-the-possible-decreasing-sequences-made-from-a-set%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Let $f(k,x)$ be the number of integer sequences $x_1,x_2,ldots, x_k$ with
- $0le x_1le x_2leldotsle x_k=x$
- $a_1-x_1ge a_2-x_2geldots ge a_k-x_kge 0$
You want to calculate $sum_{x=0}^{a_i}f(i,x)$.
You may achieve this by using the recursion
$$f(k+1,x)=begin{cases}0&text{if }x>a_{k+1}\sum_{j=0}^{x} f(k,j)&text{if} xle a_{k+1}le a_k\
sum_{j=0}^{x+a_k-a_{k+1}} f(k,j)&text{otherwise}
end{cases} $$
answered Jan 17 at 16:52
Hagen von EitzenHagen von Eitzen
283k23273508
$endgroup$
add a comment |
$begingroup$
Let $f(k,x)$ be the number of integer sequences $x_1,x_2,ldots, x_k$ with
- $0le x_1le x_2leldotsle x_k=x$
- $a_1-x_1ge a_2-x_2geldots ge a_k-x_kge 0$
You want to calculate $sum_{x=0}^{a_i}f(i,x)$.
You may achieve this by using the recursion
$$f(k+1,x)=begin{cases}0&text{if }x>a_{k+1}\sum_{j=0}^{x} f(k,j)&text{if} xle a_{k+1}le a_k\
sum_{j=0}^{x+a_k-a_{k+1}} f(k,j)&text{otherwise}
end{cases} $$
answered Jan 17 at 16:52
Hagen von EitzenHagen von Eitzen
283k23273508
$endgroup$
add a comment |
$begingroup$
Let $f(k,x)$ be the number of integer sequences $x_1,x_2,ldots, x_k$ with
- $0le x_1le x_2leldotsle x_k=x$
- $a_1-x_1ge a_2-x_2geldots ge a_k-x_kge 0$
You want to calculate $sum_{x=0}^{a_i}f(i,x)$.
You may achieve this by using the recursion
$$f(k+1,x)=begin{cases}0&text{if }x>a_{k+1}\sum_{j=0}^{x} f(k,j)&text{if} xle a_{k+1}le a_k\
sum_{j=0}^{x+a_k-a_{k+1}} f(k,j)&text{otherwise}
end{cases} $$
answered Jan 17 at 16:52
Hagen von EitzenHagen von Eitzen
283k23273508
$endgroup$
Let $f(k,x)$ be the number of integer sequences $x_1,x_2,ldots, x_k$ with
- $0le x_1le x_2leldotsle x_k=x$
- $a_1-x_1ge a_2-x_2geldots ge a_k-x_kge 0$
You want to calculate $sum_{x=0}^{a_i}f(i,x)$.
You may achieve this by using the recursion
$$f(k+1,x)=begin{cases}0&text{if }x>a_{k+1}\sum_{j=0}^{x} f(k,j)&text{if} xle a_{k+1}le a_k\
sum_{j=0}^{x+a_k-a_{k+1}} f(k,j)&text{otherwise}
end{cases} $$
answered Jan 17 at 16:52
Hagen von EitzenHagen von Eitzen
283k23273508
answered Jan 17 at 16:52
Hagen von EitzenHagen von Eitzen
283k23273508
answered Jan 17 at 16:52
Hagen von EitzenHagen von Eitzen
283k23273508
answered Jan 17 at 16:52
Hagen von EitzenHagen von Eitzen
283k23273508
283k23273508
add a comment |
add a comment |
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.
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%2f3077198%2fcalculating-the-number-of-all-the-possible-decreasing-sequences-made-from-a-set%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
$begingroup$
To clarify, the coordinates must be nonnegative integers? That is, choosing $a_1=(-1,5)$ is not allowed?
$endgroup$
– Mike Earnest
Jan 17 at 16:36
$begingroup$
Yes, this is correct, my bad for forgetting.
$endgroup$
– Sciencephile
Jan 17 at 16:38