which combination(partiotioning ) has the smallest value?
up vote
0
down vote
favorite
A positive integer can be partitioned, for example, the number 7 can be partitioned into the following:
$7=7$
$ 7=6+1$ , $ 7=5+2$,$ 7=4+3$
$ 7=4+2+1$,$ 7=3+3+1$,$ 7=3+2+2$,
$ 7=2+2+2+1$,...
I consider $n_k$ as the number of times that a number is used. For example, in partitioning $ 7 = 3 + 2 + 2$, we have $n_2 = 2$ and $ n_3 = 1$
suppose $K$ as largest element in every partiotioning , For example, in partitioning $ 7 = 3 + 2 + 2$ , $K$ is $3$ , and in the partitioning $ 7 = 5 + 2 $ , we have $K=5$ .
let $dbinom{1}{2}=0$ ,I want to know from all the above combinations Which one have smaller $P=sum_{k=1}^K dbinom{k}{2} n_k$ .(For example, in partitioning $ 7 = 3 + 2 + 2$ , this value is $P=dbinom{3}{2} +2* dbinom{2}{2} = 5 $)
I mean, which combination(partiotioning ) has the smallest value of $P=sum_{k=2}^K dbinom{k}{2} n_k$ ?
thanks
optimization linear-programming integer-programming mixed-integer-programming
asked Dec 1 at 3:43
ilen
113
add a comment |
up vote
0
down vote
favorite
A positive integer can be partitioned, for example, the number 7 can be partitioned into the following:
$7=7$
$ 7=6+1$ , $ 7=5+2$,$ 7=4+3$
$ 7=4+2+1$,$ 7=3+3+1$,$ 7=3+2+2$,
$ 7=2+2+2+1$,...
I consider $n_k$ as the number of times that a number is used. For example, in partitioning $ 7 = 3 + 2 + 2$, we have $n_2 = 2$ and $ n_3 = 1$
suppose $K$ as largest element in every partiotioning , For example, in partitioning $ 7 = 3 + 2 + 2$ , $K$ is $3$ , and in the partitioning $ 7 = 5 + 2 $ , we have $K=5$ .
let $dbinom{1}{2}=0$ ,I want to know from all the above combinations Which one have smaller $P=sum_{k=1}^K dbinom{k}{2} n_k$ .(For example, in partitioning $ 7 = 3 + 2 + 2$ , this value is $P=dbinom{3}{2} +2* dbinom{2}{2} = 5 $)
I mean, which combination(partiotioning ) has the smallest value of $P=sum_{k=2}^K dbinom{k}{2} n_k$ ?
thanks
optimization linear-programming integer-programming mixed-integer-programming
asked Dec 1 at 3:43
ilen
113
The partition that only uses ones (7=1+1+1+1+1+1+1) has value 0, right?
– LinAlg
2 days ago
Hi @LinAlg we can't use 1, because the summation start from 2.
– ilen
2 days ago
Ok, so a value of 0 cannot be attained. $2+1+1+ldots+1$ has value 1.
– LinAlg
yesterday
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
A positive integer can be partitioned, for example, the number 7 can be partitioned into the following:
$7=7$
$ 7=6+1$ , $ 7=5+2$,$ 7=4+3$
$ 7=4+2+1$,$ 7=3+3+1$,$ 7=3+2+2$,
$ 7=2+2+2+1$,...
I consider $n_k$ as the number of times that a number is used. For example, in partitioning $ 7 = 3 + 2 + 2$, we have $n_2 = 2$ and $ n_3 = 1$
suppose $K$ as largest element in every partiotioning , For example, in partitioning $ 7 = 3 + 2 + 2$ , $K$ is $3$ , and in the partitioning $ 7 = 5 + 2 $ , we have $K=5$ .
let $dbinom{1}{2}=0$ ,I want to know from all the above combinations Which one have smaller $P=sum_{k=1}^K dbinom{k}{2} n_k$ .(For example, in partitioning $ 7 = 3 + 2 + 2$ , this value is $P=dbinom{3}{2} +2* dbinom{2}{2} = 5 $)
I mean, which combination(partiotioning ) has the smallest value of $P=sum_{k=2}^K dbinom{k}{2} n_k$ ?
thanks
optimization linear-programming integer-programming mixed-integer-programming
asked Dec 1 at 3:43
ilen
113
A positive integer can be partitioned, for example, the number 7 can be partitioned into the following:
$7=7$
$ 7=6+1$ , $ 7=5+2$,$ 7=4+3$
$ 7=4+2+1$,$ 7=3+3+1$,$ 7=3+2+2$,
$ 7=2+2+2+1$,...
I consider $n_k$ as the number of times that a number is used. For example, in partitioning $ 7 = 3 + 2 + 2$, we have $n_2 = 2$ and $ n_3 = 1$
suppose $K$ as largest element in every partiotioning , For example, in partitioning $ 7 = 3 + 2 + 2$ , $K$ is $3$ , and in the partitioning $ 7 = 5 + 2 $ , we have $K=5$ .
let $dbinom{1}{2}=0$ ,I want to know from all the above combinations Which one have smaller $P=sum_{k=1}^K dbinom{k}{2} n_k$ .(For example, in partitioning $ 7 = 3 + 2 + 2$ , this value is $P=dbinom{3}{2} +2* dbinom{2}{2} = 5 $)
I mean, which combination(partiotioning ) has the smallest value of $P=sum_{k=2}^K dbinom{k}{2} n_k$ ?
thanks
optimization linear-programming integer-programming mixed-integer-programming
optimization linear-programming integer-programming mixed-integer-programming
asked Dec 1 at 3:43
ilen
113
asked Dec 1 at 3:43
ilen
113
edited 2 days ago
asked Dec 1 at 3:43
ilen
113
asked Dec 1 at 3:43
ilen
113
asked Dec 1 at 3:43
ilen
113
113
The partition that only uses ones (7=1+1+1+1+1+1+1) has value 0, right?
– LinAlg
2 days ago
Hi @LinAlg we can't use 1, because the summation start from 2.
– ilen
2 days ago
Ok, so a value of 0 cannot be attained. $2+1+1+ldots+1$ has value 1.
– LinAlg
yesterday
add a comment |
The partition that only uses ones (7=1+1+1+1+1+1+1) has value 0, right?
– LinAlg
2 days ago
Hi @LinAlg we can't use 1, because the summation start from 2.
– ilen
2 days ago
Ok, so a value of 0 cannot be attained. $2+1+1+ldots+1$ has value 1.
– LinAlg
yesterday
The partition that only uses ones (7=1+1+1+1+1+1+1) has value 0, right?
– LinAlg
2 days ago
The partition that only uses ones (7=1+1+1+1+1+1+1) has value 0, right?
– LinAlg
2 days ago
Hi @LinAlg we can't use 1, because the summation start from 2.
– ilen
2 days ago
Hi @LinAlg we can't use 1, because the summation start from 2.
– ilen
2 days ago
Ok, so a value of 0 cannot be attained. $2+1+1+ldots+1$ has value 1.
– LinAlg
yesterday
Ok, so a value of 0 cannot be attained. $2+1+1+ldots+1$ has value 1.
– LinAlg
yesterday
add a 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%2f3020957%2fwhich-combinationpartiotioning-has-the-smallest-value%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
The partition that only uses ones (7=1+1+1+1+1+1+1) has value 0, right?
– LinAlg
2 days ago
Hi @LinAlg we can't use 1, because the summation start from 2.
– ilen
2 days ago
Ok, so a value of 0 cannot be attained. $2+1+1+ldots+1$ has value 1.
– LinAlg
yesterday