which combination(partiotioning ) has the smallest value?
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
166
add a comment |
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
166
The partition that only uses ones (7=1+1+1+1+1+1+1) has value 0, right?
– LinAlg
Dec 1 at 19:50
Hi @LinAlg we can't use 1, because the summation start from 2.
– ilen
Dec 2 at 5:57
Ok, so a value of 0 cannot be attained. $2+1+1+ldots+1$ has value 1.
– LinAlg
Dec 2 at 14:17
add a comment |
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
166
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
166
asked Dec 1 at 3:43
ilen
166
edited Dec 1 at 16:48
asked Dec 1 at 3:43
ilen
166
asked Dec 1 at 3:43
ilen
166
asked Dec 1 at 3:43
ilen
166
166
The partition that only uses ones (7=1+1+1+1+1+1+1) has value 0, right?
– LinAlg
Dec 1 at 19:50
Hi @LinAlg we can't use 1, because the summation start from 2.
– ilen
Dec 2 at 5:57
Ok, so a value of 0 cannot be attained. $2+1+1+ldots+1$ has value 1.
– LinAlg
Dec 2 at 14:17
add a comment |
The partition that only uses ones (7=1+1+1+1+1+1+1) has value 0, right?
– LinAlg
Dec 1 at 19:50
Hi @LinAlg we can't use 1, because the summation start from 2.
– ilen
Dec 2 at 5:57
Ok, so a value of 0 cannot be attained. $2+1+1+ldots+1$ has value 1.
– LinAlg
Dec 2 at 14:17
The partition that only uses ones (7=1+1+1+1+1+1+1) has value 0, right?
– LinAlg
Dec 1 at 19:50
The partition that only uses ones (7=1+1+1+1+1+1+1) has value 0, right?
– LinAlg
Dec 1 at 19:50
Hi @LinAlg we can't use 1, because the summation start from 2.
– ilen
Dec 2 at 5:57
Hi @LinAlg we can't use 1, because the summation start from 2.
– ilen
Dec 2 at 5:57
Ok, so a value of 0 cannot be attained. $2+1+1+ldots+1$ has value 1.
– LinAlg
Dec 2 at 14:17
Ok, so a value of 0 cannot be attained. $2+1+1+ldots+1$ has value 1.
– LinAlg
Dec 2 at 14:17
add a comment |
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The partition that only uses ones (7=1+1+1+1+1+1+1) has value 0, right?
– LinAlg
Dec 1 at 19:50
Hi @LinAlg we can't use 1, because the summation start from 2.
– ilen
Dec 2 at 5:57
Ok, so a value of 0 cannot be attained. $2+1+1+ldots+1$ has value 1.
– LinAlg
Dec 2 at 14:17