Produce formula
Let $a_{i,j}$ any positive integers wher $1leq i,j leq 3$.
I want to write closed form for following summation:
$$a_{1,3}(a_{2,1}+a_{2, 2}+a_{2,3})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{2,3}(a_{1,1}+a_{1, 2})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{3,3}(a_{1,1}+a_{1, 2})(a_{2,1}+a_{2, 2})$$
$textbf{My attempt:}$
begin{equation*}
sum_{j=1}^3 a_{j,3}left[prod_{substack{i=1 \ ineq j}}^3left(sum_{substack{k=1 \ k<j}}^{3 text{or} 3-1} a_{i,k} right) right]
end{equation*}
We will take $3-1$ when $k<j$.
Actually, I want to generalize this. My attempt so complicated to me.
If we consider all of $a_{i,j}$ as a matrix:
begin{pmatrix}
a_{1,1} & a_{1,2} & a_{1,3} \
a_{2,1} & a_{2,2} & a_{2,3} \
a_{3,1} & a_{3,2} & a_{3,3} \
end{pmatrix}
then are there any computer program to make formula?
$$a_{1,3}(a_{2,1}+a_{2, 2}+ a_{2,3})(a_{3,1}+a_{3, 2}+a_{3,3})cdots (a_{r,1}+a_{r,2}+a_{r,3})+a_{2,3}(a_{1,1}+a_{1, 2})(a_{3,1}+a_{3, 2}+a_{3,3})+cdots (a_{r,1}+a_{r,2}+a_{r,3})+a_{3,3}(a_{1,1}+a_{1, 2})(a_{2,1}+a_{2, 2})(a_{4,1}+a_{4,2}+a_{4,3}+cdots (a_{r,1}+a_{r,2}+a_{r,3}) \ vdots \
+a_{r,3}(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})cdots (a_{r-1,1}+a_{r-1,2})
$$
combinatorics combinations
add a comment |
Let $a_{i,j}$ any positive integers wher $1leq i,j leq 3$.
I want to write closed form for following summation:
$$a_{1,3}(a_{2,1}+a_{2, 2}+a_{2,3})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{2,3}(a_{1,1}+a_{1, 2})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{3,3}(a_{1,1}+a_{1, 2})(a_{2,1}+a_{2, 2})$$
$textbf{My attempt:}$
begin{equation*}
sum_{j=1}^3 a_{j,3}left[prod_{substack{i=1 \ ineq j}}^3left(sum_{substack{k=1 \ k<j}}^{3 text{or} 3-1} a_{i,k} right) right]
end{equation*}
We will take $3-1$ when $k<j$.
Actually, I want to generalize this. My attempt so complicated to me.
If we consider all of $a_{i,j}$ as a matrix:
begin{pmatrix}
a_{1,1} & a_{1,2} & a_{1,3} \
a_{2,1} & a_{2,2} & a_{2,3} \
a_{3,1} & a_{3,2} & a_{3,3} \
end{pmatrix}
then are there any computer program to make formula?
$$a_{1,3}(a_{2,1}+a_{2, 2}+ a_{2,3})(a_{3,1}+a_{3, 2}+a_{3,3})cdots (a_{r,1}+a_{r,2}+a_{r,3})+a_{2,3}(a_{1,1}+a_{1, 2})(a_{3,1}+a_{3, 2}+a_{3,3})+cdots (a_{r,1}+a_{r,2}+a_{r,3})+a_{3,3}(a_{1,1}+a_{1, 2})(a_{2,1}+a_{2, 2})(a_{4,1}+a_{4,2}+a_{4,3}+cdots (a_{r,1}+a_{r,2}+a_{r,3}) \ vdots \
+a_{r,3}(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})cdots (a_{r-1,1}+a_{r-1,2})
$$
combinatorics combinations
add a comment |
Let $a_{i,j}$ any positive integers wher $1leq i,j leq 3$.
I want to write closed form for following summation:
$$a_{1,3}(a_{2,1}+a_{2, 2}+a_{2,3})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{2,3}(a_{1,1}+a_{1, 2})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{3,3}(a_{1,1}+a_{1, 2})(a_{2,1}+a_{2, 2})$$
$textbf{My attempt:}$
begin{equation*}
sum_{j=1}^3 a_{j,3}left[prod_{substack{i=1 \ ineq j}}^3left(sum_{substack{k=1 \ k<j}}^{3 text{or} 3-1} a_{i,k} right) right]
end{equation*}
We will take $3-1$ when $k<j$.
Actually, I want to generalize this. My attempt so complicated to me.
If we consider all of $a_{i,j}$ as a matrix:
begin{pmatrix}
a_{1,1} & a_{1,2} & a_{1,3} \
a_{2,1} & a_{2,2} & a_{2,3} \
a_{3,1} & a_{3,2} & a_{3,3} \
end{pmatrix}
then are there any computer program to make formula?
$$a_{1,3}(a_{2,1}+a_{2, 2}+ a_{2,3})(a_{3,1}+a_{3, 2}+a_{3,3})cdots (a_{r,1}+a_{r,2}+a_{r,3})+a_{2,3}(a_{1,1}+a_{1, 2})(a_{3,1}+a_{3, 2}+a_{3,3})+cdots (a_{r,1}+a_{r,2}+a_{r,3})+a_{3,3}(a_{1,1}+a_{1, 2})(a_{2,1}+a_{2, 2})(a_{4,1}+a_{4,2}+a_{4,3}+cdots (a_{r,1}+a_{r,2}+a_{r,3}) \ vdots \
+a_{r,3}(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})cdots (a_{r-1,1}+a_{r-1,2})
$$
combinatorics combinations
Let $a_{i,j}$ any positive integers wher $1leq i,j leq 3$.
I want to write closed form for following summation:
$$a_{1,3}(a_{2,1}+a_{2, 2}+a_{2,3})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{2,3}(a_{1,1}+a_{1, 2})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{3,3}(a_{1,1}+a_{1, 2})(a_{2,1}+a_{2, 2})$$
$textbf{My attempt:}$
begin{equation*}
sum_{j=1}^3 a_{j,3}left[prod_{substack{i=1 \ ineq j}}^3left(sum_{substack{k=1 \ k<j}}^{3 text{or} 3-1} a_{i,k} right) right]
end{equation*}
We will take $3-1$ when $k<j$.
Actually, I want to generalize this. My attempt so complicated to me.
If we consider all of $a_{i,j}$ as a matrix:
begin{pmatrix}
a_{1,1} & a_{1,2} & a_{1,3} \
a_{2,1} & a_{2,2} & a_{2,3} \
a_{3,1} & a_{3,2} & a_{3,3} \
end{pmatrix}
then are there any computer program to make formula?
$$a_{1,3}(a_{2,1}+a_{2, 2}+ a_{2,3})(a_{3,1}+a_{3, 2}+a_{3,3})cdots (a_{r,1}+a_{r,2}+a_{r,3})+a_{2,3}(a_{1,1}+a_{1, 2})(a_{3,1}+a_{3, 2}+a_{3,3})+cdots (a_{r,1}+a_{r,2}+a_{r,3})+a_{3,3}(a_{1,1}+a_{1, 2})(a_{2,1}+a_{2, 2})(a_{4,1}+a_{4,2}+a_{4,3}+cdots (a_{r,1}+a_{r,2}+a_{r,3}) \ vdots \
+a_{r,3}(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})cdots (a_{r-1,1}+a_{r-1,2})
$$
combinatorics combinations
combinatorics combinations
edited Nov 27 at 13:23
asked Nov 27 at 10:57
1Spectre1
999
999
add a comment |
add a comment |
1 Answer
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$$a_{1,3}(a_{2,1}+a_{2, 2}+a_{2,3})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{2,3}(a_{1,1}+a_{1, 2})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{3,3}(a_{1,1}+a_{1, 2})(a_{2,1}+a_{2, 2})$$
$$=(a_{1,1}+a_{1,2}+a_{1,3})(a_{2,1}+a_{2,2}+a_{2,3})(a_{3,1}+a_{3,2}+a_{3,3})-(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})(a_{3,1}+a_{3,2})$$
$$=prod_{i=1}^3sum_{j=1}^3a_{i,j}-prod_{i=1}^3sum_{j=1}^2a_{i,j}.$$
More generally,
$$sum_{i=1}^ra_{i,m}left(prod_{h=1}^{i-1}sum_{j=1}^{m-1}a_{h,j}right)left(prod_{h=i+1}^rsum_{j=1}^ma_{h,j}right)=prod_{i=1}^rsum_{j=1}^ma_{i,j}-prod_{i=1}^rsum_{j=1}^{m-1}a_{i,j}$$
$$=sum_{min{j_1,j_2,dots,j_r}}a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}.$$
Thank you for the answer. Is it possible to generalize it? I edited my question.
– 1Spectre1
Nov 27 at 13:24
1
@1ENİGMA1 I cleaned up my answer.
– bof
Nov 28 at 23:13
1
For example, $n!=prod_{i=1}^ni$ and in particular $0!=prod_{i=1}^0i=1$. Likewise, $x^n=prod_{i=1}^nx$, in particular, $x^0=prod_{i=1}^0x=1$.
– bof
Nov 29 at 8:59
1
$sum_{min{j_1,j_2,dots,j_r}}a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}$ is the sum of all products of the form $a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}$ where at least one of the indices $j_1,j_2,dots,j_r$ is equal to $m$, or in other words, $m$ is an element of the set ${j_1,j_2,dots,j_r}$.
– bof
Nov 29 at 9:04
1
$$prod_{i=1}^2sum_{j=1}^2a_{i,j}=(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})=a_{1,1}a_{2,1}+a_{1,1}a_{2,2}+a_{1,2}a_{2,1}+a_{1,2}a_{2,2}$$ while $$sum_{2in{j_1,j_2}}a_{1,j_1}a_{2,j_2}=a_{1,1}a_{2,2}+a_{1,2}a_{2,1}+a_{1,2}a_{2,2}.$$ The term $a_{1,1}a_{2,1}$ is gone because $2notin{j_1,j_2}={1,1}$.
– bof
Nov 30 at 6:57
|
show 8 more comments
Your Answer
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$$a_{1,3}(a_{2,1}+a_{2, 2}+a_{2,3})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{2,3}(a_{1,1}+a_{1, 2})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{3,3}(a_{1,1}+a_{1, 2})(a_{2,1}+a_{2, 2})$$
$$=(a_{1,1}+a_{1,2}+a_{1,3})(a_{2,1}+a_{2,2}+a_{2,3})(a_{3,1}+a_{3,2}+a_{3,3})-(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})(a_{3,1}+a_{3,2})$$
$$=prod_{i=1}^3sum_{j=1}^3a_{i,j}-prod_{i=1}^3sum_{j=1}^2a_{i,j}.$$
More generally,
$$sum_{i=1}^ra_{i,m}left(prod_{h=1}^{i-1}sum_{j=1}^{m-1}a_{h,j}right)left(prod_{h=i+1}^rsum_{j=1}^ma_{h,j}right)=prod_{i=1}^rsum_{j=1}^ma_{i,j}-prod_{i=1}^rsum_{j=1}^{m-1}a_{i,j}$$
$$=sum_{min{j_1,j_2,dots,j_r}}a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}.$$
Thank you for the answer. Is it possible to generalize it? I edited my question.
– 1Spectre1
Nov 27 at 13:24
1
@1ENİGMA1 I cleaned up my answer.
– bof
Nov 28 at 23:13
1
For example, $n!=prod_{i=1}^ni$ and in particular $0!=prod_{i=1}^0i=1$. Likewise, $x^n=prod_{i=1}^nx$, in particular, $x^0=prod_{i=1}^0x=1$.
– bof
Nov 29 at 8:59
1
$sum_{min{j_1,j_2,dots,j_r}}a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}$ is the sum of all products of the form $a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}$ where at least one of the indices $j_1,j_2,dots,j_r$ is equal to $m$, or in other words, $m$ is an element of the set ${j_1,j_2,dots,j_r}$.
– bof
Nov 29 at 9:04
1
$$prod_{i=1}^2sum_{j=1}^2a_{i,j}=(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})=a_{1,1}a_{2,1}+a_{1,1}a_{2,2}+a_{1,2}a_{2,1}+a_{1,2}a_{2,2}$$ while $$sum_{2in{j_1,j_2}}a_{1,j_1}a_{2,j_2}=a_{1,1}a_{2,2}+a_{1,2}a_{2,1}+a_{1,2}a_{2,2}.$$ The term $a_{1,1}a_{2,1}$ is gone because $2notin{j_1,j_2}={1,1}$.
– bof
Nov 30 at 6:57
|
show 8 more comments
$$a_{1,3}(a_{2,1}+a_{2, 2}+a_{2,3})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{2,3}(a_{1,1}+a_{1, 2})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{3,3}(a_{1,1}+a_{1, 2})(a_{2,1}+a_{2, 2})$$
$$=(a_{1,1}+a_{1,2}+a_{1,3})(a_{2,1}+a_{2,2}+a_{2,3})(a_{3,1}+a_{3,2}+a_{3,3})-(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})(a_{3,1}+a_{3,2})$$
$$=prod_{i=1}^3sum_{j=1}^3a_{i,j}-prod_{i=1}^3sum_{j=1}^2a_{i,j}.$$
More generally,
$$sum_{i=1}^ra_{i,m}left(prod_{h=1}^{i-1}sum_{j=1}^{m-1}a_{h,j}right)left(prod_{h=i+1}^rsum_{j=1}^ma_{h,j}right)=prod_{i=1}^rsum_{j=1}^ma_{i,j}-prod_{i=1}^rsum_{j=1}^{m-1}a_{i,j}$$
$$=sum_{min{j_1,j_2,dots,j_r}}a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}.$$
Thank you for the answer. Is it possible to generalize it? I edited my question.
– 1Spectre1
Nov 27 at 13:24
1
@1ENİGMA1 I cleaned up my answer.
– bof
Nov 28 at 23:13
1
For example, $n!=prod_{i=1}^ni$ and in particular $0!=prod_{i=1}^0i=1$. Likewise, $x^n=prod_{i=1}^nx$, in particular, $x^0=prod_{i=1}^0x=1$.
– bof
Nov 29 at 8:59
1
$sum_{min{j_1,j_2,dots,j_r}}a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}$ is the sum of all products of the form $a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}$ where at least one of the indices $j_1,j_2,dots,j_r$ is equal to $m$, or in other words, $m$ is an element of the set ${j_1,j_2,dots,j_r}$.
– bof
Nov 29 at 9:04
1
$$prod_{i=1}^2sum_{j=1}^2a_{i,j}=(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})=a_{1,1}a_{2,1}+a_{1,1}a_{2,2}+a_{1,2}a_{2,1}+a_{1,2}a_{2,2}$$ while $$sum_{2in{j_1,j_2}}a_{1,j_1}a_{2,j_2}=a_{1,1}a_{2,2}+a_{1,2}a_{2,1}+a_{1,2}a_{2,2}.$$ The term $a_{1,1}a_{2,1}$ is gone because $2notin{j_1,j_2}={1,1}$.
– bof
Nov 30 at 6:57
|
show 8 more comments
$$a_{1,3}(a_{2,1}+a_{2, 2}+a_{2,3})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{2,3}(a_{1,1}+a_{1, 2})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{3,3}(a_{1,1}+a_{1, 2})(a_{2,1}+a_{2, 2})$$
$$=(a_{1,1}+a_{1,2}+a_{1,3})(a_{2,1}+a_{2,2}+a_{2,3})(a_{3,1}+a_{3,2}+a_{3,3})-(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})(a_{3,1}+a_{3,2})$$
$$=prod_{i=1}^3sum_{j=1}^3a_{i,j}-prod_{i=1}^3sum_{j=1}^2a_{i,j}.$$
More generally,
$$sum_{i=1}^ra_{i,m}left(prod_{h=1}^{i-1}sum_{j=1}^{m-1}a_{h,j}right)left(prod_{h=i+1}^rsum_{j=1}^ma_{h,j}right)=prod_{i=1}^rsum_{j=1}^ma_{i,j}-prod_{i=1}^rsum_{j=1}^{m-1}a_{i,j}$$
$$=sum_{min{j_1,j_2,dots,j_r}}a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}.$$
$$a_{1,3}(a_{2,1}+a_{2, 2}+a_{2,3})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{2,3}(a_{1,1}+a_{1, 2})(a_{3,1}+a_{3, 2}+a_{3,3})+a_{3,3}(a_{1,1}+a_{1, 2})(a_{2,1}+a_{2, 2})$$
$$=(a_{1,1}+a_{1,2}+a_{1,3})(a_{2,1}+a_{2,2}+a_{2,3})(a_{3,1}+a_{3,2}+a_{3,3})-(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})(a_{3,1}+a_{3,2})$$
$$=prod_{i=1}^3sum_{j=1}^3a_{i,j}-prod_{i=1}^3sum_{j=1}^2a_{i,j}.$$
More generally,
$$sum_{i=1}^ra_{i,m}left(prod_{h=1}^{i-1}sum_{j=1}^{m-1}a_{h,j}right)left(prod_{h=i+1}^rsum_{j=1}^ma_{h,j}right)=prod_{i=1}^rsum_{j=1}^ma_{i,j}-prod_{i=1}^rsum_{j=1}^{m-1}a_{i,j}$$
$$=sum_{min{j_1,j_2,dots,j_r}}a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}.$$
edited Nov 28 at 23:11
answered Nov 27 at 12:27
bof
50.1k457119
50.1k457119
Thank you for the answer. Is it possible to generalize it? I edited my question.
– 1Spectre1
Nov 27 at 13:24
1
@1ENİGMA1 I cleaned up my answer.
– bof
Nov 28 at 23:13
1
For example, $n!=prod_{i=1}^ni$ and in particular $0!=prod_{i=1}^0i=1$. Likewise, $x^n=prod_{i=1}^nx$, in particular, $x^0=prod_{i=1}^0x=1$.
– bof
Nov 29 at 8:59
1
$sum_{min{j_1,j_2,dots,j_r}}a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}$ is the sum of all products of the form $a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}$ where at least one of the indices $j_1,j_2,dots,j_r$ is equal to $m$, or in other words, $m$ is an element of the set ${j_1,j_2,dots,j_r}$.
– bof
Nov 29 at 9:04
1
$$prod_{i=1}^2sum_{j=1}^2a_{i,j}=(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})=a_{1,1}a_{2,1}+a_{1,1}a_{2,2}+a_{1,2}a_{2,1}+a_{1,2}a_{2,2}$$ while $$sum_{2in{j_1,j_2}}a_{1,j_1}a_{2,j_2}=a_{1,1}a_{2,2}+a_{1,2}a_{2,1}+a_{1,2}a_{2,2}.$$ The term $a_{1,1}a_{2,1}$ is gone because $2notin{j_1,j_2}={1,1}$.
– bof
Nov 30 at 6:57
|
show 8 more comments
Thank you for the answer. Is it possible to generalize it? I edited my question.
– 1Spectre1
Nov 27 at 13:24
1
@1ENİGMA1 I cleaned up my answer.
– bof
Nov 28 at 23:13
1
For example, $n!=prod_{i=1}^ni$ and in particular $0!=prod_{i=1}^0i=1$. Likewise, $x^n=prod_{i=1}^nx$, in particular, $x^0=prod_{i=1}^0x=1$.
– bof
Nov 29 at 8:59
1
$sum_{min{j_1,j_2,dots,j_r}}a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}$ is the sum of all products of the form $a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}$ where at least one of the indices $j_1,j_2,dots,j_r$ is equal to $m$, or in other words, $m$ is an element of the set ${j_1,j_2,dots,j_r}$.
– bof
Nov 29 at 9:04
1
$$prod_{i=1}^2sum_{j=1}^2a_{i,j}=(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})=a_{1,1}a_{2,1}+a_{1,1}a_{2,2}+a_{1,2}a_{2,1}+a_{1,2}a_{2,2}$$ while $$sum_{2in{j_1,j_2}}a_{1,j_1}a_{2,j_2}=a_{1,1}a_{2,2}+a_{1,2}a_{2,1}+a_{1,2}a_{2,2}.$$ The term $a_{1,1}a_{2,1}$ is gone because $2notin{j_1,j_2}={1,1}$.
– bof
Nov 30 at 6:57
Thank you for the answer. Is it possible to generalize it? I edited my question.
– 1Spectre1
Nov 27 at 13:24
Thank you for the answer. Is it possible to generalize it? I edited my question.
– 1Spectre1
Nov 27 at 13:24
1
1
@1ENİGMA1 I cleaned up my answer.
– bof
Nov 28 at 23:13
@1ENİGMA1 I cleaned up my answer.
– bof
Nov 28 at 23:13
1
1
For example, $n!=prod_{i=1}^ni$ and in particular $0!=prod_{i=1}^0i=1$. Likewise, $x^n=prod_{i=1}^nx$, in particular, $x^0=prod_{i=1}^0x=1$.
– bof
Nov 29 at 8:59
For example, $n!=prod_{i=1}^ni$ and in particular $0!=prod_{i=1}^0i=1$. Likewise, $x^n=prod_{i=1}^nx$, in particular, $x^0=prod_{i=1}^0x=1$.
– bof
Nov 29 at 8:59
1
1
$sum_{min{j_1,j_2,dots,j_r}}a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}$ is the sum of all products of the form $a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}$ where at least one of the indices $j_1,j_2,dots,j_r$ is equal to $m$, or in other words, $m$ is an element of the set ${j_1,j_2,dots,j_r}$.
– bof
Nov 29 at 9:04
$sum_{min{j_1,j_2,dots,j_r}}a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}$ is the sum of all products of the form $a_{1,j_1}a_{2,j_2}cdots a_{r,j_r}$ where at least one of the indices $j_1,j_2,dots,j_r$ is equal to $m$, or in other words, $m$ is an element of the set ${j_1,j_2,dots,j_r}$.
– bof
Nov 29 at 9:04
1
1
$$prod_{i=1}^2sum_{j=1}^2a_{i,j}=(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})=a_{1,1}a_{2,1}+a_{1,1}a_{2,2}+a_{1,2}a_{2,1}+a_{1,2}a_{2,2}$$ while $$sum_{2in{j_1,j_2}}a_{1,j_1}a_{2,j_2}=a_{1,1}a_{2,2}+a_{1,2}a_{2,1}+a_{1,2}a_{2,2}.$$ The term $a_{1,1}a_{2,1}$ is gone because $2notin{j_1,j_2}={1,1}$.
– bof
Nov 30 at 6:57
$$prod_{i=1}^2sum_{j=1}^2a_{i,j}=(a_{1,1}+a_{1,2})(a_{2,1}+a_{2,2})=a_{1,1}a_{2,1}+a_{1,1}a_{2,2}+a_{1,2}a_{2,1}+a_{1,2}a_{2,2}$$ while $$sum_{2in{j_1,j_2}}a_{1,j_1}a_{2,j_2}=a_{1,1}a_{2,2}+a_{1,2}a_{2,1}+a_{1,2}a_{2,2}.$$ The term $a_{1,1}a_{2,1}$ is gone because $2notin{j_1,j_2}={1,1}$.
– bof
Nov 30 at 6:57
|
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