Notation of symmetric sum notation

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When you use the symmetric sum notation, for example, $$sum_textsymabc+a$$ if there are 3 variables, then does abc count once, 3 times or 6 times?



I am confused about repetitions of the same expression in a symmetric sum notation.




algebra-precalculus summation notation permutations symmetric-sum




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    up vote
    3
    down vote

    favorite












    When you use the symmetric sum notation, for example, $$sum_textsymabc+a$$ if there are 3 variables, then does abc count once, 3 times or 6 times?



    I am confused about repetitions of the same expression in a symmetric sum notation.




    algebra-precalculus summation notation permutations symmetric-sum




    share|cite|improve this question























      up vote
      3
      down vote

      favorite









      up vote
      3
      down vote

      favorite











      When you use the symmetric sum notation, for example, $$sum_textsymabc+a$$ if there are 3 variables, then does abc count once, 3 times or 6 times?



      I am confused about repetitions of the same expression in a symmetric sum notation.




      algebra-precalculus summation notation permutations symmetric-sum




      share|cite|improve this question













      When you use the symmetric sum notation, for example, $$sum_textsymabc+a$$ if there are 3 variables, then does abc count once, 3 times or 6 times?



      I am confused about repetitions of the same expression in a symmetric sum notation.





      algebra-precalculus summation notation permutations symmetric-sum





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Aug 3 at 7:17









      Michael Rozenberg

      87.3k1577178




      87.3k1577178









      asked Aug 2 at 0:51









      abc...

      1,175524




      1,175524




















          4 Answers
          4






          active

          oldest

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          up vote
          3
          down vote



          accepted










          I think if we want to work with symmetric sum, it's better to write $$sum_sym(abc+a)=6abc+2(a+b+c)$$ and
          $$sum_symabc+a=6abc+a.$$






          share|cite|improve this answer





















          • I've interpreted according to the first one!
            – gimusi
            Aug 2 at 1:57






          • 1




            @gimusi I think your explanation is right. Also, I think it's better $sumlimits_symQ(x_1,x_2,...,x_n)=sumlimits_sigmain S_nQleft(x_sigma(1),x_sigma(2),...,x_sigma(n)right).$
            – Michael Rozenberg
            Aug 2 at 2:22











          • Thanks for your kind suggestion! Bye
            – gimusi
            Aug 2 at 2:25

















          up vote
          1
          down vote













          The symmetric sum notation $sum_colorbluemathrmsym$ is the sum over all permutations of the elements of a predefined set $S$.




          • If $S=a,b,c$, then
            beginalign*
            colorbluesum_mathrmsymleft(abc+aright)&=(abc+a)+(acb+a)+(bac+b)+(bca+b)+(cab+c)+(cba+c)\
            &,,colorblue=6abc+2(a+b+c)
            endalign*


          • If $S=a,b$, then
            beginalign*
            colorbluesum_mathrmsymleft(abc+aright)&=(abc+a)+(bac+b)\
            &,,colorblue=2abc+a+b
            endalign*




          On the other hand the cyclic sum notation $sum_colorbluemathrmcyc$ is the sum over elements of a predefined set $S$ in a cyclic manner $ato bto ctocdotsto a$.




          • If $S=a,b,c$, then
            beginalign*
            colorbluesum_mathrmcycleft(abc+aright)&=(abc+a)+(bca+b)+(cab+c)\
            &,,colorblue=3abc+a+b+c
            endalign*

          • If $S=a,b$, then
            beginalign*
            colorbluesum_mathrmcycleft(abc+aright)&=(abc+a)+(bac+b)\
            &,,colorblue=2abc+a+b
            endalign*






          share|cite|improve this answer




























            up vote
            0
            down vote













            We have that



            $$sum_mathrmsymQ(x_i)=sum_sigma Q(x_sigma(i))$$



            for all permutations of $1, ldots , n$.



            Therefore it should be



            $$sum_textsymabc+a=Q(a,b,c)+Q(a,c,b)+Q(b,a,c)+Q(b,c,a)+Q(c,a,b)+Q(c,b,a)=$$



            $$=2abc+2a+2abc+2b+2abc+2c=6abc+2(a+b+c)$$






            share|cite|improve this answer






























              up vote
              0
              down vote













              The symetric sum of a function of $n$ arguments is the sum of all permutations for those arguments.   Sometimes written as $sumlimits_rm sym$, but preferably written a $sumlimits_sigma$



              $$sum_sigma f(a,b,c)= f(a,b,c)+f(a,c,b)+f(b,a,c)+f(b,c,a)+f(c,a,b)+f(c,b,a)$$



              So in this case $$sum_sigma abc = 3 abc$$



              The $k$-th elemental symmetric sum for a set of numbers is the sum of the products for all selections of $k$ elements from that set.   That is selection without order, so each selected tupple forms a single term.



              The first elemental symmetric sum is confussingly also indicated with the $sumlimits_rm sym$ operator.



              The $k$-th elemental symmetric sum is usually indicated with $sumlimits_rm sym^k$.



              And so, for example, $$sum_rm sym^2 a,b,c = ab + ac+ bc$$







              share|cite|improve this answer























              • If so, $sumlimits_cycab$ what is it?
                – Michael Rozenberg
                Aug 2 at 1:34










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              4 Answers
              4






              active

              oldest

              votes








              4 Answers
              4






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes








              up vote
              3
              down vote



              accepted










              I think if we want to work with symmetric sum, it's better to write $$sum_sym(abc+a)=6abc+2(a+b+c)$$ and
              $$sum_symabc+a=6abc+a.$$






              share|cite|improve this answer





















              • I've interpreted according to the first one!
                – gimusi
                Aug 2 at 1:57






              • 1




                @gimusi I think your explanation is right. Also, I think it's better $sumlimits_symQ(x_1,x_2,...,x_n)=sumlimits_sigmain S_nQleft(x_sigma(1),x_sigma(2),...,x_sigma(n)right).$
                – Michael Rozenberg
                Aug 2 at 2:22











              • Thanks for your kind suggestion! Bye
                – gimusi
                Aug 2 at 2:25














              up vote
              3
              down vote



              accepted










              I think if we want to work with symmetric sum, it's better to write $$sum_sym(abc+a)=6abc+2(a+b+c)$$ and
              $$sum_symabc+a=6abc+a.$$






              share|cite|improve this answer





















              • I've interpreted according to the first one!
                – gimusi
                Aug 2 at 1:57






              • 1




                @gimusi I think your explanation is right. Also, I think it's better $sumlimits_symQ(x_1,x_2,...,x_n)=sumlimits_sigmain S_nQleft(x_sigma(1),x_sigma(2),...,x_sigma(n)right).$
                – Michael Rozenberg
                Aug 2 at 2:22











              • Thanks for your kind suggestion! Bye
                – gimusi
                Aug 2 at 2:25












              up vote
              3
              down vote



              accepted







              up vote
              3
              down vote



              accepted






              I think if we want to work with symmetric sum, it's better to write $$sum_sym(abc+a)=6abc+2(a+b+c)$$ and
              $$sum_symabc+a=6abc+a.$$






              share|cite|improve this answer













              I think if we want to work with symmetric sum, it's better to write $$sum_sym(abc+a)=6abc+2(a+b+c)$$ and
              $$sum_symabc+a=6abc+a.$$







              share|cite|improve this answer













              share|cite|improve this answer



              share|cite|improve this answer











              answered Aug 2 at 1:41









              Michael Rozenberg

              87.3k1577178




              87.3k1577178











              • I've interpreted according to the first one!
                – gimusi
                Aug 2 at 1:57






              • 1




                @gimusi I think your explanation is right. Also, I think it's better $sumlimits_symQ(x_1,x_2,...,x_n)=sumlimits_sigmain S_nQleft(x_sigma(1),x_sigma(2),...,x_sigma(n)right).$
                – Michael Rozenberg
                Aug 2 at 2:22











              • Thanks for your kind suggestion! Bye
                – gimusi
                Aug 2 at 2:25
















              • I've interpreted according to the first one!
                – gimusi
                Aug 2 at 1:57






              • 1




                @gimusi I think your explanation is right. Also, I think it's better $sumlimits_symQ(x_1,x_2,...,x_n)=sumlimits_sigmain S_nQleft(x_sigma(1),x_sigma(2),...,x_sigma(n)right).$
                – Michael Rozenberg
                Aug 2 at 2:22











              • Thanks for your kind suggestion! Bye
                – gimusi
                Aug 2 at 2:25















              I've interpreted according to the first one!
              – gimusi
              Aug 2 at 1:57




              I've interpreted according to the first one!
              – gimusi
              Aug 2 at 1:57




              1




              1




              @gimusi I think your explanation is right. Also, I think it's better $sumlimits_symQ(x_1,x_2,...,x_n)=sumlimits_sigmain S_nQleft(x_sigma(1),x_sigma(2),...,x_sigma(n)right).$
              – Michael Rozenberg
              Aug 2 at 2:22





              @gimusi I think your explanation is right. Also, I think it's better $sumlimits_symQ(x_1,x_2,...,x_n)=sumlimits_sigmain S_nQleft(x_sigma(1),x_sigma(2),...,x_sigma(n)right).$
              – Michael Rozenberg
              Aug 2 at 2:22













              Thanks for your kind suggestion! Bye
              – gimusi
              Aug 2 at 2:25




              Thanks for your kind suggestion! Bye
              – gimusi
              Aug 2 at 2:25










              up vote
              1
              down vote













              The symmetric sum notation $sum_colorbluemathrmsym$ is the sum over all permutations of the elements of a predefined set $S$.




              • If $S=a,b,c$, then
                beginalign*
                colorbluesum_mathrmsymleft(abc+aright)&=(abc+a)+(acb+a)+(bac+b)+(bca+b)+(cab+c)+(cba+c)\
                &,,colorblue=6abc+2(a+b+c)
                endalign*


              • If $S=a,b$, then
                beginalign*
                colorbluesum_mathrmsymleft(abc+aright)&=(abc+a)+(bac+b)\
                &,,colorblue=2abc+a+b
                endalign*




              On the other hand the cyclic sum notation $sum_colorbluemathrmcyc$ is the sum over elements of a predefined set $S$ in a cyclic manner $ato bto ctocdotsto a$.




              • If $S=a,b,c$, then
                beginalign*
                colorbluesum_mathrmcycleft(abc+aright)&=(abc+a)+(bca+b)+(cab+c)\
                &,,colorblue=3abc+a+b+c
                endalign*

              • If $S=a,b$, then
                beginalign*
                colorbluesum_mathrmcycleft(abc+aright)&=(abc+a)+(bac+b)\
                &,,colorblue=2abc+a+b
                endalign*






              share|cite|improve this answer

























                up vote
                1
                down vote













                The symmetric sum notation $sum_colorbluemathrmsym$ is the sum over all permutations of the elements of a predefined set $S$.




                • If $S=a,b,c$, then
                  beginalign*
                  colorbluesum_mathrmsymleft(abc+aright)&=(abc+a)+(acb+a)+(bac+b)+(bca+b)+(cab+c)+(cba+c)\
                  &,,colorblue=6abc+2(a+b+c)
                  endalign*


                • If $S=a,b$, then
                  beginalign*
                  colorbluesum_mathrmsymleft(abc+aright)&=(abc+a)+(bac+b)\
                  &,,colorblue=2abc+a+b
                  endalign*




                On the other hand the cyclic sum notation $sum_colorbluemathrmcyc$ is the sum over elements of a predefined set $S$ in a cyclic manner $ato bto ctocdotsto a$.




                • If $S=a,b,c$, then
                  beginalign*
                  colorbluesum_mathrmcycleft(abc+aright)&=(abc+a)+(bca+b)+(cab+c)\
                  &,,colorblue=3abc+a+b+c
                  endalign*

                • If $S=a,b$, then
                  beginalign*
                  colorbluesum_mathrmcycleft(abc+aright)&=(abc+a)+(bac+b)\
                  &,,colorblue=2abc+a+b
                  endalign*






                share|cite|improve this answer























                  up vote
                  1
                  down vote










                  up vote
                  1
                  down vote









                  The symmetric sum notation $sum_colorbluemathrmsym$ is the sum over all permutations of the elements of a predefined set $S$.




                  • If $S=a,b,c$, then
                    beginalign*
                    colorbluesum_mathrmsymleft(abc+aright)&=(abc+a)+(acb+a)+(bac+b)+(bca+b)+(cab+c)+(cba+c)\
                    &,,colorblue=6abc+2(a+b+c)
                    endalign*


                  • If $S=a,b$, then
                    beginalign*
                    colorbluesum_mathrmsymleft(abc+aright)&=(abc+a)+(bac+b)\
                    &,,colorblue=2abc+a+b
                    endalign*




                  On the other hand the cyclic sum notation $sum_colorbluemathrmcyc$ is the sum over elements of a predefined set $S$ in a cyclic manner $ato bto ctocdotsto a$.




                  • If $S=a,b,c$, then
                    beginalign*
                    colorbluesum_mathrmcycleft(abc+aright)&=(abc+a)+(bca+b)+(cab+c)\
                    &,,colorblue=3abc+a+b+c
                    endalign*

                  • If $S=a,b$, then
                    beginalign*
                    colorbluesum_mathrmcycleft(abc+aright)&=(abc+a)+(bac+b)\
                    &,,colorblue=2abc+a+b
                    endalign*






                  share|cite|improve this answer













                  The symmetric sum notation $sum_colorbluemathrmsym$ is the sum over all permutations of the elements of a predefined set $S$.




                  • If $S=a,b,c$, then
                    beginalign*
                    colorbluesum_mathrmsymleft(abc+aright)&=(abc+a)+(acb+a)+(bac+b)+(bca+b)+(cab+c)+(cba+c)\
                    &,,colorblue=6abc+2(a+b+c)
                    endalign*


                  • If $S=a,b$, then
                    beginalign*
                    colorbluesum_mathrmsymleft(abc+aright)&=(abc+a)+(bac+b)\
                    &,,colorblue=2abc+a+b
                    endalign*




                  On the other hand the cyclic sum notation $sum_colorbluemathrmcyc$ is the sum over elements of a predefined set $S$ in a cyclic manner $ato bto ctocdotsto a$.




                  • If $S=a,b,c$, then
                    beginalign*
                    colorbluesum_mathrmcycleft(abc+aright)&=(abc+a)+(bca+b)+(cab+c)\
                    &,,colorblue=3abc+a+b+c
                    endalign*

                  • If $S=a,b$, then
                    beginalign*
                    colorbluesum_mathrmcycleft(abc+aright)&=(abc+a)+(bac+b)\
                    &,,colorblue=2abc+a+b
                    endalign*







                  share|cite|improve this answer













                  share|cite|improve this answer



                  share|cite|improve this answer











                  answered Aug 2 at 13:44









                  Markus Scheuer

                  55.8k450135




                  55.8k450135




















                      up vote
                      0
                      down vote













                      We have that



                      $$sum_mathrmsymQ(x_i)=sum_sigma Q(x_sigma(i))$$



                      for all permutations of $1, ldots , n$.



                      Therefore it should be



                      $$sum_textsymabc+a=Q(a,b,c)+Q(a,c,b)+Q(b,a,c)+Q(b,c,a)+Q(c,a,b)+Q(c,b,a)=$$



                      $$=2abc+2a+2abc+2b+2abc+2c=6abc+2(a+b+c)$$






                      share|cite|improve this answer



























                        up vote
                        0
                        down vote













                        We have that



                        $$sum_mathrmsymQ(x_i)=sum_sigma Q(x_sigma(i))$$



                        for all permutations of $1, ldots , n$.



                        Therefore it should be



                        $$sum_textsymabc+a=Q(a,b,c)+Q(a,c,b)+Q(b,a,c)+Q(b,c,a)+Q(c,a,b)+Q(c,b,a)=$$



                        $$=2abc+2a+2abc+2b+2abc+2c=6abc+2(a+b+c)$$






                        share|cite|improve this answer

























                          up vote
                          0
                          down vote










                          up vote
                          0
                          down vote









                          We have that



                          $$sum_mathrmsymQ(x_i)=sum_sigma Q(x_sigma(i))$$



                          for all permutations of $1, ldots , n$.



                          Therefore it should be



                          $$sum_textsymabc+a=Q(a,b,c)+Q(a,c,b)+Q(b,a,c)+Q(b,c,a)+Q(c,a,b)+Q(c,b,a)=$$



                          $$=2abc+2a+2abc+2b+2abc+2c=6abc+2(a+b+c)$$






                          share|cite|improve this answer















                          We have that



                          $$sum_mathrmsymQ(x_i)=sum_sigma Q(x_sigma(i))$$



                          for all permutations of $1, ldots , n$.



                          Therefore it should be



                          $$sum_textsymabc+a=Q(a,b,c)+Q(a,c,b)+Q(b,a,c)+Q(b,c,a)+Q(c,a,b)+Q(c,b,a)=$$



                          $$=2abc+2a+2abc+2b+2abc+2c=6abc+2(a+b+c)$$







                          share|cite|improve this answer















                          share|cite|improve this answer



                          share|cite|improve this answer








                          edited Aug 2 at 1:24


























                          answered Aug 2 at 1:19









                          gimusi

                          63.8k73480




                          63.8k73480




















                              up vote
                              0
                              down vote













                              The symetric sum of a function of $n$ arguments is the sum of all permutations for those arguments.   Sometimes written as $sumlimits_rm sym$, but preferably written a $sumlimits_sigma$



                              $$sum_sigma f(a,b,c)= f(a,b,c)+f(a,c,b)+f(b,a,c)+f(b,c,a)+f(c,a,b)+f(c,b,a)$$



                              So in this case $$sum_sigma abc = 3 abc$$



                              The $k$-th elemental symmetric sum for a set of numbers is the sum of the products for all selections of $k$ elements from that set.   That is selection without order, so each selected tupple forms a single term.



                              The first elemental symmetric sum is confussingly also indicated with the $sumlimits_rm sym$ operator.



                              The $k$-th elemental symmetric sum is usually indicated with $sumlimits_rm sym^k$.



                              And so, for example, $$sum_rm sym^2 a,b,c = ab + ac+ bc$$







                              share|cite|improve this answer























                              • If so, $sumlimits_cycab$ what is it?
                                – Michael Rozenberg
                                Aug 2 at 1:34














                              up vote
                              0
                              down vote













                              The symetric sum of a function of $n$ arguments is the sum of all permutations for those arguments.   Sometimes written as $sumlimits_rm sym$, but preferably written a $sumlimits_sigma$



                              $$sum_sigma f(a,b,c)= f(a,b,c)+f(a,c,b)+f(b,a,c)+f(b,c,a)+f(c,a,b)+f(c,b,a)$$



                              So in this case $$sum_sigma abc = 3 abc$$



                              The $k$-th elemental symmetric sum for a set of numbers is the sum of the products for all selections of $k$ elements from that set.   That is selection without order, so each selected tupple forms a single term.



                              The first elemental symmetric sum is confussingly also indicated with the $sumlimits_rm sym$ operator.



                              The $k$-th elemental symmetric sum is usually indicated with $sumlimits_rm sym^k$.



                              And so, for example, $$sum_rm sym^2 a,b,c = ab + ac+ bc$$







                              share|cite|improve this answer























                              • If so, $sumlimits_cycab$ what is it?
                                – Michael Rozenberg
                                Aug 2 at 1:34












                              up vote
                              0
                              down vote










                              up vote
                              0
                              down vote









                              The symetric sum of a function of $n$ arguments is the sum of all permutations for those arguments.   Sometimes written as $sumlimits_rm sym$, but preferably written a $sumlimits_sigma$



                              $$sum_sigma f(a,b,c)= f(a,b,c)+f(a,c,b)+f(b,a,c)+f(b,c,a)+f(c,a,b)+f(c,b,a)$$



                              So in this case $$sum_sigma abc = 3 abc$$



                              The $k$-th elemental symmetric sum for a set of numbers is the sum of the products for all selections of $k$ elements from that set.   That is selection without order, so each selected tupple forms a single term.



                              The first elemental symmetric sum is confussingly also indicated with the $sumlimits_rm sym$ operator.



                              The $k$-th elemental symmetric sum is usually indicated with $sumlimits_rm sym^k$.



                              And so, for example, $$sum_rm sym^2 a,b,c = ab + ac+ bc$$







                              share|cite|improve this answer















                              The symetric sum of a function of $n$ arguments is the sum of all permutations for those arguments.   Sometimes written as $sumlimits_rm sym$, but preferably written a $sumlimits_sigma$



                              $$sum_sigma f(a,b,c)= f(a,b,c)+f(a,c,b)+f(b,a,c)+f(b,c,a)+f(c,a,b)+f(c,b,a)$$



                              So in this case $$sum_sigma abc = 3 abc$$



                              The $k$-th elemental symmetric sum for a set of numbers is the sum of the products for all selections of $k$ elements from that set.   That is selection without order, so each selected tupple forms a single term.



                              The first elemental symmetric sum is confussingly also indicated with the $sumlimits_rm sym$ operator.



                              The $k$-th elemental symmetric sum is usually indicated with $sumlimits_rm sym^k$.



                              And so, for example, $$sum_rm sym^2 a,b,c = ab + ac+ bc$$








                              share|cite|improve this answer















                              share|cite|improve this answer



                              share|cite|improve this answer








                              edited Aug 2 at 1:25


























                              answered Aug 2 at 1:19









                              Graham Kemp

                              80k43275




                              80k43275











                              • If so, $sumlimits_cycab$ what is it?
                                – Michael Rozenberg
                                Aug 2 at 1:34
















                              • If so, $sumlimits_cycab$ what is it?
                                – Michael Rozenberg
                                Aug 2 at 1:34















                              If so, $sumlimits_cycab$ what is it?
                              – Michael Rozenberg
                              Aug 2 at 1:34




                              If so, $sumlimits_cycab$ what is it?
                              – Michael Rozenberg
                              Aug 2 at 1:34












                               

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                              Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
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                              Color the edges and diagonals of a regular polygon

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                              Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
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                              Relationship between determinant of matrix and determinant of adjoint?

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                              Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
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