A simple finite combinatorial sum I found, that seems to work, would have good reasons to work, but I can't...











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I was doing a consistency check for some calculations I'm performing for my master thesis (roughly - about a problem in discrete bayesian model selection) - and it turns out that my choice of priors is only consistent if this identity is true:



$$sum_{k=0}^{N}left[ binom{k+j}{j}binom{N-k+j}{j}right] = binom{N+2j+1}{2j+1}$$



Now, this seems to work for small numbers, but I searched for it a lot in the literature and I can't find it.(I have a physics background so probably my knowledge of proper "literature" is the problem here). Neither I can demonstrate it!
Has anyone of you seen this before? Can this be rewritten equivalently in some more commonly seen way? Can it be proven right...or wrong?
Thanks in advance! :)










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  • 2




    For an interpretable explanation of why this identity holds, see math.stackexchange.com/questions/2327556/…
    – David Foley
    yesterday








  • 4




    Just as a remark, any time I've ever needed to find a combinatorial identity, I've looked in Henry Gould's Table of Combinatorial Identities ( available on math.wvu.edu/~gould) and without fail I've either found it or found something that allowed me to prove what I needed. Yours looks like it's doable with what's in Vol 4 Sec 1.10
    – user113102
    yesterday












  • @user113102 The index of summation doesn't appear in the right position for 1.10, though.
    – chepner
    yesterday










  • Taking $alpha$, $r$ = $j$ and using standard binomial formulas e.g. ${n choose i} = {n choose n-i}$ and you're good, no? I admittedly didn't look too hard.
    – user113102
    yesterday

















up vote
20
down vote

favorite
4












I was doing a consistency check for some calculations I'm performing for my master thesis (roughly - about a problem in discrete bayesian model selection) - and it turns out that my choice of priors is only consistent if this identity is true:



$$sum_{k=0}^{N}left[ binom{k+j}{j}binom{N-k+j}{j}right] = binom{N+2j+1}{2j+1}$$



Now, this seems to work for small numbers, but I searched for it a lot in the literature and I can't find it.(I have a physics background so probably my knowledge of proper "literature" is the problem here). Neither I can demonstrate it!
Has anyone of you seen this before? Can this be rewritten equivalently in some more commonly seen way? Can it be proven right...or wrong?
Thanks in advance! :)










share|cite|improve this question




















  • 2




    For an interpretable explanation of why this identity holds, see math.stackexchange.com/questions/2327556/…
    – David Foley
    yesterday








  • 4




    Just as a remark, any time I've ever needed to find a combinatorial identity, I've looked in Henry Gould's Table of Combinatorial Identities ( available on math.wvu.edu/~gould) and without fail I've either found it or found something that allowed me to prove what I needed. Yours looks like it's doable with what's in Vol 4 Sec 1.10
    – user113102
    yesterday












  • @user113102 The index of summation doesn't appear in the right position for 1.10, though.
    – chepner
    yesterday










  • Taking $alpha$, $r$ = $j$ and using standard binomial formulas e.g. ${n choose i} = {n choose n-i}$ and you're good, no? I admittedly didn't look too hard.
    – user113102
    yesterday















up vote
20
down vote

favorite
4









up vote
20
down vote

favorite
4






4





I was doing a consistency check for some calculations I'm performing for my master thesis (roughly - about a problem in discrete bayesian model selection) - and it turns out that my choice of priors is only consistent if this identity is true:



$$sum_{k=0}^{N}left[ binom{k+j}{j}binom{N-k+j}{j}right] = binom{N+2j+1}{2j+1}$$



Now, this seems to work for small numbers, but I searched for it a lot in the literature and I can't find it.(I have a physics background so probably my knowledge of proper "literature" is the problem here). Neither I can demonstrate it!
Has anyone of you seen this before? Can this be rewritten equivalently in some more commonly seen way? Can it be proven right...or wrong?
Thanks in advance! :)










share|cite|improve this question















I was doing a consistency check for some calculations I'm performing for my master thesis (roughly - about a problem in discrete bayesian model selection) - and it turns out that my choice of priors is only consistent if this identity is true:



$$sum_{k=0}^{N}left[ binom{k+j}{j}binom{N-k+j}{j}right] = binom{N+2j+1}{2j+1}$$



Now, this seems to work for small numbers, but I searched for it a lot in the literature and I can't find it.(I have a physics background so probably my knowledge of proper "literature" is the problem here). Neither I can demonstrate it!
Has anyone of you seen this before? Can this be rewritten equivalently in some more commonly seen way? Can it be proven right...or wrong?
Thanks in advance! :)







combinatorics summation binomial-coefficients






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share|cite|improve this question













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share|cite|improve this question








edited yesterday









N. F. Taussig

42.9k93254




42.9k93254










asked yesterday









Rocco

1097




1097








  • 2




    For an interpretable explanation of why this identity holds, see math.stackexchange.com/questions/2327556/…
    – David Foley
    yesterday








  • 4




    Just as a remark, any time I've ever needed to find a combinatorial identity, I've looked in Henry Gould's Table of Combinatorial Identities ( available on math.wvu.edu/~gould) and without fail I've either found it or found something that allowed me to prove what I needed. Yours looks like it's doable with what's in Vol 4 Sec 1.10
    – user113102
    yesterday












  • @user113102 The index of summation doesn't appear in the right position for 1.10, though.
    – chepner
    yesterday










  • Taking $alpha$, $r$ = $j$ and using standard binomial formulas e.g. ${n choose i} = {n choose n-i}$ and you're good, no? I admittedly didn't look too hard.
    – user113102
    yesterday
















  • 2




    For an interpretable explanation of why this identity holds, see math.stackexchange.com/questions/2327556/…
    – David Foley
    yesterday








  • 4




    Just as a remark, any time I've ever needed to find a combinatorial identity, I've looked in Henry Gould's Table of Combinatorial Identities ( available on math.wvu.edu/~gould) and without fail I've either found it or found something that allowed me to prove what I needed. Yours looks like it's doable with what's in Vol 4 Sec 1.10
    – user113102
    yesterday












  • @user113102 The index of summation doesn't appear in the right position for 1.10, though.
    – chepner
    yesterday










  • Taking $alpha$, $r$ = $j$ and using standard binomial formulas e.g. ${n choose i} = {n choose n-i}$ and you're good, no? I admittedly didn't look too hard.
    – user113102
    yesterday










2




2




For an interpretable explanation of why this identity holds, see math.stackexchange.com/questions/2327556/…
– David Foley
yesterday






For an interpretable explanation of why this identity holds, see math.stackexchange.com/questions/2327556/…
– David Foley
yesterday






4




4




Just as a remark, any time I've ever needed to find a combinatorial identity, I've looked in Henry Gould's Table of Combinatorial Identities ( available on math.wvu.edu/~gould) and without fail I've either found it or found something that allowed me to prove what I needed. Yours looks like it's doable with what's in Vol 4 Sec 1.10
– user113102
yesterday






Just as a remark, any time I've ever needed to find a combinatorial identity, I've looked in Henry Gould's Table of Combinatorial Identities ( available on math.wvu.edu/~gould) and without fail I've either found it or found something that allowed me to prove what I needed. Yours looks like it's doable with what's in Vol 4 Sec 1.10
– user113102
yesterday














@user113102 The index of summation doesn't appear in the right position for 1.10, though.
– chepner
yesterday




@user113102 The index of summation doesn't appear in the right position for 1.10, though.
– chepner
yesterday












Taking $alpha$, $r$ = $j$ and using standard binomial formulas e.g. ${n choose i} = {n choose n-i}$ and you're good, no? I admittedly didn't look too hard.
– user113102
yesterday






Taking $alpha$, $r$ = $j$ and using standard binomial formulas e.g. ${n choose i} = {n choose n-i}$ and you're good, no? I admittedly didn't look too hard.
– user113102
yesterday












2 Answers
2






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up vote
41
down vote



accepted










Your identity is correct. I don't know a reference offhand, but here is a proof.



The right side, $binom{N+2j+1}{2j+1}$, is the number of bitstrings of length $N+2j+1$ consisting of $N$ zeroes and $2j+1$ ones.



The sum on the left counts the same set of bitstrings. Namely, for $0le kle N$, the term $binom{k+j}jbinom{N-k+j}j$ is the number of those bitstrings in which the middle one, with $j$ ones on either side, is in the $k+j+1^text{st}$ position; i.e., it has $k$ zeroes and $j$ ones to the left, $N-k$ zeroes and $j$ ones to the right.




P.S. I found your identity in László Lovász, Combinatorial Problems and Exercises, North-Holland, 1979 (the first edition), where it is Exercise 1.42(i) on p. 18, with hint on p. 96 and solution on p. 172. Lovász gives the identity in the following (more general) form:
$$sum_{k=0}^mbinom{u+k}kbinom{v-k}{m-k}=binom{u+v+1}m.$$
If we set $m=N$, $u=j$, $v=N+j$, this becomes
$$sum_{k=0}^Nbinom{j+k}kbinom{N+j-k}{N-k}=binom{N+2j+1}N$$
which is plainly equivalent to your identity
$$sum_{k=0}^Nbinom{k+j}jbinom{N-k+j}j=binom{N+2j+1}{2j+1}.$$






share|cite|improve this answer






























    up vote
    6
    down vote













    We have



    $$sum_{k=0}^N {k+jchoose j} {N-k+jchoose j}
    = sum_{k=0}^N {k+jchoose j} {N-k+jchoose N-k}
    \ = sum_{k=0}^N {k+jchoose j} [z^{N-k}] (1+z)^{N-k+j}
    = [z^N] (1+z)^{N+j} sum_{k=0}^N {k+jchoose j} z^k (1+z)^{-k}.$$



    Now we may extend $k$ beyond $N$ as there is no contribution to
    $[z^N]$ at the front in this case (we have $z^k (1+z)^{-k} = z^k
    +cdots$
    )



    $$[z^N] (1+z)^{N+j} sum_{kge 0} {k+jchoose j} z^k (1+z)^{-k}
    = [z^N] (1+z)^{N+j} frac{1}{(1-z/(1+z))^{j+1}}
    \ = [z^N] (1+z)^{N+j} frac{(1+z)^{j+1}}{(1+z-z)^{j+1}}
    = [z^N] (1+z)^{N+2j+1} = {N+2j+1choose 2j+1}.$$






    share|cite|improve this answer























    • I'm sorry, I'm not familiar with these kind of manipulations and/or notation. What's the definition of the object $[z^n]$? I mean, even though it's sort of self explanatory from what you did...it's an interesting notation that, with my tiny experience, I have never seen. Do you have any reference I can read to get a grasp on it? Thanks!
      – Rocco
      18 hours ago








    • 2




      There is this at Wikipedia on formal power series.
      – Marko Riedel
      14 hours ago











    Your Answer





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    2 Answers
    2






    active

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    2 Answers
    2






    active

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    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    41
    down vote



    accepted










    Your identity is correct. I don't know a reference offhand, but here is a proof.



    The right side, $binom{N+2j+1}{2j+1}$, is the number of bitstrings of length $N+2j+1$ consisting of $N$ zeroes and $2j+1$ ones.



    The sum on the left counts the same set of bitstrings. Namely, for $0le kle N$, the term $binom{k+j}jbinom{N-k+j}j$ is the number of those bitstrings in which the middle one, with $j$ ones on either side, is in the $k+j+1^text{st}$ position; i.e., it has $k$ zeroes and $j$ ones to the left, $N-k$ zeroes and $j$ ones to the right.




    P.S. I found your identity in László Lovász, Combinatorial Problems and Exercises, North-Holland, 1979 (the first edition), where it is Exercise 1.42(i) on p. 18, with hint on p. 96 and solution on p. 172. Lovász gives the identity in the following (more general) form:
    $$sum_{k=0}^mbinom{u+k}kbinom{v-k}{m-k}=binom{u+v+1}m.$$
    If we set $m=N$, $u=j$, $v=N+j$, this becomes
    $$sum_{k=0}^Nbinom{j+k}kbinom{N+j-k}{N-k}=binom{N+2j+1}N$$
    which is plainly equivalent to your identity
    $$sum_{k=0}^Nbinom{k+j}jbinom{N-k+j}j=binom{N+2j+1}{2j+1}.$$






    share|cite|improve this answer



























      up vote
      41
      down vote



      accepted










      Your identity is correct. I don't know a reference offhand, but here is a proof.



      The right side, $binom{N+2j+1}{2j+1}$, is the number of bitstrings of length $N+2j+1$ consisting of $N$ zeroes and $2j+1$ ones.



      The sum on the left counts the same set of bitstrings. Namely, for $0le kle N$, the term $binom{k+j}jbinom{N-k+j}j$ is the number of those bitstrings in which the middle one, with $j$ ones on either side, is in the $k+j+1^text{st}$ position; i.e., it has $k$ zeroes and $j$ ones to the left, $N-k$ zeroes and $j$ ones to the right.




      P.S. I found your identity in László Lovász, Combinatorial Problems and Exercises, North-Holland, 1979 (the first edition), where it is Exercise 1.42(i) on p. 18, with hint on p. 96 and solution on p. 172. Lovász gives the identity in the following (more general) form:
      $$sum_{k=0}^mbinom{u+k}kbinom{v-k}{m-k}=binom{u+v+1}m.$$
      If we set $m=N$, $u=j$, $v=N+j$, this becomes
      $$sum_{k=0}^Nbinom{j+k}kbinom{N+j-k}{N-k}=binom{N+2j+1}N$$
      which is plainly equivalent to your identity
      $$sum_{k=0}^Nbinom{k+j}jbinom{N-k+j}j=binom{N+2j+1}{2j+1}.$$






      share|cite|improve this answer

























        up vote
        41
        down vote



        accepted







        up vote
        41
        down vote



        accepted






        Your identity is correct. I don't know a reference offhand, but here is a proof.



        The right side, $binom{N+2j+1}{2j+1}$, is the number of bitstrings of length $N+2j+1$ consisting of $N$ zeroes and $2j+1$ ones.



        The sum on the left counts the same set of bitstrings. Namely, for $0le kle N$, the term $binom{k+j}jbinom{N-k+j}j$ is the number of those bitstrings in which the middle one, with $j$ ones on either side, is in the $k+j+1^text{st}$ position; i.e., it has $k$ zeroes and $j$ ones to the left, $N-k$ zeroes and $j$ ones to the right.




        P.S. I found your identity in László Lovász, Combinatorial Problems and Exercises, North-Holland, 1979 (the first edition), where it is Exercise 1.42(i) on p. 18, with hint on p. 96 and solution on p. 172. Lovász gives the identity in the following (more general) form:
        $$sum_{k=0}^mbinom{u+k}kbinom{v-k}{m-k}=binom{u+v+1}m.$$
        If we set $m=N$, $u=j$, $v=N+j$, this becomes
        $$sum_{k=0}^Nbinom{j+k}kbinom{N+j-k}{N-k}=binom{N+2j+1}N$$
        which is plainly equivalent to your identity
        $$sum_{k=0}^Nbinom{k+j}jbinom{N-k+j}j=binom{N+2j+1}{2j+1}.$$






        share|cite|improve this answer














        Your identity is correct. I don't know a reference offhand, but here is a proof.



        The right side, $binom{N+2j+1}{2j+1}$, is the number of bitstrings of length $N+2j+1$ consisting of $N$ zeroes and $2j+1$ ones.



        The sum on the left counts the same set of bitstrings. Namely, for $0le kle N$, the term $binom{k+j}jbinom{N-k+j}j$ is the number of those bitstrings in which the middle one, with $j$ ones on either side, is in the $k+j+1^text{st}$ position; i.e., it has $k$ zeroes and $j$ ones to the left, $N-k$ zeroes and $j$ ones to the right.




        P.S. I found your identity in László Lovász, Combinatorial Problems and Exercises, North-Holland, 1979 (the first edition), where it is Exercise 1.42(i) on p. 18, with hint on p. 96 and solution on p. 172. Lovász gives the identity in the following (more general) form:
        $$sum_{k=0}^mbinom{u+k}kbinom{v-k}{m-k}=binom{u+v+1}m.$$
        If we set $m=N$, $u=j$, $v=N+j$, this becomes
        $$sum_{k=0}^Nbinom{j+k}kbinom{N+j-k}{N-k}=binom{N+2j+1}N$$
        which is plainly equivalent to your identity
        $$sum_{k=0}^Nbinom{k+j}jbinom{N-k+j}j=binom{N+2j+1}{2j+1}.$$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited yesterday

























        answered yesterday









        bof

        49.5k455117




        49.5k455117






















            up vote
            6
            down vote













            We have



            $$sum_{k=0}^N {k+jchoose j} {N-k+jchoose j}
            = sum_{k=0}^N {k+jchoose j} {N-k+jchoose N-k}
            \ = sum_{k=0}^N {k+jchoose j} [z^{N-k}] (1+z)^{N-k+j}
            = [z^N] (1+z)^{N+j} sum_{k=0}^N {k+jchoose j} z^k (1+z)^{-k}.$$



            Now we may extend $k$ beyond $N$ as there is no contribution to
            $[z^N]$ at the front in this case (we have $z^k (1+z)^{-k} = z^k
            +cdots$
            )



            $$[z^N] (1+z)^{N+j} sum_{kge 0} {k+jchoose j} z^k (1+z)^{-k}
            = [z^N] (1+z)^{N+j} frac{1}{(1-z/(1+z))^{j+1}}
            \ = [z^N] (1+z)^{N+j} frac{(1+z)^{j+1}}{(1+z-z)^{j+1}}
            = [z^N] (1+z)^{N+2j+1} = {N+2j+1choose 2j+1}.$$






            share|cite|improve this answer























            • I'm sorry, I'm not familiar with these kind of manipulations and/or notation. What's the definition of the object $[z^n]$? I mean, even though it's sort of self explanatory from what you did...it's an interesting notation that, with my tiny experience, I have never seen. Do you have any reference I can read to get a grasp on it? Thanks!
              – Rocco
              18 hours ago








            • 2




              There is this at Wikipedia on formal power series.
              – Marko Riedel
              14 hours ago















            up vote
            6
            down vote













            We have



            $$sum_{k=0}^N {k+jchoose j} {N-k+jchoose j}
            = sum_{k=0}^N {k+jchoose j} {N-k+jchoose N-k}
            \ = sum_{k=0}^N {k+jchoose j} [z^{N-k}] (1+z)^{N-k+j}
            = [z^N] (1+z)^{N+j} sum_{k=0}^N {k+jchoose j} z^k (1+z)^{-k}.$$



            Now we may extend $k$ beyond $N$ as there is no contribution to
            $[z^N]$ at the front in this case (we have $z^k (1+z)^{-k} = z^k
            +cdots$
            )



            $$[z^N] (1+z)^{N+j} sum_{kge 0} {k+jchoose j} z^k (1+z)^{-k}
            = [z^N] (1+z)^{N+j} frac{1}{(1-z/(1+z))^{j+1}}
            \ = [z^N] (1+z)^{N+j} frac{(1+z)^{j+1}}{(1+z-z)^{j+1}}
            = [z^N] (1+z)^{N+2j+1} = {N+2j+1choose 2j+1}.$$






            share|cite|improve this answer























            • I'm sorry, I'm not familiar with these kind of manipulations and/or notation. What's the definition of the object $[z^n]$? I mean, even though it's sort of self explanatory from what you did...it's an interesting notation that, with my tiny experience, I have never seen. Do you have any reference I can read to get a grasp on it? Thanks!
              – Rocco
              18 hours ago








            • 2




              There is this at Wikipedia on formal power series.
              – Marko Riedel
              14 hours ago













            up vote
            6
            down vote










            up vote
            6
            down vote









            We have



            $$sum_{k=0}^N {k+jchoose j} {N-k+jchoose j}
            = sum_{k=0}^N {k+jchoose j} {N-k+jchoose N-k}
            \ = sum_{k=0}^N {k+jchoose j} [z^{N-k}] (1+z)^{N-k+j}
            = [z^N] (1+z)^{N+j} sum_{k=0}^N {k+jchoose j} z^k (1+z)^{-k}.$$



            Now we may extend $k$ beyond $N$ as there is no contribution to
            $[z^N]$ at the front in this case (we have $z^k (1+z)^{-k} = z^k
            +cdots$
            )



            $$[z^N] (1+z)^{N+j} sum_{kge 0} {k+jchoose j} z^k (1+z)^{-k}
            = [z^N] (1+z)^{N+j} frac{1}{(1-z/(1+z))^{j+1}}
            \ = [z^N] (1+z)^{N+j} frac{(1+z)^{j+1}}{(1+z-z)^{j+1}}
            = [z^N] (1+z)^{N+2j+1} = {N+2j+1choose 2j+1}.$$






            share|cite|improve this answer














            We have



            $$sum_{k=0}^N {k+jchoose j} {N-k+jchoose j}
            = sum_{k=0}^N {k+jchoose j} {N-k+jchoose N-k}
            \ = sum_{k=0}^N {k+jchoose j} [z^{N-k}] (1+z)^{N-k+j}
            = [z^N] (1+z)^{N+j} sum_{k=0}^N {k+jchoose j} z^k (1+z)^{-k}.$$



            Now we may extend $k$ beyond $N$ as there is no contribution to
            $[z^N]$ at the front in this case (we have $z^k (1+z)^{-k} = z^k
            +cdots$
            )



            $$[z^N] (1+z)^{N+j} sum_{kge 0} {k+jchoose j} z^k (1+z)^{-k}
            = [z^N] (1+z)^{N+j} frac{1}{(1-z/(1+z))^{j+1}}
            \ = [z^N] (1+z)^{N+j} frac{(1+z)^{j+1}}{(1+z-z)^{j+1}}
            = [z^N] (1+z)^{N+2j+1} = {N+2j+1choose 2j+1}.$$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited yesterday

























            answered yesterday









            Marko Riedel

            38.6k339106




            38.6k339106












            • I'm sorry, I'm not familiar with these kind of manipulations and/or notation. What's the definition of the object $[z^n]$? I mean, even though it's sort of self explanatory from what you did...it's an interesting notation that, with my tiny experience, I have never seen. Do you have any reference I can read to get a grasp on it? Thanks!
              – Rocco
              18 hours ago








            • 2




              There is this at Wikipedia on formal power series.
              – Marko Riedel
              14 hours ago


















            • I'm sorry, I'm not familiar with these kind of manipulations and/or notation. What's the definition of the object $[z^n]$? I mean, even though it's sort of self explanatory from what you did...it's an interesting notation that, with my tiny experience, I have never seen. Do you have any reference I can read to get a grasp on it? Thanks!
              – Rocco
              18 hours ago








            • 2




              There is this at Wikipedia on formal power series.
              – Marko Riedel
              14 hours ago
















            I'm sorry, I'm not familiar with these kind of manipulations and/or notation. What's the definition of the object $[z^n]$? I mean, even though it's sort of self explanatory from what you did...it's an interesting notation that, with my tiny experience, I have never seen. Do you have any reference I can read to get a grasp on it? Thanks!
            – Rocco
            18 hours ago






            I'm sorry, I'm not familiar with these kind of manipulations and/or notation. What's the definition of the object $[z^n]$? I mean, even though it's sort of self explanatory from what you did...it's an interesting notation that, with my tiny experience, I have never seen. Do you have any reference I can read to get a grasp on it? Thanks!
            – Rocco
            18 hours ago






            2




            2




            There is this at Wikipedia on formal power series.
            – Marko Riedel
            14 hours ago




            There is this at Wikipedia on formal power series.
            – Marko Riedel
            14 hours ago


















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