Divisibility of sum of multinomialsInteger-valued factorial ratiosPerron number distributionDesign constraint...



Divisibility of sum of multinomials


Integer-valued factorial ratiosPerron number distributionDesign constraint systems over the realsOn “The Average Height of Planted Plane Trees” by Knuth, de Bruijn and Rice (1972)Montgomery's conjecture and lower bound on certain Fourier transform.Distribution of the evaluation (at a non-trivial root of -1) of polynomials with small coefficientsCombinatorialProbabilistic Proof of Stirling's ApproximationGeneralizing Kasteleyn's formula even more?Derive a theoretical bound about coding with a partial eavesdropperTartaglia distribution













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$begingroup$


Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences
$$S(n,m,t)=sum_{k_1+cdots+k_n=m}binom{m}{k_1,dots,k_n}^t$$
where the sum runs over non-negative integers $k_1,dots,k_n$. These numbers are related to average distances (from the origin) of uniform unit-step random walks on the plane.




QUESTION. Is it always true that $n$ divides $S(n,m,t)$?




Observe that $S(n,m,1)=n^m$.






nt.number-theory co.combinatorics soft-question







share|cite|improve this question











$endgroup$

















    6












    $begingroup$


    Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences
    $$S(n,m,t)=sum_{k_1+cdots+k_n=m}binom{m}{k_1,dots,k_n}^t$$
    where the sum runs over non-negative integers $k_1,dots,k_n$. These numbers are related to average distances (from the origin) of uniform unit-step random walks on the plane.




    QUESTION. Is it always true that $n$ divides $S(n,m,t)$?




    Observe that $S(n,m,1)=n^m$.






    nt.number-theory co.combinatorics soft-question







    share|cite|improve this question











    $endgroup$















      6












      6








      6


      3



      $begingroup$


      Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences
      $$S(n,m,t)=sum_{k_1+cdots+k_n=m}binom{m}{k_1,dots,k_n}^t$$
      where the sum runs over non-negative integers $k_1,dots,k_n$. These numbers are related to average distances (from the origin) of uniform unit-step random walks on the plane.




      QUESTION. Is it always true that $n$ divides $S(n,m,t)$?




      Observe that $S(n,m,1)=n^m$.






      nt.number-theory co.combinatorics soft-question







      share|cite|improve this question











      $endgroup$




      Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences
      $$S(n,m,t)=sum_{k_1+cdots+k_n=m}binom{m}{k_1,dots,k_n}^t$$
      where the sum runs over non-negative integers $k_1,dots,k_n$. These numbers are related to average distances (from the origin) of uniform unit-step random walks on the plane.




      QUESTION. Is it always true that $n$ divides $S(n,m,t)$?




      Observe that $S(n,m,1)=n^m$.






      nt.number-theory co.combinatorics soft-question




      nt.number-theory co.combinatorics soft-question






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 1 hour ago







      T. Amdeberhan

















      asked 1 hour ago









      T. AmdeberhanT. Amdeberhan

      18.3k229132




      18.3k229132






















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          $begingroup$

          We count the number of $t$-tuples $(xi_1,ldots,xi_t)$ of the colorings of ${1,ldots,m}$ with $n$ given colors, for which any two colorings use the same multisets of colors. If $M$ is the number of such tuples in which 1 is colored red in the coloring $xi_1$ (red is one of our $n$ colors), the total number of $t$-tuples equals $ncdot M$.






          share|cite|improve this answer









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            $begingroup$

            We count the number of $t$-tuples $(xi_1,ldots,xi_t)$ of the colorings of ${1,ldots,m}$ with $n$ given colors, for which any two colorings use the same multisets of colors. If $M$ is the number of such tuples in which 1 is colored red in the coloring $xi_1$ (red is one of our $n$ colors), the total number of $t$-tuples equals $ncdot M$.






            share|cite|improve this answer









            $endgroup$


















              4












              $begingroup$

              We count the number of $t$-tuples $(xi_1,ldots,xi_t)$ of the colorings of ${1,ldots,m}$ with $n$ given colors, for which any two colorings use the same multisets of colors. If $M$ is the number of such tuples in which 1 is colored red in the coloring $xi_1$ (red is one of our $n$ colors), the total number of $t$-tuples equals $ncdot M$.






              share|cite|improve this answer









              $endgroup$
















                4












                4








                4





                $begingroup$

                We count the number of $t$-tuples $(xi_1,ldots,xi_t)$ of the colorings of ${1,ldots,m}$ with $n$ given colors, for which any two colorings use the same multisets of colors. If $M$ is the number of such tuples in which 1 is colored red in the coloring $xi_1$ (red is one of our $n$ colors), the total number of $t$-tuples equals $ncdot M$.






                share|cite|improve this answer









                $endgroup$



                We count the number of $t$-tuples $(xi_1,ldots,xi_t)$ of the colorings of ${1,ldots,m}$ with $n$ given colors, for which any two colorings use the same multisets of colors. If $M$ is the number of such tuples in which 1 is colored red in the coloring $xi_1$ (red is one of our $n$ colors), the total number of $t$-tuples equals $ncdot M$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 27 mins ago









                Fedor PetrovFedor Petrov

                51.9k6122239




                51.9k6122239






























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