Divisibility of sum of multinomials












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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







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$endgroup$

















    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






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      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 $ncdot M$.






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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 41 mins ago









                Fedor PetrovFedor Petrov

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                51.9k6122239






























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