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













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












    $begingroup$


    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






















          1 Answer
          1






          active

          oldest

          votes


















          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









          $endgroup$














            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "504"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f327456%2fdivisibility-of-sum-of-multinomials%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            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












              $begingroup$

              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






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to MathOverflow!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid …



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f327456%2fdivisibility-of-sum-of-multinomials%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    -

                    Popular posts from this blog

                    What is the “three and three hundred thousand syndrome”?Who wrote the book Arena?What five creatures were...

                    Image
                    What happens when the centripetal force is equal and opposite to the centrifugal force? Why is "la Gestapo" feminine? Nested Dynamic SOQL Query How are passwords stolen from companies if they only store hashes? Exposing a company lying about themselves in a tightly knit industry: Is my career at risk on the long run? Can "few" be used as a subject? If so, what is the rule? Why are there no stars visible in cislunar space? categorizing a variable turns it from insignificant to significant Do I need an EFI partition for each 18.04 ubuntu I have on my HD? What are the consequences of changing the number of hours in ...
                    Read more

                    Gersau Kjelder | Navigasjonsmeny46°59′0″N 8°31′0″E46°59′0″N...

                    Image
                    Tidlegare republikkarSkipingar i 1433Kommunar i SchwyzDistrikt i SchwyzNedleggingar i 1798Stader ved Vierwaldstättersee kommunekantonenSchwyzSveitskm²VierwaldstätterseeLauerz Gersau Kommunevåpenet til Gersau Plassering Styresmakter Land   Sveits Kanton kantonen Schwyz Distrikt Gersau Språk tysk Geografi Flatevidd  - kommune 23,70 km² Innbyggjarar  - kommune (desember 2010)    - folketettleik 2 094   88,4 /km² Koordinatar 46°59′0″N 8°31′0″E  Koordinatar: 46°59′0″N 8°31′0″E  Høgd over havet 435 moh Diverse annan informasjon Postnummer 6442 Nettstad: www.gersau.ch Freistaat und Republik Gersau Fristaten og republikken Gersau By-republikk og alliert protektorat i Det gamle sveitsiske eidsforundet ←   ← 1433–1798 1814–1818 →   → Hovudstad Gersau Styreform Republikk Historisk periode Tidleg moderne tid  - Kjøpte fridom frå ...
                    Read more

                    Hestehale Innhaldsliste Hestehale på kvinner | Hestehale på menn | Galleri | Sjå òg |...

                    Image
                    Frisyrar frisyrehårhårstrikkhårklemmerøyrethesthovudverkeuropeisk motejobbtreningfletterparykkarungdomsopprørmotkulturmusikararmotorsykkelentusiastarpolitietpolitiskforretningverksemdfiskeflette Hestehale sett bakfrå. Hestehale på ein mannleg indianarleiar. Hestehale er ein frisyre der ein samlar det meste eller alt av langt hår [1] i bakhovudet med ein hårstrikk eller liknande. Avhengig av mote kan det òg vere vanleg å sette opp hestehalen med hårklemmer eller å ha hestehalen hengande på den eine sida av hovudet ved å samle håret ved øyret. Hestehale er ein vanleg frisyre for jenter og for gutar med langt hår. Han har fått namnet sitt etter likskapen med halen på ein hest. For stramme hestehaler kan føre til hårtap, [2] i nokre høve også hovudverk. [3] Innhaldsliste 1 Hestehale på kvinner 2 Hestehale på menn 3 Galleri 4 Sjå òg 5 Bakgrunnsstoff 6 Kjelder Hestehale på kvinner | Ungjente med hestehale som held håret unna ansiktet...
                    Read more