Cycles on the torus












6












$begingroup$


Challenge



This challenge will have you write a program that takes in two integers n and m and outputs the number non-intersecting loops on the n by m torus made by only taking steps up and to the right. You can think of torus as the grid with wraparound both at the top and the bottom.



This is code-golf so fewest bytes wins.



Example



For example, if the input is n=m=5, one valid walk is 



(0,0) -> (0,1) -> (0,2) -> (1,2) -> (2,2) -> (2,3) -> (2,4) -> 

(2,0) -> (3,0) -> (4,0) -> (4,1) -> (4,2) -> (4,3) -> 

(0,3) -> (1,3) -> (1,4) -> 

(1,0) -> (1,1) -> (2,1) -> (3,1) -> (3,2) -> (3,3) -> (3,4) -> (4,4) -> (0,4) -> (0,0)


as shown in the graphic.



A loop on the torus.



Some example input/outputs 



f(1,1) = 2 (up or right)

f(1,2) = 2 (up or right-right)

f(2,2) = 4 (up-up, up-right-up-right, right-right, right-up-right-up)

f(2,3) = 7

f(3,3) = 22

f(2,4) = 13

f(3,4) = 66

f(4,4) = 258





code-golf combinatorics grid







share|improve this question









$endgroup$

















    6












    $begingroup$


    Challenge



    This challenge will have you write a program that takes in two integers n and m and outputs the number non-intersecting loops on the n by m torus made by only taking steps up and to the right. You can think of torus as the grid with wraparound both at the top and the bottom.



    This is code-golf so fewest bytes wins.



    Example



    For example, if the input is n=m=5, one valid walk is 



    (0,0) -> (0,1) -> (0,2) -> (1,2) -> (2,2) -> (2,3) -> (2,4) -> 
    
(2,0) -> (3,0) -> (4,0) -> (4,1) -> (4,2) -> (4,3) -> 
    
(0,3) -> (1,3) -> (1,4) -> 
    
(1,0) -> (1,1) -> (2,1) -> (3,1) -> (3,2) -> (3,3) -> (3,4) -> (4,4) -> (0,4) -> (0,0)


    as shown in the graphic.



    A loop on the torus.



    Some example input/outputs 



    f(1,1) = 2 (up or right)
    
f(1,2) = 2 (up or right-right)
    
f(2,2) = 4 (up-up, up-right-up-right, right-right, right-up-right-up)
    
f(2,3) = 7
    
f(3,3) = 22
    
f(2,4) = 13
    
f(3,4) = 66
    
f(4,4) = 258





    code-golf combinatorics grid







    share|improve this question









    $endgroup$















      6












      6








      6





      $begingroup$


      Challenge



      This challenge will have you write a program that takes in two integers n and m and outputs the number non-intersecting loops on the n by m torus made by only taking steps up and to the right. You can think of torus as the grid with wraparound both at the top and the bottom.



      This is code-golf so fewest bytes wins.



      Example



      For example, if the input is n=m=5, one valid walk is 



      (0,0) -> (0,1) -> (0,2) -> (1,2) -> (2,2) -> (2,3) -> (2,4) -> 
      
(2,0) -> (3,0) -> (4,0) -> (4,1) -> (4,2) -> (4,3) -> 
      
(0,3) -> (1,3) -> (1,4) -> 
      
(1,0) -> (1,1) -> (2,1) -> (3,1) -> (3,2) -> (3,3) -> (3,4) -> (4,4) -> (0,4) -> (0,0)


      as shown in the graphic.



      A loop on the torus.



      Some example input/outputs 



      f(1,1) = 2 (up or right)
      
f(1,2) = 2 (up or right-right)
      
f(2,2) = 4 (up-up, up-right-up-right, right-right, right-up-right-up)
      
f(2,3) = 7
      
f(3,3) = 22
      
f(2,4) = 13
      
f(3,4) = 66
      
f(4,4) = 258





      code-golf combinatorics grid







      share|improve this question









      $endgroup$




      Challenge



      This challenge will have you write a program that takes in two integers n and m and outputs the number non-intersecting loops on the n by m torus made by only taking steps up and to the right. You can think of torus as the grid with wraparound both at the top and the bottom.



      This is code-golf so fewest bytes wins.



      Example



      For example, if the input is n=m=5, one valid walk is 



      (0,0) -> (0,1) -> (0,2) -> (1,2) -> (2,2) -> (2,3) -> (2,4) -> 
      
(2,0) -> (3,0) -> (4,0) -> (4,1) -> (4,2) -> (4,3) -> 
      
(0,3) -> (1,3) -> (1,4) -> 
      
(1,0) -> (1,1) -> (2,1) -> (3,1) -> (3,2) -> (3,3) -> (3,4) -> (4,4) -> (0,4) -> (0,0)


      as shown in the graphic.



      A loop on the torus.



      Some example input/outputs 



      f(1,1) = 2 (up or right)
      
f(1,2) = 2 (up or right-right)
      
f(2,2) = 4 (up-up, up-right-up-right, right-right, right-up-right-up)
      
f(2,3) = 7
      
f(3,3) = 22
      
f(2,4) = 13
      
f(3,4) = 66
      
f(4,4) = 258





      code-golf combinatorics grid




      code-golf combinatorics grid






      share|improve this question













      share|improve this question











      share|improve this question




      share|improve this question










      asked 4 hours ago









      Peter KageyPeter Kagey

      878518




      878518






















          1 Answer
          1






          active

          oldest

          votes


















          3












          $begingroup$


          Python 2, 87 bytes





          f=lambda m,n,z=0,l=:z==0if z in l else sum(f(m,n,(z+d)%m%(n*1j),l+[z])for d in(1,1j))


          Try it online!



          The interesting thing here is using a complex number z to store the coordinate of the current position. We can move up by adding 1 and move right by adding 1j. To my surprise, modulo works on complex numbers in a way that lets us handle the wrapping for each dimension separately: doing %m acts on the real part, and %(n*1j) acts on the imaginary part.






          share|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.ifUsing("editor", function () {
            StackExchange.using("externalEditor", function () {
            StackExchange.using("snippets", function () {
            StackExchange.snippets.init();
            });
            });
            }, "code-snippets");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "200"
            };
            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: false,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: null,
            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
            },
            onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcodegolf.stackexchange.com%2fquestions%2f181203%2fcycles-on-the-torus%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









            3












            $begingroup$


            Python 2, 87 bytes





            f=lambda m,n,z=0,l=:z==0if z in l else sum(f(m,n,(z+d)%m%(n*1j),l+[z])for d in(1,1j))


            Try it online!



            The interesting thing here is using a complex number z to store the coordinate of the current position. We can move up by adding 1 and move right by adding 1j. To my surprise, modulo works on complex numbers in a way that lets us handle the wrapping for each dimension separately: doing %m acts on the real part, and %(n*1j) acts on the imaginary part.






            share|improve this answer









            $endgroup$


















              3












              $begingroup$


              Python 2, 87 bytes





              f=lambda m,n,z=0,l=:z==0if z in l else sum(f(m,n,(z+d)%m%(n*1j),l+[z])for d in(1,1j))


              Try it online!



              The interesting thing here is using a complex number z to store the coordinate of the current position. We can move up by adding 1 and move right by adding 1j. To my surprise, modulo works on complex numbers in a way that lets us handle the wrapping for each dimension separately: doing %m acts on the real part, and %(n*1j) acts on the imaginary part.






              share|improve this answer









              $endgroup$
















                3












                3








                3





                $begingroup$


                Python 2, 87 bytes





                f=lambda m,n,z=0,l=:z==0if z in l else sum(f(m,n,(z+d)%m%(n*1j),l+[z])for d in(1,1j))


                Try it online!



                The interesting thing here is using a complex number z to store the coordinate of the current position. We can move up by adding 1 and move right by adding 1j. To my surprise, modulo works on complex numbers in a way that lets us handle the wrapping for each dimension separately: doing %m acts on the real part, and %(n*1j) acts on the imaginary part.






                share|improve this answer









                $endgroup$




                Python 2, 87 bytes





                f=lambda m,n,z=0,l=:z==0if z in l else sum(f(m,n,(z+d)%m%(n*1j),l+[z])for d in(1,1j))


                Try it online!



                The interesting thing here is using a complex number z to store the coordinate of the current position. We can move up by adding 1 and move right by adding 1j. To my surprise, modulo works on complex numbers in a way that lets us handle the wrapping for each dimension separately: doing %m acts on the real part, and %(n*1j) acts on the imaginary part.







                share|improve this answer












                share|improve this answer



                share|improve this answer










                answered 2 hours ago









                xnorxnor

                91.8k18187444




                91.8k18187444






























                    draft saved

                    draft discarded




















































                    If this is an answer to a challenge…




                    • …Be sure to follow the challenge specification. However, please refrain from exploiting obvious loopholes. Answers abusing any of the standard loopholes are considered invalid. If you think a specification is unclear or underspecified, comment on the question instead.


                    • …Try to optimize your score. For instance, answers to code-golf challenges should attempt to be as short as possible. You can always include a readable version of the code in addition to the competitive one.
                      Explanations of your answer make it more interesting to read and are very much encouraged.


                    • …Include a short header which indicates the language(s) of your code and its score, as defined by the challenge.



                    More generally…




                    • …Please make sure to answer the question and provide sufficient detail.


                    • …Avoid asking for help, clarification or responding to other answers (use comments instead).





                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcodegolf.stackexchange.com%2fquestions%2f181203%2fcycles-on-the-torus%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 other Star Trek series did the main TNG cast show up in?

                    Image
                    .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0; } 4 I'm referring to Picard, Riker, Data, Worf, Geordi, Dr. Crusher, Wesley Crusher, Chief O'Brien and family, Deanna Troi, Guinan. I know Picard and Worf show up on DS9, though I haven't seen it myself. I'm curious what happened to the characters after that last episode of TNG? star-trek star-trek-tng share | improve this question edited 10 hours ago Ham Sandwich 6,051 2 25 75 ...
                    Read more

                    Berlina muro

                    Image
                    La berlina muro ĉe Bethaniendamm, 1986 La berlina muro aŭ simple la muro [1] , nomita ankaŭ "kontraŭfaŝisma remparo" en propaganda lingvo de la iama Germana Demokratia Respubliko, estis parto de la interngermana limo, kiu dividis de la 13-a de aŭgusto 1961 ĝis la 9-a de novembro 1989 okcidentan Berlinon de la orienta parto de la urbo kaj de la ĉirkaŭa tereno de la GDR. Ĝi estis unu el la plej famaj simboloj de la malvarma milito kaj de la disdivido de Germanio. Pli ol simpla muro, la berlina muro estis fakte tuta defendokonstruaĵo kun du muroj, ena malplena strio, rondirvojo, gardturoj kaj alarmsistemoj. Dum la provo transiri la severe garditan limon al okcidenta Berlino multaj homoj estis mortigitaj. La preciza nombro de viktimoj estas pridisputita kaj ne certa; la nombroj anoncataj varias inter 86 kaj 238 mortintoj. La malplifortiĝo de Sovetunio kaj la politiko de liberigo gvidita de Miĥail Gorbaĉov ebligis al la orientgermanoj faligi la 9-an de novembro 1989 l...
                    Read more

                    Berlina aerponto

                    Image
                    Flugkoridoroj dum la blokado de Berlino La Berlina aerponto estas la provizigo de okcidenta Berlino per aviadiloj fare de la aliancanoj inter la 23-a de junio 1948 kaj la 12-a de majo 1949 dum la blokado de Berlino fare de la sovetia armeo. Enhavo 1 Historio 1.1 La aerkoridoroj 1.2 Unuaj flugoj 1.3 Plibonigo de la aerponto 1.4 Fino de la aerponto 1.5 Rozinbombiloj 1.6 Bilanco 2 Flughavenoj 2.1 Berlino 2.2 Okcidenta Germanio 3 Aviadiloj de la berlina aerponto 4 15-a datreveno de la aerponto 5 Bibliografio 6 Referencoj 7 Eksteraj ligiloj Historio | General Lucius D. Clay, militguberniestro de la usona okupadzono Jam antaŭ la berlina aerponto estiĝis la malgranda aerponto . La estro de la sovetia militadministracio en Germanio, Vasili Daniloviĉ Sokolovski, ordonis la 1-an de aprilo 1948, kiel reago al la konferenco de Londono inter britoj, francoj kaj reprezentantoj de Benelukso pri aligo de Germa...
                    Read more