Pixel coloring problem












3














We have an images $m$ pixels length and $n$ pixels width. There are at least $k$ pixels with different colors. Image and its reverse, reflect and rotation are considered to be equivalent. Permuting the colors will not change the structure. We select one concrete pixel and start to change its color, before all pixels in image will be same colored. Next we do the same for each other pixels, so for each pixel exist shortest way to make this image one-colored. There are two different pixels in any image, which have biggest and smallest values of shortest way (of course they may be equal).

If $m=1$ and $n,k$ variable (or $n=1$ and $m,k$ variable) we have OEIS A284949 different images. So my questions are:




  • How can I calculate quantity of different images for $m,n,k$ variable?


  • How can I calculate minimum quantity of coloring choosing pixel (with smallest value of shortest way to make selected image one-colored) for any image of set, bounded by $m,n,k$ variable?


  • How can I calculate minimum quantity of coloring any pixel (include pixel with biggest value of shortest way to make selected image one-colored) for any image of set, bounded by $m,n,k$ variable?











share|cite|improve this question





























    3














    We have an images $m$ pixels length and $n$ pixels width. There are at least $k$ pixels with different colors. Image and its reverse, reflect and rotation are considered to be equivalent. Permuting the colors will not change the structure. We select one concrete pixel and start to change its color, before all pixels in image will be same colored. Next we do the same for each other pixels, so for each pixel exist shortest way to make this image one-colored. There are two different pixels in any image, which have biggest and smallest values of shortest way (of course they may be equal).

    If $m=1$ and $n,k$ variable (or $n=1$ and $m,k$ variable) we have OEIS A284949 different images. So my questions are:




    • How can I calculate quantity of different images for $m,n,k$ variable?


    • How can I calculate minimum quantity of coloring choosing pixel (with smallest value of shortest way to make selected image one-colored) for any image of set, bounded by $m,n,k$ variable?


    • How can I calculate minimum quantity of coloring any pixel (include pixel with biggest value of shortest way to make selected image one-colored) for any image of set, bounded by $m,n,k$ variable?











    share|cite|improve this question



























      3












      3








      3


      1





      We have an images $m$ pixels length and $n$ pixels width. There are at least $k$ pixels with different colors. Image and its reverse, reflect and rotation are considered to be equivalent. Permuting the colors will not change the structure. We select one concrete pixel and start to change its color, before all pixels in image will be same colored. Next we do the same for each other pixels, so for each pixel exist shortest way to make this image one-colored. There are two different pixels in any image, which have biggest and smallest values of shortest way (of course they may be equal).

      If $m=1$ and $n,k$ variable (or $n=1$ and $m,k$ variable) we have OEIS A284949 different images. So my questions are:




      • How can I calculate quantity of different images for $m,n,k$ variable?


      • How can I calculate minimum quantity of coloring choosing pixel (with smallest value of shortest way to make selected image one-colored) for any image of set, bounded by $m,n,k$ variable?


      • How can I calculate minimum quantity of coloring any pixel (include pixel with biggest value of shortest way to make selected image one-colored) for any image of set, bounded by $m,n,k$ variable?











      share|cite|improve this question















      We have an images $m$ pixels length and $n$ pixels width. There are at least $k$ pixels with different colors. Image and its reverse, reflect and rotation are considered to be equivalent. Permuting the colors will not change the structure. We select one concrete pixel and start to change its color, before all pixels in image will be same colored. Next we do the same for each other pixels, so for each pixel exist shortest way to make this image one-colored. There are two different pixels in any image, which have biggest and smallest values of shortest way (of course they may be equal).

      If $m=1$ and $n,k$ variable (or $n=1$ and $m,k$ variable) we have OEIS A284949 different images. So my questions are:




      • How can I calculate quantity of different images for $m,n,k$ variable?


      • How can I calculate minimum quantity of coloring choosing pixel (with smallest value of shortest way to make selected image one-colored) for any image of set, bounded by $m,n,k$ variable?


      • How can I calculate minimum quantity of coloring any pixel (include pixel with biggest value of shortest way to make selected image one-colored) for any image of set, bounded by $m,n,k$ variable?








      combinatorics rotations coloring reflection






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 20 '18 at 6:12

























      asked Nov 10 '18 at 19:29









      user514787

      690210




      690210



























          active

          oldest

          votes











          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: "69"
          };
          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%2fmath.stackexchange.com%2fquestions%2f2993043%2fpixel-coloring-problem%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown






























          active

          oldest

          votes













          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes
















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • 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.





          Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


          Please pay close attention to the following guidance:


          • 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.


          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%2fmath.stackexchange.com%2fquestions%2f2993043%2fpixel-coloring-problem%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

          android studio warns about leanback feature tag usage required on manifest while using Unity exported app?

          SQL update select statement

          WPF add header to Image with URL pettitions [duplicate]