{"version":9,"randomSeed":"86c819015b022980669a56b463974dab","graph":{"viewport":{"xmin":-1.992463466716299,"ymin":-5.030801883096851,"xmax":11.197179020116732,"ymax":7.350057799723576},"showGrid":false,"showXAxis":false,"showYAxis":false},"expressions":{"list":[{"type":"expression","id":"1","color":"#c74440","latex":"w=3","slider":{"hardMin":true,"hardMax":true,"min":"1","step":"1"}},{"type":"expression","id":"2","color":"#2d70b3","latex":"h=3","slider":{"hardMin":true,"hardMax":true,"min":"1","step":"1"}},{"type":"folder","id":"62","title":"implementation","collapsed":true},{"type":"expression","id":"4","folderId":"62","color":"#6042a6","latex":"B\\left(k,i\\right)=\\operatorname{mod}\\left(\\operatorname{floor}\\left(\\frac{k}{2^{i}}\\right),2\\right)"},{"type":"expression","id":"3","folderId":"62","color":"#388c46","latex":"C\\left(x,y,k\\right)=B\\left(k,yw+x\\right)","labelSize":"medium"},{"type":"text","id":"39","folderId":"62","text":"is this image-number k bound-irreducible within w-by-h?"},{"type":"expression","id":"17","folderId":"62","color":"#c74440","latex":"I\\left(k\\right)=\\left\\{\\left(\\sum_{i=0}^{w-1}C\\left(i,0,k\\right)\\right)\\left(\\sum_{i=0}^{w-1}C\\left(i,h-1,k\\right)\\right)\\left(\\sum_{i=0}^{h-1}C\\left(0,i,k\\right)\\right)\\left(\\sum_{i=0}^{h-1}C\\left(w-1,i,k\\right)\\right)>0,0\\right\\}","hidden":true},{"type":"text","id":"41","folderId":"62","text":"rotation of image-number k by a quarter-turn"},{"type":"expression","id":"42","folderId":"62","color":"#388c46","latex":"R_{q}\\left(k\\right)=\\left(\\sum_{i=0}^{h-1}\\left(\\sum_{j=0}^{w-1}2^{\\left(wi+j\\right)}C\\left(w-i-1,j,k\\right)\\right)\\right)","hidden":true},{"type":"text","id":"45","folderId":"62","text":"rotation of image-number k by a half-turn"},{"type":"expression","id":"46","folderId":"62","color":"#c74440","latex":"R_{h}\\left(k\\right)=\\left(\\sum_{i=0}^{h-1}\\left(\\sum_{j=0}^{w-1}2^{\\left(wi+j\\right)}C\\left(w-j-1,h-i-1,k\\right)\\right)\\right)","hidden":true},{"type":"text","id":"78","folderId":"62","text":"binary weight of image-number k (number of on pixels)"},{"type":"expression","id":"79","folderId":"62","color":"#6042a6","latex":"W\\left(k\\right)=\\sum_{i=0}^{wh-1}B\\left(k,i\\right)","hidden":true},{"type":"text","id":"81","folderId":"62","text":"binary subimage of k derived by expanding subimage-number m by 4-adjacencies (with helper functions for each direction's additions"},{"type":"text","id":"93","folderId":"62","text":"terms in the product are, respectively: positioning of bit to add, presence of original cell in m as on-pixel to extend from, presence of new cell in k to extend into, absence of new cell in m"},{"type":"expression","id":"83","folderId":"62","color":"#2d70b3","latex":"E_{u}\\left(k,m\\right)=\\left(\\sum_{i=0}^{h-2}\\left(\\sum_{j=0}^{w-1}\\left(2^{\\left(w\\left(i+1\\right)+j\\right)}\\cdot C\\left(j,i,m\\right)\\cdot C\\left(j,i+1,k\\right)\\cdot\\left(1-C\\left(j,i+1,m\\right)\\right)\\right)\\right)\\right)","labelSize":"medium"},{"type":"expression","id":"89","folderId":"62","color":"#388c46","latex":"E_{d}\\left(k,m\\right)=\\left(\\sum_{i=1}^{h-1}\\left(\\sum_{j=0}^{w-1}\\left(2^{\\left(w\\left(i-1\\right)+j\\right)}\\cdot C\\left(j,i,m\\right)\\cdot C\\left(j,i-1,k\\right)\\cdot\\left(1-C\\left(j,i-1,m\\right)\\right)\\right)\\right)\\right)"},{"type":"expression","id":"90","folderId":"62","color":"#6042a6","latex":"E_{l}\\left(k,m\\right)=\\left(\\sum_{i=0}^{h-1}\\left(\\sum_{j=1}^{w-1}\\left(2^{\\left(wi+j-1\\right)}\\cdot C\\left(j,i,m\\right)\\cdot C\\left(j-1,i,k\\right)\\cdot\\left(1-C\\left(j-1,i,m\\right)\\right)\\right)\\right)\\right)"},{"type":"expression","id":"91","folderId":"62","color":"#000000","latex":"E_{r}\\left(k,m\\right)=\\left(\\sum_{i=0}^{h-1}\\left(\\sum_{j=0}^{w-2}\\left(2^{\\left(wi+j+1\\right)}\\cdot C\\left(j,i,m\\right)\\cdot C\\left(j+1,i,k\\right)\\cdot\\left(1-C\\left(j+1,i,m\\right)\\right)\\right)\\right)\\right)"},{"type":"expression","id":"82","folderId":"62","color":"#c74440","latex":"E_{4}\\left(k,m\\right)=m+U\\left(\\left[E_{u}\\left(k,m\\right),E_{d}\\left(k,m\\right),E_{l}\\left(k,m\\right),E_{r}\\left(k,m\\right)\\right]\\right)"},{"type":"text","id":"101","folderId":"62","text":"top bit of k"},{"type":"expression","id":"102","folderId":"62","color":"#2d70b3","latex":"T\\left(k\\right)=2^{\\operatorname{floor}\\left(\\log_{2}k\\right)}","hidden":true},{"type":"text","id":"108","folderId":"62","text":"is binary image-number k simply connected (a polyomino)?"},{"type":"expression","id":"109","folderId":"62","color":"#388c46","latex":"S\\left(k\\right)=\\left\\{k=0:0,k=E_{4}\\left(k,E_{4}\\left(k,E_{4}\\left(k,E_{4}\\left(k,E_{4}\\left(k,E_{4}\\left(k,E_{4}\\left(k,T\\left(k\\right)\\right)\\right)\\right)\\right)\\right)\\right)\\right),0\\right\\}","hidden":true},{"type":"text","id":"120","folderId":"62","text":"union (not same as sum) of list of image-numbers"},{"type":"expression","id":"118","folderId":"62","color":"#000000","latex":"U\\left(L\\right)=\\sum_{i=0}^{wh-1}\\left(2^{i}\\cdot\\left(1-\\prod_{j=1}^{\\operatorname{length}\\left(L\\right)}\\left(1-B\\left(L\\left[j\\right],i\\right)\\right)\\right)\\right)"},{"type":"text","id":"49","text":"total bound-irreducible images w-by-h, counting rotations distinctly"},{"type":"expression","id":"22","color":"#c74440","latex":"\\sum_{j=0}^{2^{wh}-1}I\\left(j\\right)","labelSize":"medium"},{"type":"text","id":"52","text":"total bound-irreducible images w-by-h, reducing rotations"},{"type":"expression","id":"53","color":"#c74440","latex":"\\sum_{s=0}^{2^{wh}-1}\\left(I\\left(s\\right)\\cdot\\left\\{w=h:\\left\\{s\\le R_{q}\\left(s\\right),0\\right\\},1\\right\\}\\cdot\\left\\{w=h:\\left\\{s\\le R_{q}\\left(R_{h}\\left(s\\right)\\right),0\\right\\},1\\right\\}\\cdot\\left\\{s\\le R_{h}\\left(s\\right),0\\right\\}\\right)"},{"type":"text","id":"95","text":"total simply connected images (polyominoes), bound-irreducible in w-by-h, reducting rotations"},{"type":"expression","id":"96","color":"#388c46","latex":"\\sum_{s=0}^{2^{wh}-1}\\left(S\\left(s\\right)\\cdot I\\left(s\\right)\\cdot\\left\\{w=h:\\left\\{s\\le R_{q}\\left(s\\right),0\\right\\},1\\right\\}\\cdot\\left\\{w=h:\\left\\{s\\le R_{q}\\left(R_{h}\\left(s\\right)\\right),0\\right\\},1\\right\\}\\cdot\\left\\{s\\le R_{h}\\left(s\\right),0\\right\\}\\right)"},{"type":"folder","id":"55","title":"results (distinct rotations)","collapsed":true},{"type":"text","id":"59","folderId":"55","text":"for h=1, w=n"},{"type":"expression","id":"60","folderId":"55","color":"#6042a6","latex":"I_{T1}=\\left[1,1,2,4,8,16,32,64\\right]"},{"type":"text","id":"24","folderId":"55","text":"for h=2, w=n"},{"type":"expression","id":"25","folderId":"55","color":"#388c46","latex":"I_{T2}=\\left[1,7,32,136,560,2272,9152,36736\\right]","labelSize":"medium"},{"type":"text","id":"27","folderId":"55","text":"for h=3, w=n"},{"type":"expression","id":"28","folderId":"55","color":"#000000","latex":"I_{T3}=\\left[2,32,322,2852,23944\\right]"},{"type":"text","id":"30","folderId":"55","text":"for h=4, w=n"},{"type":"expression","id":"31","folderId":"55","color":"#2d70b3","latex":"I_{T4}=\\left[4,136,2852,51472\\right]"},{"type":"folder","id":"57","title":"results (collapsed rotations)","collapsed":true},{"type":"text","id":"68","folderId":"57","text":"for h=1, w=n"},{"type":"expression","id":"69","folderId":"57","color":"#000000","latex":"U_{T1}=\\left[1,1,2,3,6,10,20,36,72,136\\right]"},{"type":"text","id":"71","folderId":"57","text":"for h=2, w=n"},{"type":"expression","id":"72","folderId":"57","color":"#2d70b3","latex":"U_{T2}=\\left[1,3,19,74,292,1160,4624\\right]"},{"type":"text","id":"75","folderId":"57","text":"for h=3, w=n"},{"type":"expression","id":"76","folderId":"57","color":"#2d70b3","latex":"U_{T3}=\\left[2,19,90,1453,12082\\right]"},{"type":"text","id":"73","folderId":"57","text":"for h=4, w=n"},{"type":"expression","id":"74","folderId":"57","color":"#2d70b3","latex":"U_{T4}=\\left[3,74,1453,12932\\right]"},{"type":"text","id":"37","text":"see https://oeis.org/A163437 (values are for h=n, w=n)"},{"type":"text","id":"35","text":"display and manual counting"},{"type":"expression","id":"10","color":"#000000","latex":"n=13","slider":{"hardMin":true,"hardMax":true,"animationPeriod":80000,"playDirection":-1,"min":"0","max":"2^{wh}-1","step":"1"}},{"type":"expression","id":"43","color":"#6042a6","latex":"S\\left(n\\right)","labelSize":"medium"},{"type":"expression","id":"123","color":"#6042a6","latex":"E_{4}\\left(47,32\\right)"},{"type":"expression","id":"14","color":"#388c46","latex":"n\\to n+1"},{"type":"expression","id":"15","color":"#6042a6","latex":"p=135","slider":{"hardMin":true,"min":"0","step":"1"}},{"type":"expression","id":"16","color":"#000000","latex":"p\\to p+1,n\\to n+1"},{"type":"folder","id":"64","title":"display","collapsed":true},{"type":"expression","id":"20","folderId":"64","color":"#388c46","latex":"I\\left(n+0x\\right)>0\\left\\{-\\frac{1}{2}\\le x\\le w+\\frac{1}{2}\\right\\}\\left\\{-\\frac{1}{2}\\le y\\le h+\\frac{1}{2}\\right\\}","fillOpacity":"1"},{"type":"expression","id":"9","folderId":"64","color":"#000000","latex":"C\\left(\\operatorname{floor}\\left(x\\right),\\operatorname{floor}\\left(y\\right),n\\right)\\ge0.5\\left\\{0\\le x\\le w\\right\\}\\left\\{0\\le y\\le h\\right\\}","lines":false,"fillOpacity":"1"},{"type":"expression","id":"11","folderId":"64","color":"#c74440","latex":"x=\\left[0,w\\right]\\left\\{0\\le y\\le h\\right\\}","lineOpacity":"1","lineWidth":"6"},{"type":"expression","id":"13","folderId":"64","color":"#c74440","latex":"y=\\left[0,h\\right]\\left\\{0\\le x\\le w\\right\\}","lineOpacity":"1","lineWidth":"6"}]}}