Jonathan Burns 10/09/2009 jtburns@mail.usf.edu =========================================================== Assembly Graph Words - Single Transverse Component (Counts) =========================================================== The table below records values obtained through a search using Mathematica. The values correspond to simple assembly graphs with one transversal component and two endpoints. We consider an assembly word to be a permutation of {1,1,2,2,...,n,n} such that when the ith term appears for the first time in the word, it is preceeded by 1,2,...,i-1. Ex: 112332 is an assembly word 113232 is not an assemby word (first 3 not preceeded by 2) Assembly words are associated with a transversal orientation. The assembly word generated by reversing the orientation of the knot is considered isomorphic to the original word. Assembly words which are invariant under orientation reversal are called palindromes. Ex: 12341243 is the orientation reversal of 12342134 (reverse word and map 4->1, 3->2, 1->3, 2->4) Ex: 121323 is a palindrome An assembly word is reducible if it can be written as the composistion of assembly words by matching endpoints. A assembly word that cannot be reduced is called irreducible. Ex: 122133 can be reduced to the composition of 1221 and 11 (33 gets relabeled to 11). 123213 is irreducible An assembly word is called prime if it contains no sub-assembly graphs. By definition, a prime assembly word is also irreducible. Ex: 123231 can be reduced to the graphs 11 and 1212 (2323 relabeled). 123213 is prime. Below are the results of the search. ================================================================================================================= | Rigid Vert. | Assembly Words Palindromes Isomorphisms* Realizable Irreducible Prime | ================================================================================================================= | 1 | 1 1 1 1 1 1 | | 2 | 3 3 3 3 2 1 | | 3 | 15 7 11 11 8 3 | | 4 | 105 25 65 64 48 17 | | 5 | 945 81 513 504 397 135 | | 6 | 10395 331 5363 5241 4323 1463 | | 7 | 135135 1303 68219 66515 56937 19306 | ================================================================================================================= | OEIS | A001147 A047974 ** ---- ---- ---- | ================================================================================================================= * Number of unique words considering orientation reversals to be isomorphic. ** (Assembly Words + Palindromes)/2 ================================================================================================================= | | Number of Rigid Verticies | | Assembly |-------------------------------------------------------------------------------------------------| | Number | 2 3 4 5 6 7 | ================================================================================================================= | 1 | 3 11 64 504 5241 66515 | | 2 | 0 0 1 9 122 1701 | | 3 | 0 0 0 0 0 3 | ================================================================================================================= Irreducible Words ==================== n = 1 ---------- {11} n = 2 ---------- {1212, 1221} n = 3 ---------- {122313, 122331, 121323, 123123, 123213, 123231, 123312, 123321} n = 4 ---------- {12233414, 12233441, 12323414, 12323441, 12332414, 12332441, 12133424, 12313424, 12331424, 12334124, 12334214, 12334241, 12313442, 12331442, 12334142, 12334412, 12334421, 12123434, 12231434, 12234134, 12132434, 12312434, 12321434, 12324134, 12324314, 12324341, 12314234, 12341234, 12342134, 12342314, 12342341, 12314324, 12341324, 12343124, 12343214, 12343241, 12343142, 12343412, 12343421, 12231443, 12234143, 12324143, 12342143, 12344123, 12344213, 12344231, 12344312, 12344321} n = 5 ---------- {1223344515, 1223344551, 1232344515, 1232344551, 1233244515, 1233244551, 1233424515, 1233424551, 1233442515, 1233442551, 1213344525, 1231344525, 1233144525, 1233414525, 1233441525, 1233445125, 1233445215, 1233445251, 1231344552, 1233144552, 1233414552, 1233441552, 1233445152, 1233445512, 1233445521, 1223434515, 1223434551, 1232434515, 1232434551, 1234234515, 1234234551, 1234324515, 1234324551, 1234342515, 1234342551, 1213434525, 1231434525, 1234134525, 1234314525, 1234341525, 1234345125, 1234345215, 1234345251, 1231434552, 1234134552, 1234314552, 1234341552, 1234345152, 1234345512, 1234345521, 1223443515, 1223443551, 1232443515, 1232443551, 1234243515, 1234243551, 1234423515, 1234423551, 1234432515, 1234432551, 1213443525, 1231443525, 1234143525, 1234413525, 1234431525, 1234435125, 1234435215, 1234435251, 1231443552, 1234143552, 1234413552, 1234431552, 1234435152, 1234435512, 1234435521, 1212344535, 1223144535, 1223414535, 1223441535, 1223445135, 1213244535, 1231244535, 1232144535, 1232414535, 1232441535, 1232445135, 1232445315, 1232445351, 1213424535, 1231424535, 1234124535, 1234214535, 1234241535, 1234245135, 1234245315, 1234245351, 1213442535, 1231442535, 1234142535, 1234412535, 1234421535, 1234425135, 1234425315, 1234425351, 1231445235, 1234145235, 1234415235, 1234451235, 1234452135, 1234452315, 1234452351, 1231445325, 1234145325, 1234415325, 1234451325, 1234453125, 1234453215, 1234453251, 1234145352, 1234415352, 1234451352, 1234453152, 1234453512, 1234453521, 1223144553, 1223414553, 1223441553, 1223445153, 1213244553, 1231244553, 1232144553, 1232414553, 1232445153, 1213424553, 1231424553, 1234124553, 1234214553, 1234241553, 1234245153, 1234245513, 1234245531, 1234142553, 1234412553, 1234421553, 1234425153, 1234425513, 1234425531, 1234145253, 1234415253, 1234451253, 1234452153, 1234452513, 1234452531, 1234145523, 1234415523, 1234451523, 1234455123, 1234455213, 1234455231, 1234145532, 1234415532, 1234451532, 1234455312, 1234455321, 1212334545, 1223134545, 1213234545, 1231234545, 1232134545, 1232341545, 1232345145, 1232345415, 1232345451, 1231324545, 1233124545, 1233241545, 1233245145, 1233245415, 1233245451, 1231342545, 1233142545, 1233412545, 1233421545, 1233425145, 1233425415, 1233425451, 1231345245, 1233145245, 1233415245, 1233451245, 1233452145, 1233452415, 1233452451, 1231345425, 1233145425, 1233415425, 1233451425, 1233145452, 1233415452, 1233451452, 1223143545, 1223413545, 1213243545, 1231243545, 1232143545, 1232413545, 1232431545, 1232435145, 1232435415, 1232435451, 1213423545, 1231423545, 1234123545, 1234213545, 1234231545, 1234235145, 1234235415, 1234235451, 1231432545, 1234132545, 1234312545, 1234321545, 1234325145, 1234325415, 1234325451, 1231435245, 1234135245, 1234315245, 1234351245, 1234352145, 1234352415, 1234352451, 1231435425, 1234135425, 1234315425, 1234351425, 1234354125, 1234354215, 1234354251, 1234135452, 1234315452, 1234351452, 1234354152, 1234354512, 1234354521, 1223145345, 1223415345, 1223451345, 1231245345, 1232145345, 1232415345, 1232451345, 1232453145, 1231425345, 1234125345, 1234215345, 1234251345, 1234253145, 1234253415, 1234253451, 1234152345, 1234512345, 1234521345, 1234523145, 1234523415, 1234523451, 1231453245, 1234153245, 1234513245, 1234531245, 1234532145, 1234532415, 1234532451, 1234153425, 1234513425, 1234531425, 1234534125, 1234534215, 1234534251, 1234531452, 1234534152, 1234534512, 1234534521, 1223145435, 1223415435, 1223451435, 1232145435, 1232415435, 1232451435, 1234215435, 1234251435, 1234254135, 1234254315, 1234254351, 1234521435, 1234524135, 1234524315, 1234524351, 1234154235, 1234514235, 1234541235, 1234542135, 1234542315, 1234542351, 1234154325, 1234514325, 1234541325, 1234543125, 1234543215, 1234543251, 1234541352, 1234543152, 1234543512, 1234543521, 1223145453, 1223415453, 1223451453, 1232145453, 1232415453, 1232451453, 1234215453, 1234251453, 1234254153, 1234521453, 1234524153, 1234541253, 1234542153, 1234542513, 1234542531, 1234541523, 1234545123, 1234545213, 1234545231, 1234541532, 1234545312, 1234545321, 1233241554, 1233245154, 1233245514, 1233245541, 1233142554, 1233412554, 1233421554, 1233425154, 1233425514, 1233425541, 1233145254, 1233415254, 1233451254, 1233452154, 1233452514, 1233452541, 1233145524, 1233415524, 1233451524, 1233451542, 1223143554, 1223413554, 1232143554, 1232413554, 1234213554, 1234312554, 1234321554, 1234325154, 1234315254, 1234351254, 1234352154, 1234352514, 1234352541, 1234351524, 1232145354, 1232415354, 1232451354, 1234251354, 1234521354, 1234531254, 1234532154, 1234532514, 1234532541, 1234531524, 1234551234, 1234552134, 1234552314, 1234552341, 1234551324, 1234553124, 1234553214, 1234553241, 1234553142, 1234553412, 1234553421, 1234552143, 1234554123, 1234554213, 1234554231, 1234554312, 1234554321} Prime Words ==================== n = 1 ---------- {11} n = 2 ---------- {1212} n = 3 ---------- {121323, 123123, 123213} n = 4 ---------- {12313424, 12132434, 12312434, 12321434, 12324134, 12324314, 12314234, 12341234, 12342134, 12342314, 12314324, 12341324, 12343124, 12343214, 12343142, 12324143, 12342143} n = 5 ---------- {1231434525, 1234134525, 1234314525, 1234143525, 1232414535, 1213424535, 1231424535, 1234124535, 1234214535, 1234241535, 1234245135, 1234245315, 1234142535, 1234145235, 1234145325, 1234145352, 1234245153, 1234145253, 1231342545, 1231345245, 1231345425, 1213243545, 1231243545, 1232143545, 1232413545, 1232431545, 1232435145, 1232435415, 1213423545, 1231423545, 1234123545, 1234213545, 1234231545, 1234235145, 1234235415, 1231432545, 1234132545, 1234312545, 1234321545, 1234325145, 1234325415, 1231435245, 1234135245, 1234315245, 1234351245, 1234352145, 1234352415, 1231435425, 1234135425, 1234315425, 1234351425, 1234354125, 1234354215, 1234135452, 1234315452, 1234351452, 1234354152, 1231245345, 1232145345, 1232415345, 1232451345, 1232453145, 1231425345, 1234125345, 1234215345, 1234251345, 1234253145, 1234253415, 1234152345, 1234512345, 1234521345, 1234523145, 1234523415, 1231453245, 1234153245, 1234513245, 1234531245, 1234532145, 1234532415, 1234153425, 1234513425, 1234531425, 1234534125, 1234534215, 1234531452, 1234534152, 1232145435, 1232415435, 1232451435, 1234215435, 1234251435, 1234254135, 1234254315, 1234521435, 1234524135, 1234524315, 1234154235, 1234514235, 1234541235, 1234542135, 1234542315, 1234154325, 1234514325, 1234541325, 1234543125, 1234543215, 1234541352, 1234543152, 1232415453, 1232451453, 1234215453, 1234251453, 1234254153, 1234521453, 1234524153, 1234541253, 1234542153, 1234542513, 1234541523, 1234541532, 1234325154, 1234315254, 1234351254, 1234352154, 1234352514, 1234351524, 1232145354, 1232415354, 1232451354, 1234251354, 1234521354, 1234531254, 1234532154, 1234532514, 1234531524} n = 6 ---------- {123414535626, 123424515636, 123414525636, 123143545626, 123413545626, 123431545626, 123435145626, 123435415626, 123145345626, 123415345626, 123451345626, 123453145626, 123453415626, 123145435626, 123415435626, 123451435626, 123454135626, 123454315626, 123415453626, 123451453626, 123454153626, 123241545636, 123245145636, 123245415636, 121342545636, 123142545636, 123412545636, 123421545636, 123425145636, 123425415636, 123425451636, 123425456136, 123425456316, 121345245636, 123145245636, 123415245636, 123451245636, 123452145636, 123452415636, 123452451636, 123452456136, 123452456316, 121345425636, 123145425636, 123415425636, 123451425636, 123454125636, 123454215636, 123454251636, 123454256136, 123454256316, 123415452636, 123451452636, 123454152636, 123415456236, 123451456236, 123454156236, 123415456326, 123451456326, 123454156326, 123415456362, 123451456362, 123454156362, 123425456163, 123452456163, 123454256163, 123415456263, 123451456263, 123454156263, 123435154626, 123145354626, 123415354626, 123451354626, 123453154626, 123453514626, 123451534626, 123451543626, 123245154636, 123425154636, 121345254636, 123145254636, 123415254636, 123451254636, 123452154636, 123452514636, 123452541636, 123452546136, 123452546316, 123451524636, 123451542636, 123451546236, 123451546326, 123451546362, 123452546163, 123451546263, 123134525646, 123243515646, 123423515646, 123432515646, 121343525646, 123143525646, 123413525646, 123431525646, 123435125646, 123435215646, 123435251646, 123435256146, 123435256416, 123435152646, 123435156246, 123435156426, 123435156462, 121324535646, 123124535646, 123214535646, 123241535646, 123245135646, 123245315646, 123245351646, 123245356146, 123245356416, 121342535646, 123142535646, 123412535646, 123421535646, 123425135646, 123425315646, 123425351646, 123425356146, 123425356416, 121345235646, 123145235646, 123415235646, 123451235646, 123452135646, 123452315646, 123452351646, 123452356146, 123452356416, 121345325646, 123145325646, 123415325646, 123451325646, 123453125646, 123453215646, 123453251646, 123453256146, 123453256416, 123145352646, 123415352646, 123451352646, 123453152646, 123453512646, 123453521646, 123453526146, 123453526416, 123145356246, 123415356246, 123451356246, 123453156246, 123453516246, 123453561246, 123453562146, 123453562416, 123145356426, 123415356426, 123451356426, 123453156426, 123453516426, 123453561426, 123453564126, 123453564216, 123145356462, 123415356462, 123451356462, 123453156462, 123453516462, 123453561462, 123453564162, 123245153646, 123425153646, 121345253646, 123145253646, 123415253646, 123451253646, 123452153646, 123452513646, 123452531646, 123452536146, 123452536416, 123451523646, 123451532646, 123451536246, 123451536426, 123451536462, 123245156346, 123425156346, 121345256346, 123145256346, 123415256346, 123451256346, 123452156346, 123452516346, 123452561346, 123452563146, 123452563416, 123451526346, 123451562346, 123451563246, 123451563426, 123451563462, 123245156436, 123425156436, 121345256436, 123145256436, 123415256436, 123451256436, 123452156436, 123452516436, 123452561436, 123452564136, 123452564316, 123451526436, 123451562436, 123451564236, 123451564326, 123451564362, 123245156463, 123425156463, 121345256463, 123145256463, 123415256463, 123451256463, 123452156463, 123452516463, 123452561463, 123452564163, 123452564613, 123451526463, 123451562463, 123451564263, 123451564623, 123451564632, 123435256164, 123435156264, 123245356164, 123425356164, 123452356164, 123453256164, 123453526164, 123145356264, 123415356264, 123451356264, 123453156264, 123453516264, 123453561264, 123453562164, 123453562614, 123453561624, 123453561642, 123452536164, 123451536264, 123245156364, 123425156364, 121345256364, 123145256364, 123415256364, 123451256364, 123452156364, 123452516364, 123452561364, 123452563164, 123452563614, 123451526364, 123451562364, 123451563264, 123451563624, 123451563642, 123452561634, 123451562634, 123452561643, 123451562643, 123143452656, 123413452656, 123431452656, 123143456256, 123413456256, 123431456256, 123143456526, 123413456526, 123431456526, 123414352656, 123414356256, 123414356526, 123241453656, 121342453656, 123142453656, 123412453656, 123421453656, 123424153656, 123424513656, 123424531656, 123424536156, 123424536516, 123414253656, 123414523656, 123414532656, 123414536256, 123414536526, 123414536562, 123241456356, 121342456356, 123142456356, 123412456356, 123421456356, 123424156356, 123424516356, 123424561356, 123424563156, 123424563516, 123414256356, 123414526356, 123414562356, 123414563256, 123414563526, 123414563562, 123241456536, 121342456536, 123142456536, 123412456536, 123421456536, 123424156536, 123424516536, 123424561536, 123424565136, 123424565316, 123414256536, 123414526536, 123414562536, 123414565236, 123414565326, 123424516563, 123424561563, 123424565163, 123414526563, 123414562563, 123134254656, 123134524656, 121324354656, 123124354656, 123214354656, 123241354656, 123243154656, 123243514656, 123243541656, 123243546156, 123243546516, 121342354656, 123142354656, 123412354656, 123421354656, 123423154656, 123423514656, 123423541656, 123423546156, 123423546516, 123143254656, 123413254656, 123431254656, 123432154656, 123432514656, 123432541656, 123432546156, 123432546516, 123143524656, 123413524656, 123431524656, 123435124656, 123435214656, 123435241656, 123435246156, 123435246516, 123143542656, 123413542656, 123431542656, 123435142656, 123435412656, 123435421656, 123435426156, 123435426516, 123143546256, 123413546256, 123431546256, 123435146256, 123435416256, 123435461256, 123435462156, 123435462516, 123143546526, 123413546526, 123431546526, 123435146526, 123435416526, 123435461526, 123435465126, 123435465216, 123413546562, 123431546562, 123435146562, 123435416562, 123435461562, 123435465162, 123124534656, 123214534656, 123241534656, 123245134656, 123245314656, 123245341656, 123245346156, 123245346516, 121342534656, 123142534656, 123412534656, 123421534656, 123425134656, 123425314656, 123425341656, 123425346156, 123425346516, 121345234656, 123145234656, 123415234656, 123451234656, 123452134656, 123452314656, 123452341656, 123452346156, 123452346516, 123145324656, 123415324656, 123451324656, 123453124656, 123453214656, 123453241656, 123453246156, 123453246516, 123145342656, 123415342656, 123451342656, 123453142656, 123453412656, 123453421656, 123453426156, 123453426516, 123145346256, 123415346256, 123451346256, 123453146256, 123453416256, 123453461256, 123453462156, 123453462516, 123145346526, 123415346526, 123451346526, 123453146526, 123453416526, 123453461526, 123453465126, 123453465216, 123415346562, 123451346562, 123453146562, 123453416562, 123453461562, 123453465162, 123214543656, 123241543656, 123245143656, 121342543656, 123142543656, 123412543656, 123421543656, 123425143656, 123425413656, 123425431656, 123425436156, 123425436516, 121345243656, 123145243656, 123415243656, 123451243656, 123452143656, 123452413656, 123452431656, 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