01243_01342_01432_02143_10243_10342_10432_12043_20143_20341_20431_21043_30142_30241_30421_31042_40132_40231_40321_41032
Counting sequence:
1, 1, 2, 6, 24, 100, 426, 1848, 8120, 36018, 160940, 723338, 3266496, 14809366, 67365298, 307306440, 1405320928, 6440392540, 29571332346, 136006198392, 626466806840, 2889502457730, 13343732125152, 61689631953736, 285486410996184, 1322391984220500, 6130615189030436, 28443847272374898, 132064962716775984, 613591297937735806, 2852612296598351250, 13269666620025262304, 61761023458067375072, 287601555381767525250, 1339913251375786478492, 6245374367648248494410, 29122276245657777721872, 135852256228234223659114, 633977124235230850589950, 2959625523180554977152144, 13821281508729628455390000, 64565537484194926510071170, 301708276700592642075360816, 1410266715810565131421358736, 6593817142627792892765305168, 30838169831427850086738542880, 144261738098506372943144705904, 675024828715280898778482494168, 3159292394831208537386480088960, 14789624059306941106377790661518, 69249668213088103368986629096450, 324315541432818259736899345668288, 1519160990856950200436345838929656, 7117418328447849783817806271680418, 33351909584234022265963006229036472, 156313136999603463871238577636497400, 732730076571099286368918347258814968, 3435292156474447071152252890704901764, 16108383139521391740964513977304735740, 75545128052088307987627670986983122434, 354343896015677678850285929230930512720, 1662284258489809801195584346732496498454, 7799116988393595420992875395874228345778, 36596828197315948943878984957547293717536, 171750292399700146471868836207654372492160, 806131313100582214699624379688579192798140, 3784135588974232750596058305631823032286778, 17765546833980827986459176839803045753938808, 83414189736189777495161313930414236241327544, 391696143924807885601375088279105613387666546, 1839523182457188966792957024394076859975786880, 8639856055492493534777701686139698333551336632, 40583709565353159253528009774677308253804489336, 190651287424606286905153575923097604984029470436, 895713942436278879821883616074084978970112209508, 4208616851255230765794139908937326256293568447950, 19776472330242459192286287261645146337465040979568, 92938702139404091599049435291207869130847418093330, 436799088084831179916747577682896266174643557697350, 2053067481995242429162850105066378236309903198080480, 9650730403024349649535175676123666574198072021714400, 45368217003414528168527943864716873662673863867946610, 213293171655971055740945015655247541744910396766791200, 1002847926348366384820847604202012374427929417376266320, 4715473435520245506313283296017109567719267222039870240, 22174143373611170426561032652299439144269706733497971600, 104279511390311157413051988116652887858521516848498313840, 490434482893704603097144608722554837125890764495081291920, 2306705730413597347408503207428656073667234314674867879936, 10850054291958771197503908505270017355078325359457992828792, 51038688484972660607170108390080337207482865901231250890040, 240101214801986371634602122189587034804720988776529115585888, 1129577062332627015083344652223121467924520046231496340252256, 5314512676838943351824588092434944172356509877783978958516792, 25005557560702117949388875710930699798276169316124922556804480, 117661561217589095122219029134586739917897989417946414675711520, 553677802050826244256168622918320107396307119009514414530669664, 2605574822026357127819793242702615393824960133461601508350766180, 12262341932580293903688040520405748193617087863208346419766255796, 57712016933022660333807242405378812465863021374609652696321064962, 271632402436163467445499509178367779571902628292213425480022905200
Implicit equation for the generating function in Maple syntax:
(-4*x^4+12*x^3-4*x^2+24*x-5)*F(x)^3+(x+3)*(4*x^4-12*x^3+4*x^2-24*x+5)*F(x)^2+(-8*x^5+12*x^4+36*x^3+37*x^2+62*x-15)*F(x)+(4*x^2+4*x-1)*(x^3-3*x^2-x-5) = 0
Implicit equation for the generating function in latex syntax:
\left(-4 x^{4}+12 x^{3}-4 x^{2}+24 x -5\right) F \! \left(x \right)^{3}+\left(x +3\right) \left(4 x^{4}-12 x^{3}+4 x^{2}-24 x +5\right) F \! \left(x \right)^{2}+\left(-8 x^{5}+12 x^{4}+36 x^{3}+37 x^{2}+62 x -15\right) F \! \left(x \right)+\left(4 x^{2}+4 x -1\right) \left(x^{3}-3 x^{2}-x -5\right) = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 100
a(6) = 426
a(7) = 1848
a(n+7) = 4/5*(n-1)*(n+1)/(n+6)/(n+5)*a(n)-3/5*n*(7*n+11)/(n+6)/(n+5)*a(n+1)+7/10*(14*n+45)*(n+1)/(n+6)/(n+5)*a(n+2)-1/5*(91*n^2+459*n+590)/(n+6)/(n+5)*a(n+3)+1/5*(89*n^2+493*n+660)/(n+6)/(n+5)*a(n+4)-1/20*(493*n^2+3625*n+6414)/(n+6)/(n+5)*a(n+5)+3/10*(31*n+133)/(n+6)*a(n+6), n >= 8
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(2\right) = 2
a \! \left(3\right) = 6
a \! \left(4\right) = 24
a \! \left(5\right) = 100
a \! \left(6\right) = 426
a \! \left(7\right) = 1848
a \! \left(n +7\right) = \frac{4 \left(n -1\right) \left(n +1\right) a \! \left(n \right)}{5 \left(n +6\right) \left(n +5\right)}-\frac{3 n \left(7 n +11\right) a \! \left(n +1\right)}{5 \left(n +6\right) \left(n +5\right)}+\frac{7 \left(14 n +45\right) \left(n +1\right) a \! \left(n +2\right)}{10 \left(n +6\right) \left(n +5\right)}-\frac{\left(91 n^{2}+459 n +590\right) a \! \left(n +3\right)}{5 \left(n +6\right) \left(n +5\right)}+\frac{\left(89 n^{2}+493 n +660\right) a \! \left(n +4\right)}{5 \left(n +6\right) \left(n +5\right)}-\frac{\left(493 n^{2}+3625 n +6414\right) a \! \left(n +5\right)}{20 \left(n +6\right) \left(n +5\right)}+\frac{3 \left(31 n +133\right) a \! \left(n +6\right)}{10 \left(n +6\right)}, \quad n \geq 8
Specification 1
Strategy pack name: point_placements_tracked_fusion
Tree: http://permpal.com/tree/24327/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[3,x]
F[3,x] = F[4,x]*F[5,x]
F[4,x] = x
F[5,x] = F[6,x,1]
F[6,x,k[0]] = (F[7,x,k[0]]*k[0]-F[7,x,1])/(-1+k[0])
F[7,x,k[0]] = F[1,x]+F[8,x,k[0]]
F[8,x,k[0]] = F[9,x,k[0]]
F[9,x,k[0]] = F[10,x,k[0]]*F[16,x,k[0]]
F[10,x,k[0]] = F[11,x,k[0]]+F[7,x,k[0]]
F[11,x,k[0]] = F[12,x,k[0]]
F[12,x,k[0]] = F[13,x,k[0]]*F[16,x,k[0]]*F[7,x,k[0]]
F[13,x,k[0]] = F[10,x,k[0]]+F[14,x,k[0]]
F[14,x,k[0]] = F[15,x,k[0]]
F[15,x,k[0]] = F[16,x,k[0]]*F[17,x,k[0]]
F[16,x,k[0]] = k[0]*x
F[17,x,k[0]] = F[18,x,k[0]]+F[21,x,k[0]]
F[18,x,k[0]] = F[19,x,k[0]]+F[7,x,k[0]]
F[19,x,k[0]] = F[20,x,k[0]]
F[20,x,k[0]] = F[16,x,k[0]]*F[17,x,k[0]]
F[21,x,k[0]] = F[22,x,k[0]]
F[22,x,k[0]] = F[13,x,k[0]]*F[16,x,k[0]]*F[18,x,k[0]]
System of equations in latex syntax:
F_{0}\! \left(x \right) = F_{1}\! \left(x \right)+F_{2}\! \left(x \right)
F_{1}\! \left(x \right) = 1
F_{2}\! \left(x \right) = F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{4}\! \left(x \right) F_{5}\! \left(x \right)
F_{4}\! \left(x \right) = x
F_{5}\! \left(x \right) = F_{6}\! \left(x , 1\right)
F_{6}\! \left(x , y\right) = \frac{y F_{7}\! \left(x , y\right)-F_{7}\! \left(x , 1\right)}{-1+y}
F_{7}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{16}\! \left(x , y\right)
F_{10}\! \left(x , y\right) = F_{11}\! \left(x , y\right)+F_{7}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right)
F_{12}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{7}\! \left(x , y\right)
F_{13}\! \left(x , y\right) = F_{10}\! \left(x , y\right)+F_{14}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right)
F_{15}\! \left(x , y\right) = F_{16}\! \left(x , y\right) F_{17}\! \left(x , y\right)
F_{16}\! \left(x , y\right) = y x
F_{17}\! \left(x , y\right) = F_{18}\! \left(x , y\right)+F_{21}\! \left(x , y\right)
F_{18}\! \left(x , y\right) = F_{19}\! \left(x , y\right)+F_{7}\! \left(x , y\right)
F_{19}\! \left(x , y\right) = F_{20}\! \left(x , y\right)
F_{20}\! \left(x , y\right) = F_{16}\! \left(x , y\right) F_{17}\! \left(x , y\right)
F_{21}\! \left(x , y\right) = F_{22}\! \left(x , y\right)
F_{22}\! \left(x , y\right) = F_{13}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{18}\! \left(x , y\right)
System of equations in sympy syntax:
Eq(F_0(x), F_1(x) + F_2(x))
Eq(F_1(x), 1)
Eq(F_2(x), F_3(x))
Eq(F_3(x), F_4(x)*F_5(x))
Eq(F_4(x), x)
Eq(F_5(x), F_6(x, 1))
Eq(F_6(x, k_0), (-k_0*F_7(x, k_0) + F_7(x, 1))/(1 - k_0))
Eq(F_7(x, k_0), F_1(x) + F_8(x, k_0))
Eq(F_8(x, k_0), F_9(x, k_0))
Eq(F_9(x, k_0), F_10(x, k_0)*F_16(x, k_0))
Eq(F_10(x, k_0), F_11(x, k_0) + F_7(x, k_0))
Eq(F_11(x, k_0), F_12(x, k_0))
Eq(F_12(x, k_0), F_13(x, k_0)*F_16(x, k_0)*F_7(x, k_0))
Eq(F_13(x, k_0), F_10(x, k_0) + F_14(x, k_0))
Eq(F_14(x, k_0), F_15(x, k_0))
Eq(F_15(x, k_0), F_16(x, k_0)*F_17(x, k_0))
Eq(F_16(x, k_0), k_0*x)
Eq(F_17(x, k_0), F_18(x, k_0) + F_21(x, k_0))
Eq(F_18(x, k_0), F_19(x, k_0) + F_7(x, k_0))
Eq(F_19(x, k_0), F_20(x, k_0))
Eq(F_20(x, k_0), F_16(x, k_0)*F_17(x, k_0))
Eq(F_21(x, k_0), F_22(x, k_0))
Eq(F_22(x, k_0), F_13(x, k_0)*F_16(x, k_0)*F_18(x, k_0))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_insertion", "extra_basis": [], "ignore_parent": false, "maxreqlen": 1, "one_cell_only": false, "strategy_class": "CellInsertionFactory"}, {"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "point_only": false, "strategy_class": "PatternPlacementFactory"}]], "inferral_strats": [{"class_module": "tilings.strategies.row_and_col_separation", "ignore_parent": true, "inferrable": true, "possibly_empty": false, "strategy_class": "RowColumnSeparationStrategy", "workable": true}, {"class_module": "tilings.strategies.obstruction_inferral", "strategy_class": "ObstructionTransitivityFactory"}], "initial_strats": [{"class_module": "tilings.strategies.rearrange_assumption", "strategy_class": "RearrangeAssumptionFactory"}, {"class_module": "tilings.strategies.assumption_insertion", "strategy_class": "AddAssumptionFactory"}, {"class_module": "tilings.strategies.factor", "ignore_parent": true, "interleaving": null, "strategy_class": "FactorFactory", "tracked": false, "unions": false, "workable": true}, {"class_module": "tilings.strategies.fusion.fusion", "strategy_class": "FusionFactory", "tracked": true}], "iterative": false, "name": "point_placements_tracked_fusion", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": false, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 4, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 1, 0, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 0, 3, 2, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [4, 1, 0, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rules": [{"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 1, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 2, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 3, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 0, 4, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 0, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 1, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 0, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 1, 0, 4, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 1, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 2, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [3, 0, 4, 2, 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