02341_02431_03142_03241_12340_12430_13042_13240_23041_23140
Counting sequence:
1, 1, 2, 6, 24, 110, 542, 2800, 14966, 82074, 459208, 2610938, 15042218, 87621664, 515190026, 3053559598, 18225049304, 109441034198, 660742323670, 4008361264080, 24421204151438, 149364648063826, 916748821025288, 5644655601477746, 34856968779353874, 215825236366301760, 1339624576181967122, 8333945571129071190, 51955591538427515160, 324536199857266542590, 2030883565843017320590, 12730501800158721556400, 79927692281814928997990, 502570890430761177122250, 3164511112655428189542600, 19952141757944194738785770, 125954422647091267564802490, 796064813980419699024815200, 5036943263814648078102791834, 31903915631584606770699227646, 202281343846460045364399908568, 1283751443454287129056970423654, 8154536264652527633428131864518, 51843243356274702158878946922640, 329869595204979822194810386237310, 2100552183775122147039073228793218, 13385994252805882826203057926573576, 85364962282279104154645805105416418, 544762640140142505624523005656680226, 3478734077892079746408606684603708544, 22228479834308013965250890194794655778, 142122215680098298140650601333935089318, 909217007639694348447405387389442842648, 5819917114647837537644892440428935242894, 37273532612143527333340520773460754204926, 238841705770574532156837505706743320987760, 1531219474183908100222783192138023650234198, 9821428244665472875545069189168125541195770, 63025229659406730242498087091100483609827784, 404622332173945154690502707050511225489822426, 2598808482126678735542632840567901865391261642, 16698652896241708689736121068439821912876490912, 107340932301661507506058610549769960445639792682, 690272177542606644523562068902880686147039969486, 4440593729010880714925757597258118628057793707224, 28577367486892155022699678191098363817582130002422, 183975071581873903085714580891035761624466106524086, 1184804306453250682713196139273738883447536158325072, 7632743354149387664008142432983411366460779239517550, 49187731637656338028440192473847356023398330632848050, 317081513726598254769797060006319401505105061571359752, 2044650637749470774774059131511618440590969145574419794, 13188566632256798659363427432370688242808875482241969202, 85094750717487164924093656368768027866293875740445753024, 549200809486887516478563089141683982486556390623247718706, 3545516179666409667253194086978013890879478689632117036278, 22895202566135178017245238592696225770248103497843615345944, 147884671750379942816711333035398805400436071923634116049438, 955460069021631039572852937973884022700954569028664526827374, 6174613792405890441313004715150247437950068336367033952187696, 39912801213543907707873453061345157953008495868368381681199430, 258057895190116429094440175682003100562610063609230096547085098, 1668868426541287667959082899618149013100043420823591602990679240, 10795049020234811804356390720962328584646148195553580920291619274, 69842916592778143082245522864139169906325901025327933104248532186, 451973613182750325103925308104598541772576586595739506947968263392, 2925462789823183126014896026599241378710439342564479611858793395642, 18939336208965465935366242952157703982593095049809191740238218383262, 122637015394119277353405880114177312747079297815633166029697421799640, 794260693655132119045999320540410792808714175070589446374568537538950, 5145023344149177479251735839566234099645356714312677673804847493918630, 33334394730954468352582066882359853416942543369126400030196408975199760, 216011547633958210333098513867303652315997842106707410304608958746344030, 1400034752915305735587898816657843647568368355171004415008408648215033570, 9075623171363016979897255082578105019197258901111698467311146333019292680, 58842115268425152230281261428230370046957977076543638664191008776914575810, 381568681610517527254258647029597549726573397686135927006918192578959327810, 2474732607114753155593374983575050453215265570519942336975100677938121058560, 16052899647401773153815115189711613147328729780521179606171247186011631396930, 104147040532505180360984508006922937937597758045005800369849163636407271507270, 675782954323736559733061503052494538525990616239303614272227428105681136891928
Implicit equation for the generating function in Maple syntax:
-F(x)^3+(x+2)*F(x)^2-3*x*F(x)+x-1 = 0
Implicit equation for the generating function in latex syntax:
-F \! \left(x \right)^{3}+\left(x +2\right) F \! \left(x \right)^{2}-3 x F \! \left(x \right)+x -1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(2) = 2
a(3) = 6
a(4) = 24
a(5) = 110
a(n+6) = 5/7*(n-1)*n/(n+6)/(n+5)*a(n)-3/7*n*(15*n+11)/(n+6)/(n+5)*a(n+1)+6/35*(71*n+120)*(n+1)/(n+6)/(n+5)*a(n+2)-1/35*(197*n^2+301*n-606)/(n+6)/(n+5)*a(n+3)+6/35*(3*n^2-55*n-254)/(n+6)/(n+5)*a(n+4)+3/5*(11*n+52)/(n+6)*a(n+5), n >= 6
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) = 110
a \! \left(n +6\right) = \frac{5 \left(n -1\right) n a \! \left(n \right)}{7 \left(n +6\right) \left(n +5\right)}-\frac{3 n \left(15 n +11\right) a \! \left(n +1\right)}{7 \left(n +6\right) \left(n +5\right)}+\frac{6 \left(71 n +120\right) \left(n +1\right) a \! \left(n +2\right)}{35 \left(n +6\right) \left(n +5\right)}-\frac{\left(197 n^{2}+301 n -606\right) a \! \left(n +3\right)}{35 \left(n +6\right) \left(n +5\right)}+\frac{6 \left(3 n^{2}-55 n -254\right) a \! \left(n +4\right)}{35 \left(n +6\right) \left(n +5\right)}+\frac{3 \left(11 n +52\right) a \! \left(n +5\right)}{5 \left(n +6\right)}, \quad n \geq 6
Specification 1
Strategy pack name: col_placements_tracked_fusion
Tree: http://permpal.com/tree/24232/
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[4,x]
F[3,x] = x
F[4,x] = F[1,x]+F[21,x]+F[5,x]
F[5,x] = F[6,x,1]
F[6,x,k[0]] = F[3,x]*F[7,x,k[0]]
F[7,x,k[0]] = F[1,x]+F[18,x,k[0]]+F[19,x,k[0]]+F[8,x,k[0]]
F[8,x,k[0]] = F[3,x]*F[9,x,k[0]]
F[9,x,k[0]] = 1/(-1+k[0])*(F[10,x,k[0]]*k[0]-F[10,x,1])
F[10,x,k[0]] = F[1,x]+F[11,x,k[0]]+F[13,x,k[0]]+F[14,x,k[0]]
F[11,x,k[0]] = F[12,x,k[0]]*F[3,x]
F[12,x,k[0]] = 1/(-1+k[0])*(F[7,x,k[0]]*k[0]-F[7,x,1])
F[13,x,k[0]] = F[3,x]*F[9,x,k[0]]
F[14,x,k[0]] = F[15,x,k[0]]
F[15,x,k[0]] = F[10,x,k[0]]*F[16,x,k[0]]*F[17,x,k[0]]
F[16,x,k[0]] = F[1,x]+F[17,x,k[0]]
F[17,x,k[0]] = k[0]*x
F[18,x,k[0]] = F[12,x,k[0]]*F[3,x]
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[7,x,k[0]]
F[21,x] = F[22,x]*F[3,x]
F[22,x] = F[10,x,1]
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_{4}\! \left(x \right)
F_{3}\! \left(x \right) = x
F_{4}\! \left(x \right) = F_{1}\! \left(x \right)+F_{21}\! \left(x \right)+F_{5}\! \left(x \right)
F_{5}\! \left(x \right) = F_{6}\! \left(x , 1\right)
F_{6}\! \left(x , y\right) = F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{18}\! \left(x , y\right)+F_{19}\! \left(x , y\right)+F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = F_{3}\! \left(x \right) F_{9}\! \left(x , y\right)
F_{9}\! \left(x , y\right) = \frac{F_{10}\! \left(x , y\right) y -F_{10}\! \left(x , 1\right)}{-1+y}
F_{10}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{13}\! \left(x , y\right)+F_{14}\! \left(x , y\right)
F_{11}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{3}\! \left(x \right)
F_{12}\! \left(x , y\right) = \frac{F_{7}\! \left(x , y\right) y -F_{7}\! \left(x , 1\right)}{-1+y}
F_{13}\! \left(x , y\right) = F_{3}\! \left(x \right) F_{9}\! \left(x , y\right)
F_{14}\! \left(x , y\right) = F_{15}\! \left(x , y\right)
F_{15}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{17}\! \left(x , y\right)
F_{16}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{17}\! \left(x , y\right)
F_{17}\! \left(x , y\right) = y x
F_{18}\! \left(x , y\right) = F_{12}\! \left(x , y\right) F_{3}\! \left(x \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_{7}\! \left(x , y\right)
F_{21}\! \left(x \right) = F_{22}\! \left(x \right) F_{3}\! \left(x \right)
F_{22}\! \left(x \right) = F_{10}\! \left(x , 1\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)*F_4(x))
Eq(F_3(x), x)
Eq(F_4(x), F_1(x) + F_21(x) + F_5(x))
Eq(F_5(x), F_6(x, 1))
Eq(F_6(x, k_0), F_3(x)*F_7(x, k_0))
Eq(F_7(x, k_0), F_1(x) + F_18(x, k_0) + F_19(x, k_0) + F_8(x, k_0))
Eq(F_8(x, k_0), F_3(x)*F_9(x, k_0))
Eq(F_9(x, k_0), (k_0*F_10(x, k_0) - F_10(x, 1))/(k_0 - 1))
Eq(F_10(x, k_0), F_1(x) + F_11(x, k_0) + F_13(x, k_0) + F_14(x, k_0))
Eq(F_11(x, k_0), F_12(x, k_0)*F_3(x))
Eq(F_12(x, k_0), (k_0*F_7(x, k_0) - F_7(x, 1))/(k_0 - 1))
Eq(F_13(x, k_0), F_3(x)*F_9(x, k_0))
Eq(F_14(x, k_0), F_15(x, k_0))
Eq(F_15(x, k_0), F_10(x, k_0)*F_16(x, k_0)*F_17(x, k_0))
Eq(F_16(x, k_0), F_1(x) + F_17(x, k_0))
Eq(F_17(x, k_0), k_0*x)
Eq(F_18(x, k_0), F_12(x, k_0)*F_3(x))
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)*F_7(x, k_0))
Eq(F_21(x), F_22(x)*F_3(x))
Eq(F_22(x), F_10(x, 1))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": true, "place_row": false, "strategy_class": "RowAndColumnPlacementFactory"}]], "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": "col_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, 2, 3, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 2, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 3, 4, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 2, 4, 3, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 0, 4, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [1, 3, 2, 4, 0], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 0, 4, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [2, 3, 1, 4, 0], "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], "pos": [[0, 0]]}, {"patt": [0], "pos": [[0, 2]]}, {"patt": [0], "pos": [[1, 1]]}, {"patt": [0, 1], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 0], "pos": [[0, 1], [0, 1]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 2], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 2, 3, 0], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 2], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 3, 2, 0], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 2], [1, 0], [1, 2], [1, 0]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 2], [1, 0], [1, 2], [1, 2]]}, {"patt": [2, 0, 3, 1], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 2], [1, 0], [1, 2], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 2], [1, 2], [1, 2], [1, 0]]}, {"patt": [2, 1, 3, 0], "pos": [[1, 2], [1, 2], [1, 2], [1, 2]]}, {"patt": [0, 2, 3, 4, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 4, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 2, 3, 4, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 0], [1, 0]]}, {"patt": [0, 2, 4, 3, 1], "pos": [[1, 0], [1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [0, 3, 1, 4, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 4, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [0, 3, 2, 4, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 2, 4, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 2, 3, 4, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [1, 2, 3, 4, 0], "pos": [[1, 0], [1, 0], [1, 0], [1, 2], [1, 0]]}, {"patt": [1, 2, 3, 4, 0], "pos": [[1, 0], [1, 0], [1, 2], [1, 2], [1, 0]]}, {"patt": [1, 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