0123_0132_0231_0312
Counting sequence:
1, 1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520, 601080390, 2333606220, 9075135300, 35345263800, 137846528820, 538257874440, 2104098963720, 8233430727600, 32247603683100, 126410606437752, 495918532948104, 1946939425648112, 7648690600760440, 30067266499541040, 118264581564861424, 465428353255261088, 1832624140942590534, 7219428434016265740, 28453041475240576740, 112186277816662845432, 442512540276836779204, 1746130564335626209832, 6892620648693261354600, 27217014869199032015600, 107507208733336176461620, 424784580848791721628840, 1678910486211891090247320, 6637553085023755473070800, 26248505381684851188961800, 103827421287553411369671120, 410795449442059149332177040, 1625701140345170250548615520, 6435067013866298908421603100, 25477612258980856902730428600, 100891344545564193334812497256, 399608854866744452032002440112, 1583065848125949175357548128136, 6272525058612251449529907677520, 24857784491537440929618523018320, 98527218530093856775578873054432, 390590044887157789360330532465784, 1548655265692941410446222812934512, 6141219157058215937976400809912720, 24356699707654619143838606602026720, 96614908840363322603893139521372656, 383291933432261050330199012527412832, 1520803477811874490019821888415218656, 6034934435761406706427864636568328000, 23951146041928082866135587776380551750, 95067625827960698145584333020095113100, 377389666165540953244592352291892721700, 1498293301493043187508381577755872596600, 5949105755928259715106809205795376486500, 23623985175715118288974865541854103729000, 93820969697840041204785894580506297666600, 372641034574519600278163693967644731577200, 1480212998448786189993816895482588794876100, 5880298213015725960249409584793845897453000, 23362265873332749085315221863910685052043000, 92826069736708789698985814872605121940117520, 368861487637974401172285738046404563498888040, 1465865132691170996866486179768828525073503120, 5825874245311064218315521996517139009907512400, 23156006494021191956342707682359261381151378400, 92045125813734238026462263037378063990076729140, 365907784099042279561985786395502921046971688680, 1454706556296192477283016662986999417820887445240, 5783773055153536355462596370912166360010757312400, 22997383338348585032434609379579328145757058837400, 91448418451315549893681152591738975450186892788720, 363666966399417651902778537050868948883301364345840, 1446307705450557558142084756547133980616347954754720, 5752360192132899378974200736267010150178656638229000, 22880174247360071687155809670095748237789263482394000, 91012248672832285155575331798825309656983959185522800, 362048725489728431058442528694228154899210914562189600, 1440324277491745714862934407631385920577295594888710800, 5730322394321999080637480976597986995845154517299172000, 22799367824217315491046998779230288685596678611381812000, 90717484605833107953850163563674201296374152790550788800, 360979990827377575399695442513786925991822149645733347100, 1436477076900698392827654028972389210647869585188175999800, 5716592448890534420436582360196242777068052430850904489000, 22750883079422934966181954039568885395604168260154104734000
Generating function in Maple syntax:
(4*x-1-x*(-4*x+1)^(1/2))/(4*x-1)
Generating function in latex syntax:
\frac{4 x -1-x \sqrt{-4 x +1}}{4 x -1}
Generating function in sympy syntax:
(-x*sqrt(1 - 4*x) + 4*x - 1)/(4*x - 1)
Implicit equation for the generating function in Maple syntax:
(4*x-1)*F(x)^2+(-8*x+2)*F(x)+x^2+4*x-1 = 0
Implicit equation for the generating function in latex syntax:
\left(4 x -1\right) F \! \left(x \right)^{2}+\left(-8 x +2\right) F \! \left(x \right)+x^{2}+4 x -1 = 0
Recurrence in maple format:
a(0) = 1
a(1) = 1
a(n+1) = 2*(-1+2*n)*a(n)/n, n >= 2
Recurrence in latex format:
a \! \left(0\right) = 1
a \! \left(1\right) = 1
a \! \left(n +1\right) = \frac{2 \left(-1+2 n \right) a \! \left(n \right)}{n}, \quad n \geq 2
Specification 1
Strategy pack name: row_placements_tracked_fusion
Tree: http://permpal.com/tree/21127/
System of equations in Maple syntax:
F[0,x] = F[1,x]+F[2,x]
F[1,x] = 1
F[2,x] = F[11,x]*F[3,x]
F[3,x] = F[1,x]+F[12,x]+F[4,x]
F[4,x] = F[11,x]*F[5,x]
F[5,x] = F[6,x,1]
F[6,x,k[0]] = F[1,x]+F[12,x]+F[7,x,k[0]]+F[9,x,k[0]]
F[7,x,k[0]] = F[6,x,k[0]]*F[8,x,k[0]]
F[8,x,k[0]] = k[0]*x
F[9,x,k[0]] = F[10,x,k[0]]*F[11,x]
F[10,x,k[0]] = (F[6,x,k[0]]*k[0]-F[6,x,1])/(-1+k[0])
F[11,x] = x
F[12,x] = F[11,x]*F[13,x]
F[13,x] = F[14,x,1]
F[14,x,k[0]] = F[1,x]+F[15,x,k[0]]+F[16,x,k[0]]+F[4,x]
F[15,x,k[0]] = F[14,x,k[0]]*F[8,x,k[0]]
F[16,x,k[0]] = F[11,x]*F[17,x,k[0]]
F[17,x,k[0]] = (F[14,x,k[0]]*k[0]-F[14,x,1])/(-1+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_{11}\! \left(x \right) F_{3}\! \left(x \right)
F_{3}\! \left(x \right) = F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{4}\! \left(x \right)
F_{4}\! \left(x \right) = F_{11}\! \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_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{7}\! \left(x , y\right)+F_{9}\! \left(x , y\right)
F_{7}\! \left(x , y\right) = F_{6}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{8}\! \left(x , y\right) = y x
F_{9}\! \left(x , y\right) = F_{10}\! \left(x , y\right) F_{11}\! \left(x \right)
F_{10}\! \left(x , y\right) = -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}
F_{11}\! \left(x \right) = x
F_{12}\! \left(x \right) = F_{11}\! \left(x \right) F_{13}\! \left(x \right)
F_{13}\! \left(x \right) = F_{14}\! \left(x , 1\right)
F_{14}\! \left(x , y\right) = F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)+F_{16}\! \left(x , y\right)+F_{4}\! \left(x \right)
F_{15}\! \left(x , y\right) = F_{14}\! \left(x , y\right) F_{8}\! \left(x , y\right)
F_{16}\! \left(x , y\right) = F_{11}\! \left(x \right) F_{17}\! \left(x , y\right)
F_{17}\! \left(x , y\right) = -\frac{-y F_{14}\! \left(x , y\right)+F_{14}\! \left(x , 1\right)}{-1+y}
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_11(x)*F_3(x))
Eq(F_3(x), F_1(x) + F_12(x) + F_4(x))
Eq(F_4(x), F_11(x)*F_5(x))
Eq(F_5(x), F_6(x, 1))
Eq(F_6(x, k_0), F_1(x) + F_12(x) + F_7(x, k_0) + F_9(x, k_0))
Eq(F_7(x, k_0), F_6(x, k_0)*F_8(x, k_0))
Eq(F_8(x, k_0), k_0*x)
Eq(F_9(x, k_0), F_10(x, k_0)*F_11(x))
Eq(F_10(x, k_0), (-k_0*F_6(x, k_0) + F_6(x, 1))/(1 - k_0))
Eq(F_11(x), x)
Eq(F_12(x), F_11(x)*F_13(x))
Eq(F_13(x), F_14(x, 1))
Eq(F_14(x, k_0), F_1(x) + F_15(x, k_0) + F_16(x, k_0) + F_4(x))
Eq(F_15(x, k_0), F_14(x, k_0)*F_8(x, k_0))
Eq(F_16(x, k_0), F_11(x)*F_17(x, k_0))
Eq(F_17(x, k_0), (-k_0*F_14(x, k_0) + F_14(x, 1))/(1 - k_0))
Pack JSON:
{"expansion_strats": [[{"class_module": "tilings.strategies.requirement_placement", "dirs": [0, 1, 2, 3], "ignore_parent": false, "partial": false, "place_col": false, "place_row": true, "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": "row_placements_tracked_fusion", "symmetries": [], "ver_strats": [{"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}, {"class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "InsertionEncodingVerificationStrategy"}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "OneByOneVerificationStrategy", "symmetry": false}, {"basis": [], "class_module": "tilings.strategies.verification", "ignore_parent": true, "strategy_class": "LocallyFactorableVerificationStrategy", "symmetry": false}]}
Specification JSON:
{"root": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[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, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2, 3], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[0, 0], [0, 0], [0, 0], [0, 0]]}], "requirements": []}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.requirement_placement", "direction": 1, "gps": [{"patt": [0], "pos": [[0, 0]]}], "ignore_parent": false, "include_empty": true, "indices": [0], "own_col": true, "own_row": true, "strategy_class": "RequirementPlacementStrategy"}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 0]]}], "requirements": []}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"children": [{"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [1, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[1, 0], [1, 0], [1, 0], [1, 0]]}], "requirements": []}, {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}], "class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0], "pos": [[0, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [0, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 1], "pos": [[0, 0], [0, 0], [2, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[2, 0], [2, 0], [2, 0], [2, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 1]]}]]}, "rule_class": "Rule", "strategy": {"class_module": "tilings.strategies.factor", "ignore_parent": true, "partition": [[[0, 0], [2, 0]], [[1, 1]]], "strategy_class": "FactorStrategy", "workable": true}}, {"class_module": "comb_spec_searcher.strategies.rule", "comb_class": {"assumptions": [], "class_module": "tilings.tiling", "comb_class": "Tiling", "obstructions": [{"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [1, 0], "pos": [[0, 0], [0, 0]]}], "requirements": [[{"patt": [0], "pos": [[0, 0]]}]]}, "rule_class": "VerificationRule", "strategy": {"class_module": "tilings.strategies.verification", "strategy_class": "BasicVerificationStrategy"}}, {"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, 1]]}, {"patt": [0], "pos": [[1, 0]]}, {"patt": [0], "pos": [[2, 1]]}, {"patt": [0], "pos": [[3, 1]]}, {"patt": [0, 1], "pos": [[0, 0], [0, 0]]}, {"patt": [0, 1], "pos": [[0, 0], [3, 0]]}, {"patt": [0, 1], "pos": [[1, 1], [1, 1]]}, {"patt": [1, 0], "pos": [[1, 1], [1, 1]]}, {"patt": [0, 1, 2], "pos": [[0, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[2, 0], [2, 0], [2, 0]]}, {"patt": [0, 1, 2], "pos": [[2, 0], [2, 0], [3, 0]]}, {"patt": [0, 1, 2], "pos": [[2, 0], [3, 0], [3, 0]]}, {"patt": [0, 2, 1], "pos": [[2, 0], [2, 0], [3, 0]]}, {"patt": [0, 1, 2, 3], "pos": [[3, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [0, 1, 3, 2], "pos": [[3, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [0, 2, 3, 1], "pos": [[3, 0], [3, 0], [3, 0], [3, 0]]}, {"patt": [0, 3, 1, 2], "pos": [[3, 0], [3, 0], [3, 0], [3, 0]]}], "requirements": [[{"patt": [0], "pos": [[1, 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