︠035717db-bd14-470e-af0c-df685ab1eeffs︠ S = ClusterSeed(['A',2]); S ︡56794022-ad69-42e4-9edd-58c9f2c2f648︡{"stdout":"A seed for a cluster algebra of rank 2 of type ['A', 2]\n"}︡{"done":true}︡ ︠f61664bc-d240-4762-aaed-c38661131f48s︠ S.show() ︡25105eeb-4277-4c1c-a6d6-ff12b4990f46︡{"file":{"filename":"/home/user/.sage/temp/project-9d9553fd-aa65-444a-ab4d-b13f6613d595/237/tmp_zhnsVC.svg","show":true,"text":null,"uuid":"acee493e-a519-4d4a-a04a-09e9d0c222a9"},"once":false}︡{"done":true}︡ ︠a8dc30e0-2c75-43f0-afea-82fd0590d314s︠ S.cluster() ︡e294fc64-fca7-463b-b22c-6c9cb18da3c8︡{"stdout":"[x0, x1]\n"}︡{"done":true}︡ ︠52e9ba1b-9dba-4446-9524-423e46fa9fbfs︠ S.mutate(0); S.cluster() ︡15b0c599-1441-457d-bd0d-29ec3972b7e7︡{"stdout":"[(x1 + 1)/x0, x1]\n"}︡{"done":true}︡ ︠350df520-131f-420d-9661-9ccb0a1f690bs︠ pretty_print(_) ︡2273a1f8-d18f-4242-b8bc-4e3d82298f5f︡{"html":"
[$\\displaystyle \\frac{x_{1} + 1}{x_{0}}$, $\\displaystyle x_{1}$]
"}︡{"done":true}︡ ︠a7050eef-95d6-4438-8609-c15284cf3d22s︠ S.mutate(1); S.cluster() ︡149dd1dd-e000-4fd7-943b-f3205111a2d9︡{"stdout":"[(x1 + 1)/x0, (x0 + x1 + 1)/(x0*x1)]\n"}︡{"done":true}︡ ︠b082b4c3-94fd-4c38-a5ec-8af9ae3920e1s︠ S.mutate(0); S.cluster() ︡c439f1fb-7b25-4a7d-a146-e55fd454d09d︡{"stdout":"[(x0 + 1)/x1, (x0 + x1 + 1)/(x0*x1)]\n"}︡{"done":true}︡ ︠a7a59395-a124-4fa5-a59b-66f6ef905dd4s︠ S.mutate(1); S.cluster() ︡4ab447af-674b-4c30-856f-a073c5cc5a73︡{"stdout":"[(x0 + 1)/x1, x0]\n"}︡{"done":true}︡ ︠dfe690f5-ebae-4a01-9bf5-0c26ecd7dfcfs︠ S.mutate(0); S.cluster() ︡688b5109-778a-474e-807f-a67e5fccc663︡{"stdout":"[x1, x0]\n"}︡{"done":true}︡ ︠dc4d4aeb-afef-4b8b-98bb-f62a9b10a601s︠ S2 = ClusterSeed([[0,1,2]]); S2 ︡16bff94d-c164-4c8e-a06b-94f071e35608︡{"stdout":"A seed for a cluster algebra of rank 2\n"}︡{"done":true}︡ ︠5e9b4a58-c6d6-408a-b290-ed40f28a29d9s︠ S2.show() ︡3feda8f7-3b7b-4332-8573-4455e3433fa2︡{"file":{"filename":"/home/user/.sage/temp/project-9d9553fd-aa65-444a-ab4d-b13f6613d595/237/tmp_e0O2gT.svg","show":true,"text":null,"uuid":"1e63e935-3620-47f2-bfef-f0d1e6389465"},"once":false}︡{"done":true}︡ ︠d12e2a46-4ad9-42d3-a747-9f49bb50158cs︠ S2.mutate(0); S2.cluster() ︡a9a0e83c-f181-41e8-94eb-5e291780cd7d︡{"stdout":"[(x1^2 + 1)/x0, x1]\n"}︡{"done":true}︡ ︠d0e58644-1d6e-4f3d-b979-7bb7a1fb519d︠ ︡dbe05189-7aa1-4902-9e97-7e5d3979210d︡ ︠34b420a3-7e65-45c6-9859-16bd7392317ds︠ S4 = ClusterSeed(['A',4]); S4.show() ︡944c31ce-3ca5-4f5a-a30f-2e8e08f6230c︡{"file":{"filename":"/home/user/.sage/temp/project-9d9553fd-aa65-444a-ab4d-b13f6613d595/237/tmp_BV8o__.svg","show":true,"text":null,"uuid":"80e2e9b9-e5df-4473-8c7e-73cc866dddc3"},"once":false}︡{"done":true}︡ ︠7a26693d-a806-421e-8533-9c0fddca4b79s︠ S4.mutate([0,2]); S4.show() ︡075b6719-246d-48e5-a20d-29d3edd99ab1︡{"file":{"filename":"/home/user/.sage/temp/project-9d9553fd-aa65-444a-ab4d-b13f6613d595/237/tmp_YuGRVs.svg","show":true,"text":null,"uuid":"44181717-5461-4e30-8f83-b871df61990a"},"once":false}︡{"done":true}︡ ︠bd3074d8-16a4-44c8-8538-4f2f777e33efs︠ S4.mutate([1,3]); S4.show() ︡f28d5b7a-8aa5-473f-874b-895e2de789f5︡{"file":{"filename":"/home/user/.sage/temp/project-9d9553fd-aa65-444a-ab4d-b13f6613d595/237/tmp_TnkJy2.svg","show":true,"text":null,"uuid":"faf2b2f4-4167-4030-b59e-2cc442190657"},"once":false}︡{"done":true}︡ ︠c2438c10-2205-4f3f-bb17-002f7a9c7eb5s︠ S4.cluster() ︡6ae0f433-91c3-49c5-9f82-15d662d34467︡{"stdout":"[(x1 + 1)/x0, (x1^2*x3 + x0*x2 + x1*x3 + x1 + 1)/(x0*x1*x2), (x1*x3 + 1)/x2, (x1*x3 + x2 + 1)/(x2*x3)]\n"}︡{"done":true}︡ ︠93bf3758-a1ff-4d4b-b070-ef002bffa2f6s︠ pretty_print(_) ︡0060390d-ec23-484c-a706-f98a4b83a5e3︡{"html":"
[$\\displaystyle \\frac{x_{1} + 1}{x_{0}}$, $\\displaystyle \\frac{x_{1}^{2} x_{3} + x_{0} x_{2} + x_{1} x_{3} + x_{1} + 1}{x_{0} x_{1} x_{2}}$, $\\displaystyle \\frac{x_{1} x_{3} + 1}{x_{2}}$, $\\displaystyle \\frac{x_{1} x_{3} + x_{2} + 1}{x_{2} x_{3}}$]
"}︡{"done":true}︡ ︠5d0d97e2-e407-446b-8593-636587dc40c5s︠ S5 = ClusterSeed([[0,1],[1,2],[2,3]]); S5.show() ︡5a0612da-64b9-4afd-a895-1d0e0af6d2b2︡{"file":{"filename":"/home/user/.sage/temp/project-9d9553fd-aa65-444a-ab4d-b13f6613d595/237/tmp_Mg3N9L.svg","show":true,"text":null,"uuid":"5f77204c-702b-469c-9dd3-32c298443a0a"},"once":false}︡{"done":true}︡ ︠937dfc03-b11c-4618-87a9-d2a932721dffs︠ S4.mutate([0,2,1,3]); S4.cluster() ︡3dfd1aea-5cfb-4da3-94ad-9c57ae9a5d30︡{"stdout":"[(x0*x2 + x1*x3 + 1)/(x1*x2), (x0*x2^2 + x0*x2 + x1*x3 + x2 + 1)/(x1*x2*x3), (x0*x2^2 + x1^2*x3 + x0*x2 + x1*x2 + x1*x3 + x1 + x2 + 1)/(x0*x1*x2*x3), (x0*x2 + x1 + 1)/(x0*x1)]\n"}︡{"done":true}︡ ︠1737d4b7-8fe0-47ae-ac4a-2c14159af56as︠ S4.mutate([0,2,1,3]); S4.cluster() ︡68b16204-4ce2-4ca6-b678-bca578995198︡{"stdout":"[(x2 + 1)/x3, x2, (x0*x2 + 1)/x1, x0]\n"}︡{"done":true}︡ ︠592f718a-2385-467a-b95f-3f37ea0c43e4s︠ S4.mutate([0,2,1,3]); S4.cluster() ︡4815fade-30a6-4363-8b75-08d371eb0a5d︡{"stdout":"[x3, (x1*x3 + 1)/x2, x1, (x1 + 1)/x0]\n"}︡{"done":true}︡ ︠6cd1edc3-03f5-4e97-8006-bf2803f3964as︠ S6 = ClusterSeed([[0,1],[1,2],[1,3]]); S6.show() ︡31cdcf57-c51a-4083-89f2-771886445040︡{"file":{"filename":"/home/user/.sage/temp/project-9d9553fd-aa65-444a-ab4d-b13f6613d595/237/tmp_myVU0h.svg","show":true,"text":null,"uuid":"9c17b6b2-8ee8-41b8-9ad6-9a09786018c3"},"once":false}︡{"done":true}︡ ︠afccc50d-425f-4f62-b888-1699b06e4a18s︠ S6.mutate([0,1,2,3]); S6.show() ︡b4739890-8697-4acd-a535-5bf3951657ca︡{"file":{"filename":"/home/user/.sage/temp/project-9d9553fd-aa65-444a-ab4d-b13f6613d595/237/tmp_u2Bt5y.svg","show":true,"text":null,"uuid":"3e704376-c3c9-4bb6-83f2-c68478c64964"},"once":false}︡{"done":true}︡ ︠2e9315ec-74ad-4a12-89a1-4504869e733es︠ S6.cluster() ︡2fc8a9a0-825f-415d-8ec2-f8f921050fee︡{"stdout":"[(x1 + 1)/x0, (x1*x2*x3 + x2*x3 + x0)/(x0*x1), (x1*x2*x3 + x0*x1 + x2*x3 + x0)/(x0*x1*x2), (x1*x2*x3 + x0*x1 + x2*x3 + x0)/(x0*x1*x3)]\n"}︡{"done":true}︡ ︠377c7565-b748-4aa3-9874-f1b2a0667d1as︠ S6.mutate([0,1,2,3]); S6.cluster() ︡b664f0cc-0006-47f4-9b09-846a504a0183︡{"stdout":"[(x2*x3 + x0)/x1, (x1*x2^2*x3^2 + x0^2*x1^2 + 3*x0*x1*x2*x3 + x2^2*x3^2 + 2*x0^2*x1 + 2*x0*x2*x3 + x0^2)/(x0*x1^2*x2*x3), (x0*x1 + x2*x3 + x0)/(x1*x3), (x0*x1 + x2*x3 + x0)/(x1*x2)]\n"}︡{"done":true}︡ ︠cb208e42-1800-4291-b14d-f2a2a2256fd0s︠ S6.mutate([0,1,2,3]); S6.cluster() ︡1e731a27-eba0-4432-abbf-db18d2a00768︡{"stdout":"[(x0*x1^2 + x1*x2*x3 + 2*x0*x1 + x2*x3 + x0)/(x0*x1*x2*x3), (x0*x1^2 + 2*x0*x1 + x2*x3 + x0)/(x1*x2*x3), (x1 + 1)/x2, (x1 + 1)/x3]\n"}︡{"done":true}︡ ︠83ae58d3-60f9-40c5-9ec5-82ddca754416s︠ S6.mutate([0,1,2,3]); S6.cluster() ︡22345d02-7256-4181-a755-e13f4beb4cd9︡{"stdout":"[x0, x1, x2, x3]\n"}︡{"done":true}︡ ︠0418ca29-b247-4a70-8e6b-12ce6620cd39s︠ var('x1','x2','x3','x0') ︡4190227c-ad8a-4c99-a60a-02fd3a822e35︡{"stdout":"(x1, x2, x3, x0)\n"}︡{"done":true}︡ ︠4e39721f-1177-4ae9-8d85-f4b1fef6db7bs︠ (x1*x2^2*x3^2 + x0^2*x1^2 + 3*x0*x1*x2*x3 + x2^2*x3^2 + 2*x0^2*x1 + 2*x0*x2*x3 + x0^2)/(x0*x1^2*x2*x3) ︡e73c0ea2-0554-4b1e-bdee-28b44fb4bcf0︡{"stdout":"(x1*x2^2*x3^2 + x0^2*x1^2 + 3*x0*x1*x2*x3 + x2^2*x3^2 + 2*x0^2*x1 + 2*x0*x2*x3 + x0^2)/(x0*x1^2*x2*x3)\n"}︡{"done":true}︡ ︠349cc26a-7534-4b87-93db-9ec8b04614aas︠ pretty_print(_) ︡3088dd75-671b-4051-880a-6d99a6c525e4︡{"html":"
$\\displaystyle \\frac{x_{1} x_{2}^{2} x_{3}^{2} + x_{0}^{2} x_{1}^{2} + 3 \\, x_{0} x_{1} x_{2} x_{3} + x_{2}^{2} x_{3}^{2} + 2 \\, x_{0}^{2} x_{1} + 2 \\, x_{0} x_{2} x_{3} + x_{0}^{2}}{x_{0} x_{1}^{2} x_{2} x_{3}}$
"}︡{"done":true}︡ ︠2a2af0db-900e-42dc-96b4-d980bf89c0f4s︠ S7 = ClusterSeed([[0,1],[1,2],[2,3],[2,4]]); S7.show() ︡ba94bc3c-0e97-437d-9927-737c532d20a2︡{"file":{"filename":"/home/user/.sage/temp/project-9d9553fd-aa65-444a-ab4d-b13f6613d595/237/tmp_GhrTkW.svg","show":true,"text":null,"uuid":"c06e5a0b-15b3-41aa-80a6-493f2b334c27"},"once":false}︡{"done":true}︡ ︠5a80a5e2-84ac-470f-b8a6-30deb77f2438s︠ S8 = S7.principal_extension(); S8.show() ︡d257f86e-1a06-40cb-9aba-acad25abbfe8︡{"stderr":"/ext/sage/sage-8.2/local/lib/python2.7/site-packages/sage/combinat/cluster_algebra_quiver/cluster_seed.py:361:\n********************************************************************************\nHaving frozen nodes is known to produce wrong answers\nThis issue is being tracked at https://trac.sagemath.org/sage_trac/ticket/22381.\n********************************************************************************\n"}︡{"file":{"filename":"/home/user/.sage/temp/project-9d9553fd-aa65-444a-ab4d-b13f6613d595/237/tmp_i2z3L0.svg","show":true,"text":null,"uuid":"7512fa9e-9408-404e-8fe4-8740c595a823"},"once":false}︡{"done":true}︡ ︠6009e136-e593-40d7-9e70-e11d585fa234s︠ S8.mutate([0,1,2,3,4]); S8.cluster() ︡605153e2-a32c-463a-bb5a-4f33e3ef9a68︡{"stdout":"[(x1 + y0)/x0, (x0*y0*y1 + x1*x2 + x2*y0)/(x0*x1), (x0*x3*x4*y0*y1 + x0*x1*y0*y1*y2 + x1*x2*x3*x4 + x2*x3*x4*y0)/(x0*x1*x2), (x0*x1*x2*y0*y1*y2*y3 + x0*x3*x4*y0*y1 + x0*x1*y0*y1*y2 + x1*x2*x3*x4 + x2*x3*x4*y0)/(x0*x1*x2*x3), (x0*x1*x2*y0*y1*y2*y4 + x0*x3*x4*y0*y1 + x0*x1*y0*y1*y2 + x1*x2*x3*x4 + x2*x3*x4*y0)/(x0*x1*x2*x4)]\n"}︡{"done":true}︡ ︠cb592275-c13b-455f-bb2f-987f01f7f8d2s︠ S8.f_polynomials() ︡a8900334-1087-4550-9aad-065b8e5d3bb7︡{"stdout":"[y0 + 1, y0*y1 + y0 + 1, y0*y1*y2 + y0*y1 + y0 + 1, y0*y1*y2*y3 + y0*y1*y2 + y0*y1 + y0 + 1, y0*y1*y2*y4 + y0*y1*y2 + y0*y1 + y0 + 1]\n"}︡{"done":true}︡ ︠6617c109-ec6a-4e92-babe-411df0893a72︠ ︡a0abddf3-c54d-4b8d-80cb-fa7a82e3c3c6︡ ︠b32a245d-96ec-40b8-b2f8-b17000dcbf68s︠ S9 = S6.principal_extension() ︡761f82ae-c1af-45f4-ad4e-231dfdfec68e︡{"done":true}︡ ︠884c6099-2a6c-4072-ac9b-9a75abed2445s︠ S9.cluster() ︡5465695f-f03a-4927-be42-146278d45fc4︡{"stdout":"[x0, x1, x2, x3]\n"}︡{"done":true}︡ ︠4c9b12ae-7bce-4164-bc73-78dce9031f5cs︠ S9.f_polynomials() ︡12216da4-50e7-4a92-a9fd-61ab3314b415︡{"stdout":"[1, 1, 1, 1]\n"}︡{"done":true}︡ ︠5c125670-ba8b-4362-9fc6-23394155be63s︠ S9.mutate([0,1,2,3]); S9.cluster(); S9.f_polynomials() ︡f6a1e0b4-2e76-4151-a298-af2ff7b28081︡{"stdout":"[(x1 + y0)/x0, (x1*x2*x3 + x2*x3*y0 + x0*y0*y1)/(x0*x1), (x0*x1*y0*y1*y2 + x1*x2*x3 + x2*x3*y0 + x0*y0*y1)/(x0*x1*x2), (x0*x1*y0*y1*y3 + x1*x2*x3 + x2*x3*y0 + x0*y0*y1)/(x0*x1*x3)]\n[y0 + 1, y0*y1 + y0 + 1, y0*y1*y2 + y0*y1 + y0 + 1, y0*y1*y3 + y0*y1 + y0 + 1]\n"}︡{"done":true}︡ ︠39259e63-838d-480d-ae4e-dba2b62cc04es︠ S9.mutate([0,1,2,3]); S9.cluster(); S9.f_polynomials() ︡a6ce8839-c922-427d-81fd-1f1208ad2955︡{"stdout":"[(x2*x3 + x0*y1)/x1, (x0^2*x1^2*y0*y1^2*y2*y3 + x0*x1*x2*x3*y0*y1*y2 + x0^2*x1*y0*y1^2*y2 + x0*x1*x2*x3*y0*y1*y3 + x0^2*x1*y0*y1^2*y3 + x1*x2^2*x3^2 + x2^2*x3^2*y0 + x0*x1*x2*x3*y1 + 2*x0*x2*x3*y0*y1 + x0^2*y0*y1^2)/(x0*x1^2*x2*x3), (x0*x1*y1*y3 + x2*x3 + x0*y1)/(x1*x3), (x0*x1*y1*y2 + x2*x3 + x0*y1)/(x1*x2)]\n[y1 + 1, y0*y1^2*y2*y3 + y0*y1^2*y2 + y0*y1^2*y3 + y0*y1^2 + y0*y1*y2 + y0*y1*y3 + 2*y0*y1 + y0 + y1 + 1, y1*y3 + y1 + 1, y1*y2 + y1 + 1]\n"}︡{"done":true}︡ ︠92cbee69-b7fe-44eb-aaaa-3250aaf8dbcbs︠ var('y0','y1','y2','y3') ︡1bbaa0f6-f7da-4e4a-be3a-6503a6a480a9︡{"stdout":"(y0, y1, y2, y3)\n"}︡{"done":true}︡ ︠4d7bfe26-a68d-44d9-b023-84af0e6699e4s︠ pretty_print(y0*y1^2*y2*y3 + y0*y1^2*y2 + y0*y1^2*y3 + y0*y1^2 + y0*y1*y2 + y0*y1*y3 + 2*y0*y1 + y0 + y1 + 1) ︡d65292aa-b2f0-4290-bc77-88d666542b35︡{"html":"
$\\displaystyle y_{0} y_{1}^{2} y_{2} y_{3} + y_{0} y_{1}^{2} y_{2} + y_{0} y_{1}^{2} y_{3} + y_{0} y_{1}^{2} + y_{0} y_{1} y_{2} + y_{0} y_{1} y_{3} + 2 \\, y_{0} y_{1} + y_{0} + y_{1} + 1$
"}︡{"done":true}︡ ︠44943462-0889-4256-bb9a-416940275b6e︠ ︡6edefb9d-3995-491a-aa24-89361852c55f︡ ︠e454df49-dbeb-40ae-b31e-52412047b3c6︠ ︡41a0fedf-72ab-4da6-ac2b-8e1b873f9a31︡