Program to show that no perfect 1-factorization for K_{4,4,4}<\/p>\n\n\n\n
def cond_perm(X, cond):\n L = [[]]\n for c in cond:\n L_tmp = L\n L = []\n for l in L_tmp:\n for x in set(X)-set(c)-set(l):\n L.append(l+[x])\n return L\n\ndef one_factor(L3AD, i):\n n = len(L3AD[0])\n x_axis = ['x'+str(k) for k in range(1,n+1)]\n y_axis = ['y'+str(k) for k in range(1,n+1)]\n z_axis = ['z'+str(k) for k in range(1,n+1)]\n one_f = {}\n for r in range(n):\n c = (L3AD[1][r]+[i]).index(i)\n if c < n:\n x, y = x_axis[-c-1], y_axis[-r-1]\n one_f[x], one_f[y] = y, x\n z, xy = z_axis[r], (x_axis[::-1]+y_axis)[L3AD[0][r].index(i)]\n one_f[z], one_f[xy] = xy, z\n return one_f\n\ndef cycle(L3AD, i1, i2, vertex):\n one_factor_1 = one_factor(L3AD, i1)\n one_factor_2 = one_factor(L3AD, i2)\n cyc = []\n while vertex not in cyc:\n cyc.append(vertex)\n vertex = one_factor_1[vertex]\n cyc.append(vertex)\n vertex = one_factor_2[vertex]\n return cyc\n\ndef check_P1F(L3AD):\n n = len(L3AD[0])\n for i in range(1,2*n+1):\n for j in range(i+1,2*n+1):\n if len(cycle(L3AD, i, j, 'x1')) < 3*n:\n return False\n return True\n\n\n\nY = list(range(1,9))\n\nfor t_row1 in cond_perm(set(Y)-{8},zip(Y[:4])):\n c1 = list(zip(Y[:4],t_row1))\n for t_row2 in cond_perm(set(Y)-{7},c1):\n c2 = list(zip(Y[:4],t_row1,t_row2))\n Y2_ = {i for i in Y if (t_row1+t_row2).count(i)>=2}\n for t_row3 in cond_perm(set(Y)-{6}-Y2_,c2):\n c3 = list(zip(Y[:4],t_row1,t_row2,t_row3))\n Y3_ = {i for i in Y if (t_row1+t_row2+t_row3).count(i)>=2}\n for t_row4 in cond_perm(set(Y)-{5}-Y3_,c3):\n c4_1 = list(zip(Y[:4],t_row1,t_row2,t_row3,t_row4))\n c4_2 = [[5]+t_row4,\n [6]+t_row3,\n [7]+t_row2,\n [8]+t_row1]\n \n for s_row2 in cond_perm(set(Y),c4_1+c4_2):\n c5_1 = list(zip(Y[:4],t_row1,t_row2,t_row3,t_row4,s_row2[:4]))\n c5_2 = [[5]+t_row4+[s_row2[4]],\n [6]+t_row3+[s_row2[5]],\n [7]+t_row2+[s_row2[6]],\n [8]+t_row1+[s_row2[7]]]\n for s_row3 in cond_perm(set(Y),c5_1+c5_2):\n c6_1 = list(zip(Y[:4],t_row1,t_row2,t_row3,t_row4,s_row2[:4],s_row3[:4]))\n c6_2 = [[5]+t_row4+[s_row2[4]]+[s_row3[4]],\n [6]+t_row3+[s_row2[5]]+[s_row3[5]],\n [7]+t_row2+[s_row2[6]]+[s_row3[6]],\n [8]+t_row1+[s_row2[7]]+[s_row3[7]]]\n for s_row4 in cond_perm(set(Y),c6_1+c6_2):\n L3AD = [[Y,s_row2,s_row3,s_row4],\n [t_row1,t_row2,t_row3,t_row4]]\n \n print(*L3AD[0],*L3AD[1], sep=\"\u00a5n\")\n\n if(check_P1F(L3AD)):\n print('perfect')\n exit()\n else:\n print('non-perfect')\n \n\n\n\n<\/code><\/pre>\n\n\n\nProgram to count the number of Latin regular hexahedra of order 2<\/p>\n\n\n\n
N = 2\nX = list(range(1,4*N+1))\n\n\ndef cond_perm(X, cond):\n L = [[]]\n for c in cond:\n L_tmp = L\n L = []\n for l in L_tmp:\n for x in set(X)-set(c)-set(l):\n L.append(l+[x])\n return L\n \ndef contra(circuit):\n c = []\n for i in [2,3,0,1]:\n c += circuit[(i+1)*N-1:i*N-1 if i>0 else None:-1]\n return c\n\ndef top_bottom(side):\n tb_list = [[]]\n cond = list(map(list,zip(*side,*map(contra,side))))\n c = cond[N:2*N]*2\n for i in range(N):\n tb_list_tmp = tb_list\n tb_list = []\n for tb in tb_list_tmp:\n if tb_list_tmp != [[]]:\n add_c = list(map(list,zip(*tb)))\n c = [c[j]+add_c[j]+add_c[j+N] for j in range(N)]*2\n for row in cond_perm(set(X)-set(cond[i]),c):\n tb_list.append(tb+[row])\n return tb_list\n\ndef formalize_top_bottom(tb):\n return [[tb[i][:N] for i in range(N)], [tb[i][N:] for i in reversed(range(N))]]\n\ndef L_list(n, X):\n side_list = [[X]]\n for _ in range(n-1):\n side_list_tmp = side_list\n side_list = []\n for s in side_list_tmp:\n for circ in cond_perm(X,zip(*s,*map(contra,s))):\n side_list.append(s+[circ])\n \n L_list, L_list_upto_tb_trans = [], []\n for side in side_list:\n tb_list = top_bottom(side)\n L_list += [side+[tb] for tb in tb_list]\n if len(tb_list)>0:\n L_list_upto_tb_trans += [side+[tb_list[0]]]\n \n L_list_upto_trans = []\n for L in L_list_upto_tb_trans:\n flag = False\n a = list(map(set,zip(L[1][:4],contra(L[1])[:4])))\n for i in range(len(L_list_upto_trans)):\n if a == list(map(set,zip(L_list_upto_tb_trans[i][1][:4],contra(L_list_upto_tb_trans[i][1])[:4]))):\n flag = True\n break\n if flag is False:\n L_list_upto_trans.append(L)\n \n return L_list, L_list_upto_tb_trans, L_list_upto_trans\n\ndef display(L):\n print(' '*N+'+'+'-'*(2*N-1)+'+')\n for t in formalize_top_bottom(L[2])[0]:\n print(' '*N+'|'+' '.join(map(str, t))+'|')\n print(('+'+'-'*(2*N-1))*4+'+')\n for i in range(N):\n print('|'+(('{} '*(N-1)+'{}|')*4).format(*L[i]))\n print(('+'+'-'*(2*N-1))*4+'+')\n for b in formalize_top_bottom(L[2])[1]:\n print(' '*N+'|'+' '.join(map(str, b))+'|')\n print(' '*N+'+'+'-'*(2*N-1)+'+')\n\n\n\nL_set, _, L_f = L_list(N, X)\n\nprint(len(L_f))\nfor L in L_f:\n display(L)\n\nprint(len(L_set)*8*7*6*5*4*3*2)\n\n<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"Program to show that no perfect 1-factorization for K_{4,4,4} Program to count the number of Latin regular hex […]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-855","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/www.math.akita-u.ac.jp\/yamamura\/index.php?rest_route=\/wp\/v2\/pages\/855","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.math.akita-u.ac.jp\/yamamura\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/www.math.akita-u.ac.jp\/yamamura\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/www.math.akita-u.ac.jp\/yamamura\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/www.math.akita-u.ac.jp\/yamamura\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=855"}],"version-history":[{"count":1,"href":"http:\/\/www.math.akita-u.ac.jp\/yamamura\/index.php?rest_route=\/wp\/v2\/pages\/855\/revisions"}],"predecessor-version":[{"id":856,"href":"http:\/\/www.math.akita-u.ac.jp\/yamamura\/index.php?rest_route=\/wp\/v2\/pages\/855\/revisions\/856"}],"wp:attachment":[{"href":"http:\/\/www.math.akita-u.ac.jp\/yamamura\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=855"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}