30.10.12

Vorlesung1 -- Sage

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#Arithmetik

2+2 4 2^3 8 10 % 3 # berechnet 10 mod 3 1 10 // 4 #berechnet den ganzzahligen Quotienten 2 3^2*4 + 2%5 38 pi pi n(pi) # gibt eine numerische Approximation zurück 3.14159265358979 pi.n(digits=5) #gibt eine Genauigkeit von 5 Stellen an 3.1416 2.5*2 5.00000000000000 5/2*2 5 exp(2) e^2 n(exp(2)) 7.38905609893065 exp(1).n(digits=20) 2.7182818284590452354 exp(1).n(digits=200) 2.7182818284590452353602874713526624977572470936999595749669676277240766\ 303535475945713821785251664274274663919320030599218174135966290435729003\ 342952605956307381323286279434907632338298807531952510190 infinity +Infinity exp(-infinity) 0 ln(1)

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0 log(1) 0 log(10,2) log(10)/log(2) n(log(2,2)) 1.00000000000000 n(log(10,10)) 1.00000000000000 n(log(10,3)) 2.09590327428938

# Die Hilfe aufrufen: exp?

File: /home/sage/sage-5.1/local/lib/python2.7/site-packages/sage/functions/log.py Type: Definition: exp(x, coerce=True, hold=False, prec=None, dont_call_method_on_arg=False) Docstring: The exponential function, exp(x) = ex . EXAMPLES: sage: exp(-1) e^(-1) sage: exp(2) e^2 sage: exp(2).n(100) 7.3890560989306502272304274606 sage: exp(x^2 + log(x)) e^(x^2 + log(x)) sage: exp(x^2 + log(x)).simplify() x*e^(x^2) sage: exp(2.5) 12.1824939607035 sage: exp(float(2.5)) 12.182493960703473 sage: exp(RDF('2.5')) 12.1824939607 To prevent automatic evaluation, use the holdparameter: sage: exp(I*pi,hold=True) e^(I*pi) sage: exp(0,hold=True) e^0 To then evaluate again, we currently must use Maxima via sage.symbolic.expression.Expression.simplify(): sage: exp(0,hold=True).simplify() 1 sage: exp(pi*I/2) I sage: exp(pi*I) -1 sage: exp(8*pi*I) 1 sage: exp(7*pi*I/2) -I TEST: sage: latex(exp(x)) e^{x} localhost:8000/home/admin/4/

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sage: latex(exp(sqrt(x))) e^{\sqrt{x}} sage: latex(exp) \exp sage: latex(exp(sqrt(x))^x) \left(e^{\sqrt{x}}\right)^{x} sage: latex(exp(sqrt(x)^x)) e^{\left(\sqrt{x}^{x}\right)} Test conjugates: sage: conjugate(exp(x)) e^conjugate(x) Test simplifications when taking powers of exp, #7264: sage: var('a,b,c,II') (a, b, c, II) sage: model_exp = exp(II)**a*(b) sage: sol1_l={b: 5.0, a: 1.1} sage: model_exp.subs(sol1_l) 5.00000000000000*(e^II)^1.10000000000000 sage: exp(3)^II*exp(x) (e^3)^II*e^x sage: exp(x)*exp(x) e^(2*x) sage: exp(x)*exp(a) e^(a + x) sage: exp(x)*exp(a)^2 e^(2*a + x) Another instance of the same problem, #7394: sage: 2*sqrt(e) 2*sqrt(e)

exp(x^2 + log(x)) e^(x^2 + log(x)) exp(x^2 + log(x)).simplify() x*e^(x^2) n(tan(3)) -0.142546543074278 abs(-2) 2

ceil(3.44) 4 floor(2.222) 2 true True is_even(2) True is_even(3) False is_odd(3) True 6%2==0 localhost:8000/home/admin/4/

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True 6%4==0 False not(true) False true|false True true&false False #Variablen 5==5 True 5==3 False 2 4) True (x= 5

# undefinierte Variablen

y = var('y')

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y y k Traceback (click to the left of this block for traceback) ... NameError: name 'k' is not defined #alle Variablen löschen reset()

y Traceback (click to the left of this block for traceback) ... NameError: name 'y' is not defined

a,b,c = var('a,b,c') (3*a+5*b)+(6*a + 7*c) 9*a + 5*b + 7*c

# Trigonometrisches Vereinfachen x = var('x') s = (sin(x)^2 + cos(x)^2) s.simplify() # zum Vereinfachen zum Ausdrücken, hilft hier nicht s.simplify_trig() # kann den trig. Ausdruck s vereinfachen 1

# Expandieren (eines Ausdrucks) p, q = var('p,q') p = (3*x^2 + 7*x + 5) q = (7*x^3+5*x-4) p*q (3*x^2 + 7*x + 5)*(7*x^3 + 5*x - 4) expand(_) 21*x^5 + 49*x^4 + 50*x^3 + 23*x^2 - 3*x - 20 factor(_) (3*x^2 + 7*x + 5)*(7*x^3 + 5*x - 4)

# Mengen M1 = set([1,5,3,2]) M2 = set([1,6,7,2])

M1 set([1, 2, 3, 5]) M2 set([1, 2, 6, 7]) M1.union(M2) set([1, 2, 3, 5, 6, 7]) localhost:8000/home/admin/4/

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M1.intersection(M2) set([1, 2]) M1.difference(M2) set([3, 5]) 6 in M2 True 9 in M2 False M1 == M2 False # Vektoren und Matrizen

a = vector([1,2,3])

a (1, 2, 3) b = vector([4,5,6])

a+b (5, 7, 9) c=2.11

a*c (2.11000000000000, 4.22000000000000, 6.33000000000000) a*b 32 a (1, 2, 3) a.get(1) 2 a.outer_product(b) [ 4 5 6] [ 8 10 12] [12 15 18]

M = Matrix([[2,1,0],[3,2,1],[4,3,2]])

M [2 1 0] [3 2 1] [4 3 2] M[1] (3, 2, 1) localhost:8000/home/admin/4/

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M[1][1] 2 M.transpose() [2 3 4] [1 2 3] [0 1 2] M*c [ 4.22000000000000 2.11000000000000 0.000000000000000] [ 6.33000000000000 4.22000000000000 2.11000000000000] [ 8.44000000000000 6.33000000000000 4.22000000000000] c*M [ 4.22000000000000 2.11000000000000 0.000000000000000] [ 6.33000000000000 4.22000000000000 2.11000000000000] [ 8.44000000000000 6.33000000000000 4.22000000000000] M*a (4, 10, 16) a*M (20, 14, 8) x=M.solve_right(a) #Lösen der Gleichung Mx=y

x (0, 1, 0) M\a # kürzer mit Backslash (0, 1, 0) #Strings

a='hello'

a 'hello' b='world'

a+b 'helloworld' a+''+b 'hello world' a+1 Traceback (click to the left of this block for traceback) ... TypeError: unsupported operand parent(s) for '+': '' and 'Integer Ring' type(a) type(1)

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type(1.1) type(true) #Listen

v = [2, 3, 5]

v [2, 3, 5] vector(v) (2, 3, 5) v[2] 5 v[0] 2 v=v+[2]

v [2, 3, 5, 2] v[0]=1

v # Listen sind veränderbar [1, 3, 5, 2] w=vector(v)

w (1, 3, 5, 2) w[1]=8

v = v + ['hallo'];

v [1, 3, 5, 2, 'hallo'] v[4] 'hallo' w=vector(v) Traceback (click to the left of this block for traceback) ... TypeError: unable to find a common ring for all elements type(v)

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range(1,15) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] L=[2^n for n in range(1,15)]

L [2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384] L=[factor(n) for n in range(1,15)]

L [1, 2, 3, 2^2, 5, 2 * 3, 7, 2^3, 3^2, 2 * 5, 11, 2^2 * 3, 13, 2 * 7] L=[factor(n) for n in range(1, 15) if is_odd(n)]

L [1, 3, 5, 7, 3^2, 11, 13] L[1:4] [3, 5, 7] L[2:] [5, 7, 3^2, 11, 13] #Tupel

v = (1,2,3,4)

v (1, 2, 3, 4) type(v) v[1] 2 v[1]=5 #Tupel sind unveränderbar Traceback (click to the left of this block for traceback) ... TypeError: 'tuple' object does not support item assignment #Folgen

v = Sequence([1,2,3,4/5])

v [1, 2, 3, 4/5] v[1]=8 #Folgen sind veränderbar

v [1, 8, 3, 4/5]

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type(v) list(v) [1, 8, 3, 4/5] type(list(v))

#Rechenbeispiele

n, j, i = var('n', 'j', 'i')

# Summe über i für i von 1 bis n

sum(i, i, 1, n) 1/2*n^2 + 1/2*n sum(i^2,i,1,n) 1/3*n^3 + 1/2*n^2 + 1/6*n sum(i^5,i,1,n) 1/6*n^6 + 1/2*n^5 + 5/12*n^4 - 1/12*n^2

x1,x2,x3 = var('x1,x2,x3')

solve([x1^2-2*x2+x3==-2,x1+x2+4*x3==1,5*x1-2*x2+6*x3==-2],x1,x2,x3) [[x1 == 0, x2 == 1, x3 == 0], [x1 == (5/2), x2 == (7/2), x3 == (-5/4)]]

sudoku?

File: /usr/local/lib/sage/local/lib/python2.6/site-packages/sage/games/sudoku.py Type: Definition: sudoku(m) Docstring: Solves Sudoku puzzles described by matrices. INPUT: m- a square Sage matrix over Z, where zeros are blank entries OUTPUT: A Sage matrix over Z containing the first solution found, otherwise None.

This function matches the behavior of the prior Sudoku solver and is included only to replicate that behavior. It could be safely deprecated, since all of EXAMPLE: An example that was used in previous doctests.

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sage: A = matrix(ZZ,9,[5,0,0, 0,8,0, 0,4,9, 0,0,0, 5,0,0, 0,3,0, 0,6,7, 3,0,0, 0,0,1, 1,5,0, 0,0 sage: A [5 0 0 0 8 0 0 4 9] [0 0 0 5 0 0 0 3 0] [0 6 7 3 0 0 0 0 1] [1 5 0 0 0 0 0 0 0] [0 0 0 2 0 8 0 0 0] [0 0 0 0 0 0 0 1 8] [7 0 0 0 0 4 1 5 0] [0 3 0 0 0 2 0 0 0] [4 9 0 0 5 0 0 0 3] sage: sudoku(A) [5 1 3 6 8 7 2 4 9] [8 4 9 5 2 1 6 3 7] [2 6 7 3 4 9 5 8 1] [1 5 8 4 6 3 9 7 2] [9 7 4 2 1 8 3 6 5] [3 2 6 7 9 5 4 1 8] [7 8 2 9 3 4 1 5 6] [6 3 5 1 7 2 8 9 4] [4 9 1 8 5 6 7 2 3] Using inputs that are possible with the Sudokuclass, other than a matrix, will cause an error. sage: sudoku('.4..32....14..3.') Traceback (click to the left of this block for traceback) ...

A = matrix(ZZ,9,[5,0,0, 0,8,0, 0,4,9, 0,0,0, 5,0,0, 0,3,0, 0,6,7, 3,0,0, 0,0,1, 1,5,0, 0,0,0, 0,0,0, 0,0,0, 2,0,8, 0,0,0, 0,0,0, 0,0,0, 0,1,8, 7,0,0, 0,0,4, 1,5,0, 0,3,0, 0,0,2, 0,0,0, 4,9,0, 0,5,0, 0,0,3]) A [5 0 0 0 8 0 0 4 9] [0 0 0 5 0 0 0 3 0] [0 6 7 3 0 0 0 0 1] [1 5 0 0 0 0 0 0 0] [0 0 0 2 0 8 0 0 0] [0 0 0 0 0 0 0 1 8] [7 0 0 0 0 4 1 5 0] [0 3 0 0 0 2 0 0 0] [4 9 0 0 5 0 0 0 3] sudoku(A) [5 1 3 6 8 7 2 4 9] [8 4 9 5 2 1 6 3 7] [2 6 7 3 4 9 5 8 1] [1 5 8 4 6 3 9 7 2] [9 7 4 2 1 8 3 6 5] [3 2 6 7 9 5 4 1 8] [7 8 2 9 3 4 1 5 6] [6 3 5 1 7 2 8 9 4] [4 9 1 8 5 6 7 2 3] L=[2^n for n in range(1,5)]

Permutations?

File: /home/sage/sage-5.1/local/lib/python2.7/site-packages/sage/combinat/permutation.py Type: Definition: Permutations(n=None, k=None, **kwargs) Docstring: Returns a combinatorial class of permutations. Permutations(n) returns the class of permutations of n, if n is an integer, list, set, or string. Permutations(n, k) returns the class of permutations of n (where n is any of the above things) of length k; k must be an integer. Valid keyword arguments are: ‘descents’, ‘bruhat_smaller’, ‘bruhat_greater’, ‘recoils_finer’, ‘recoils_fatter’, ‘recoils’, and ‘avoiding’. With the exception of ‘avoiding’, you cannot specify n or k along with a keyword.

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Permutations(descents=list) returns the class of permutations with descents in the positions specified by ‘list’. Permutations(bruhat_smaller,greater=p) returns the class of permutations smaller or greater, respectively, than the given permutation in Bruhat order. Permutations(recoils=p) returns the class of permutations whose recoils composition is p. Permutations(recoils_fatter,finer=p) returns the class of permutations whose recoils composition is fatter or finer, respectively, than the given permutation. Permutations(n, avoiding=P) returns the class of permutations of n avoiding P. Here P may be a single permutation or a list of permutations; the returned class will avoid all patterns in P. EXAMPLES: sage: p = Permutations(3); p Standard permutations of 3 sage: p.list() [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]] sage: p = Permutations(3, 2); p Permutations of {1,...,3} of length 2 sage: p.list() [[1, 2], [1, 3], [2, 1], [2, 3], [3, 1], [3, 2]] sage: p = Permutations(['c', 'a', 't']); p Permutations of the set ['c', 'a', 't'] sage: p.list() [['c', 'a', 't'], ['c', 't', 'a'], ['a', 'c', 't'], ['a', 't', 'c'], ['t', 'c', 'a'], ['t', 'a', 'c']] sage: p = Permutations(['c', 'a', 't'], 2); p Permutations of the set ['c', 'a', 't'] of length 2 sage: p.list() [['c', 'a'], ['c', 't'], ['a', 'c'], ['a', 't'], ['t', 'c'], ['t', 'a']] sage: p = Permutations([1,1,2]); p Permutations of the multi-set [1, 1, 2] sage: p.list() [[1, 1, 2], [1, 2, 1], [2, 1, 1]] sage: p = Permutations([1,1,2], 2); p Permutations of the multi-set [1, 1, 2] of length 2 sage: p.list() [[1, 1], [1, 2], [2, 1]] sage: p = Permutations(descents=[1,3]); p Standard permutations of 4 with descents [1, 3] sage: p.list() [[1, 3, 2, 4], [1, 4, 2, 3], [2, 3, 1, 4], [2, 4, 1, 3], [3, 4, 1, 2]] sage: p = Permutations(bruhat_smaller=[1,3,2,4]); p Standard permutations that are less than or equal to [1, 3, 2, 4] in the Bruhat order sage: p.list() [[1, 2, 3, 4], [1, 3, 2, 4]] sage: p = Permutations(bruhat_greater=[4,2,3,1]); p Standard permutations that are greater than or equal to [4, 2, 3, 1] in the Bruhat order sage: p.list() [[4, 2, 3, 1], [4, 3, 2, 1]] sage: p = Permutations(recoils_finer=[2,1]); p Standard permutations whose recoils composition is finer than [2, 1] sage: p.list() [[1, 2, 3], [1, 3, 2], [3, 1, 2]] sage: p = Permutations(recoils_fatter=[2,1]); p Standard permutations whose recoils composition is fatter than [2, 1] sage: p.list() [[1, 3, 2], [3, 1, 2], [3, 2, 1]] sage: p = Permutations(recoils=[2,1]); p Standard permutations whose recoils composition is [2, 1] sage: p.list() [[1, 3, 2], [3, 1, 2]] sage: p = Permutations(4, avoiding=[1,3,2]); p Standard permutations of 4 avoiding [1, 3, 2] sage: p.list() [[4, 1, 2, 3], [4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 1, 2], localhost:8000/home/admin/4/

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[4, 3, 2, 1], [3, 4, 1, 2], [3, 4, 2, 1], [2, 3, 4, 1], [3, 2, 4, 1], [1, 2, 3, 4], [2, 1, 3, 4], [2, 3, 1, 4], [3, 1, 2, 4], [3, 2, 1, 4]] sage: p = Permutations(5, avoiding=[[3,4,1,2], [4,2,3,1]]); p Standard permutations of 5 avoiding [[3, 4, 1, 2], [4, 2, 3, 1]] sage: p.cardinality() 88 sage: p.random_element() [5, 1, 2, 4, 3]

LL=Permutations(L)

LL.list() [[2, 4, 8, 16], [2, 4, 16, 8], [2, 8, 4, 16], [2, 8, 16, 4], [2, 16, 4, 8], [2, 16, 8, 4], [4, 2, 8, 16], [4, 2, 16, 8], [4, 8, 2, 16], [4, 8, 16, 2], [4, 16, 2, 8], [4, 16, 8, 2], [8, 2, 4, 16], [8, 2, 16, 4], [8, 4, 2, 16], [8, 4, 16, 2], [8, 16, 2, 4], [8, 16, 4, 2], [16, 2, 4, 8], [16, 2, 8, 4], [16, 4, 2, 8], [16, 4, 8, 2], [16, 8, 2, 4], [16, 8, 4, 2]] b=LL[5]

b [2, 16, 8, 4] B=vector(b).outer_product(vector(LL[2]))

B [ 4 16 8 32] [ 32 128 64 256] [ 16 64 32 128] [ 8 32 16 64] LL=Permutations(range(1,8))

LL.list() WARNING: Output truncated! full_output.txt

[[1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 7, 6], [1, 2, 3, 4, 6, 5, 7], [1, 2, 3, 4, 6, 7, 5], [1, 2, 3, 4, 7, 5, 6], [1, 2, 3, 4, 7, 6, 5], [1, 2, 3, 5, 4, 6, 7], [1, 2, 3, 5, 4, 7, 6], [1, 2, 3, 5, 6, 4, 7], [1, 2, 3, 5, 6, 7, 4], [1, 2, 3, 5, 7, 4, 6], [1, 2, 3, 5, 7, 6, 4], [1, 2, 3, 6, 4, 5, 7], [1, 2, 3, 6, 4, 7, 5], [1, 2, 3, 6, 5, 4, 7], [1, 2, 3, 6, 5, 7, 4], [1, 2, 3, 6, 7, 4, 5], [1, 2, 3, 6, 7, 5, 4], [1, 2, 3, 7, 4, 5, 6], [1, 2, 3, 7, 4, 6, 5], [1, 2, 3, 7, 5, 4, 6], [1, 2, 3, 7, 5, 6, 4], [1, 2, 3, 7, 6, 4, 5], [1, 2, 3, 7, 6, 5, 4], [1, 2, 4, 3, 5, 6, 7], [1, 2, 4, 3, 5, 7, 6], [1, 2, 4, 3, 6, 5, 7], [1, 2, 4, 3, 6, 7, 5], [1, 2, 4, 3, 7, 5, 6], [1, 2, 4, 3, 7, 6, 5], [1, 2, 4, 5, 3, 6, 7], [1, 2, 4, 5, 3, 7, 6], [1, 2, 4, 5, 6, 3, 7], [1, 2, 4, 5, 6, 7, 3], [1, 2, 4, 5, 7, 3, 6], [1, 2, 4, 5, 7, 6, 3], [1, 2, 4, 6, 3, 5, 7], [1, 2, 4, 6, 3, 7, 5], [1, 2, 4, 6, 5, 3, 7], [1, 2, 4, 6, 5, 7, 3], [1, 2, 4, 6, 7, 3, 5], [1, 2, 4, 6, 7, 5, 3], [1, 2, 4, 7, 3, 5, 6], [1, 2, 4, 7, 3, 6, 5], [1, 2, 4, 7, 5, 3, 6], [1, 2, 4, 7, 5, 6, 3], [1, 2, 4, 7, 6, 3, 5], [1, 2, 4, 7, 6, 5, 3], [1, 2, 5, 3, 4, 6, 7], [1, 2, 5, 3, 4, 7, 6], [1, 2, 5, 3, 6, 4, 7], [1, 2, 5, 3, 6, 7, 4], [1, 2, 5, 3, 7, 4, 6], [1, 2, 5, 3, 7, 6, 4], [1, 2, 5, 4, 3, 6, 7], [1, 2, 5, 4, 3, 7, 6], [1, 2, 5, 4, 6, 3, 7], [1, 2, 5, 4, 6, 7, 3], [1, 2, 5, 4, 7, 3, 6], [1, 2, 5, 4, 7, 6, 3], [1, 2, 5, 6, 3, 4, 7], [1, 2, 5, 6, 3, 7, 4], [1, 2, 5, 6, 4, 3, 7], [1, 2, 5, 6, 4, 7, 3], [1, 2, 5, 6, 7, 3, 4], [1, 2, 5, 6, 7, 4, 3], [1, 2, 5, 7, 3, 4, 6], [1, 2, 5, 7, 3, 6, 4], [1, 2, 5, 7, 4, 3, 6], [1, 2, 5, 7, 4, 6, 3], [1, 2, 5, 7, 6, 3, 4], [1, 2, 5, 7, 6, 4, 3], [1, localhost:8000/home/admin/4/

13/20

2, 6, 3, 4, 5, 7], [1, 2, 6, 3, 4, 7, 5], [1, 2, 6, 3, 5, 4, 7], [1, 2, 6, 3, 5, 7, 4], [1, 2, 6, 3, 7, 4, 5], [1, 2, 6, 3, 7, 5, 4], [1, 2, 6, 4, 3, 5, 7], [1, 2, 6, 4, 3, 7, 5], [1, 2, 6, 4, 5, 3, 7], [1, 2, 6, 4, 5, 7, 3], [1, 2, 6, 4, 7, 3, 5], [1, 2, 6, 4, 7, 5, 3], [1, 2, 6, 5, 3, 4, 7], [1, 2, 6, 5, 3, 7, 4], [1, 2, 6, 5, 4, 3, 7], [1, 2, 6, 5, 4, 7, 3], [1, 2, 6, 5, 7, 3, 4], [1, 2, 6, 5, 7, 4, 3], [1, 2, 6, 7, 3, 4, 5], [1, 2, 6, 7, 3, 5, 4], [1, 2, 6, 7, 4, 3, 5], [1, 2, 6, 7, 4, 5, 3], [1, 2, 6, 7, 5, 3, 4], [1, 2, 6, 7, 5, 4, 3], [1, 2, 7, 3, 4, 5, 6], [1, 2, 7, 3, 4, 6, 5], [1, 2, 7, 3, 5, 4, 6], [1, 2, 7, 3, 5, 6, 4], [1, 2, 7, 3, 6, 4, 5], [1, 2, 7, 3, 6, 5, 4], [1, 2, 7, 4, 3, 5, 6], [1, 2, 7, 4, 3, 6, 5], [1, 2, 7, 4, 5, 3, 6], [1, 2, 7, 4, 5, 6, 3], [1, 2, 7, 4, 6, 3, 5], [1, 2, 7, 4, 6, 5, 3], [1, 2, 7, 5, 3, 4, 6], [1, 2, 7, 5, 3, 6, 4], [1, 2, 7, 5, 4, 3, 6], [1, 2, 7, 5, 4, 6, 3], [1, 2, 7, 5, 6, 3, 4], [1, 2, 7, 5, 6, 4, 3], [1, 2, 7, 6, 3, 4, 5], [1, 2, 7, 6, 3, 5, 4], [1, 2, 7, 6, 4, 3, 5], [1, 2, 7, 6, 4, 5, 3], [1, 2, 7, 6, 5, 3, 4], [1, 2, 7, 6, 5, 4, 3], [1, 3, 2, 4, 5, 6, 7], [1, 3, 2, 4, 5, 7, 6], [1, 3, 2, 4, 6, 5, 7], [1, 3, 2, 4, 6, 7, 5], [1, 3, 2, 4, 7, 5, 6], [1, 3, 2, 4, 7, 6, 5], [1, 3, 2, 5, 4, 6, 7], [1, 3, 2, 5, 4, 7, 6], [1, 3, 2, 5, 6, 4, 7], [1, 3, 2, 5, 6, 7, 4], [1, 3, 2, 5, 7, 4, 6], [1, 3, 2, 5, 7, 6, 4], [1, 3, 2, 6, 4, 5, 7], [1, 3, 2, 6, 4, 7, 5], [1, 3, 2, 6, 5, 4, 7], [1, 3, 2, 6, 5, 7, 4], [1, 3, 2, 6, 7, 4, 5], [1, 3, 2, 6, 7, 5, 4], [1, 3, 2, 7, 4, 5, 6], [1, 3, 2, 7, 4, 6, 5], [1, 3, 2, 7, 5, 4, 6], [1, 3, 2, 7, 5, 6, 4], [1, 3, 2, 7, 6, 4, 5], [1, 3, 2, 7, 6, 5, 4], [1, 3, 4, 2, 5, 6, 7], [1, 3, 4, 2, 5, 7, 6], [1, 3, 4, 2, 6, 5, 7], [1, 3, 4, 2, 6, 7, 5], [1, 3, 4, 2, 7, 5, 6], [1, 3, 4, 2, 7, 6, 5], [1, 3, 4, 5, 2, 6, 7], [1, 3, 4, 5, 2, 7, 6], [1, 3, 4, 5, 6, 2, 7], [1, 3, 4, 5, 6, 7, 2], [1, 3, 4, 5, 7, 2, 6], [1, 3, 4, 5, 7, 6, 2], [1, 3, 4, 6, 2, 5, 7], [1, 3, 4, 6, 2, 7, 5], [1, 3, 4, 6, 5, 2, 7], [1, 3, 4, 6, 5, 7, 2], [1, 3, 4, 6, 7, 2, 5], [1, 3, 4, 6, 7, 5, 2], [1, 3, 4, 7, 2, 5, 6], [1, 3, 4, 7, 2, 6, 5], [1, 3, 4, 7, 5, 2, 6], [1, 3, 4, 7, 5, 6, 2], [1, 3, 4, 7, 6, 2, 5], [1, 3, 4, 7, 6, 5, 2], [1, 3, 5, 2, 4, 6, 7], [1, 3, 5, 2, 4, 7, 6], [1, 3, 5, 2, 6, 4, 7], [1, 3, 5, 2, 6, 7, 4], [1, 3, 5, 2, 7, 4, 6], [1, 3, 5, 2, 7, 6, 4], [1, 3, 5, 4, 2, 6, 7], [1, 3, 5, 4, 2, 7, 6], [1, 3, 5, 4, 6, 2, 7], [1, 3, 5, 4, 6, 7, 2], [1, 3, 5, 4, 7, 2, 6], [1, 3, 5, 4, 7, 6, 2], [1, 3, 5, 6, 2, 4, 7], [1, 3, 5, 6, 2, 7, 4], [1, 3, 5, 6, 4, 2, 7], [1, 3, 5, 6, 4, 7, 2], [1, 3, 5, 6, 7, 2, 4], [1, 3, 5, 6, 7, 4, 2], [1, 3, 5, 7, 2, 4, 6], [1, 3, 5, 7, 2, 6, 4], [1, 3, 5, 7, 4, 2, 6], [1, 3, 5, 7, 4, 6, 2], [1, 3, 5, 7, 6, 2, 4], [1, 3, 5, 7, 6, 4, 2], [1, 3, 6, 2, 4, 5, 7], [1, 3, 6, 2, 4, 7, 5], [1, 3, 6, 2, 5, 4, 7], [1, 3, 6, 2, 5, 7, 4], [1, 3, 6, 2, 7, 4, 5], [1, 3, 6, 2, 7, 5, 4], [1, 3, 6, 4, 2, 5, 7], [1, 3, 6, 4, 2, 7, 5], [1, 3, 6, 4, 5, 2, 7], [1, 3, 6, 4, 5, 7, 2], [1, 3, 6, 4, 7, 2, 5], [1, 3, 6, 4, 7, 5, 2], [1, 3, 6, 5, 2, 4, 7], [1, 3, 6, 5, 2, 7, 4], [1, 3, 6, 5, 4, 2, 7], [1, 3, 6, 5, 4, 7, 2], [1, 3, 6, 5, 7, 2, 4], [1, 3, 6, 5, 7, 4, 2], [1, 3, 6, 7, 2, 4, 5], [1, 3, 6, 7, 2, 5, 4], [1, 3, 6, 7, 4, 2, 5], [1, 3, 6, 7, 4, 5, 2], [1, 3, 6, 7, 5, 2, 4], [1, 3, 6, 7, 5, 4, 2], [1, 3, 7, 2, 4, 5, 6], [1, 3, 7, 2, 4, 6, 5], [1, 3, 7, 2, 5, 4, 6], [1, 3, 7, 2, 5, 6, 4], [1, 3, 7, 2, 6, 4, 5], [1, 3, 7, 2, 6, 5, 4], [1, 3, 7, 4, 2, 5, 6], [1, 3, 7, 4, 2, 6, 5], [1, 3, 7, 4, 5, 2, 6], [1, 3, 7, 4, 5, 6, 2], [1, 3, 7, 4, 6, 2, 5], [1, 3, 7, 4, 6, 5, 2], [1, 3, 7, 5, 2, 4, 6], [1, 3, 7, 5, 2, 6, 4], [1, 3, 7, 5, 4, 2, 6], [1, 3, 7, 5, 4, 6, 2], [1, 3, 7, 5, 6, 2, 4], [1, 3, 7, 5, 6, 4, 2], [1, 3, 7, 6, 2, 4, 5], [1, 3, 7, 6, 2, 5, 4], [1, 3, 7, 6, 4, 2, 5], [1, 3, 7, 6, 4, 5, 2], [1, 3, 7, 6, 5, 2, 4], [1, 3, 7, 6, 5, 4, 2], [1, 4, 2, 3, 5, 6, 7], [1, 4, 2, 3, 5, 7, 6], [1, 4, 2, 3, 6, 5, 7], [1, 4, 2, 3, 6, 7, 5], [1, 4, 2, 3, 7, 5, 6], [1, 4, 2, 3, 7, 6, 5], [1, 4, 2, 5, 3, 6, 7], [1, 4, 2, 5, 3, 7, 6], [1, 4, 2, 5, 6, 3, 7], [1, 4, 2, 5, 6, 7, 3], [1, 4, 2, 5, 7, 3, 6], [1, 4, 2, 5, 7, 6, 3], [1, 4, 2, 6, 3, 5, 7], [1, 4, 2, 6, 3, 7, 5], [1, 4, 2, 6, 5, 3, 7], [1, 4, 2, 6, 5, 7, 3], [1, 4, 2, 6, 7, 3, 5], [1, 4, 2, 6, 7, 5, 3], [1, 4, 2, 7, 3, 5, 6], [1, 4, 2, 7, 3, 6, 5], [1, 4, 2, 7, 5, 3, 6], [1, 4, 2, 7, 5, 6, 3], [1, 4, 2, 7, 6, 3, 5], [1, 4, 2, 7, 6, 5, 3], [1, 4, 3, 2, 5, 6, 7], [1, 4, 3, 2, 5, 7, 6], [1, 4, 3, 2, 6, 5, 7], [1, 4, 3, 2, 6, 7, 5], [1, 4, 3, 2, 7, 5, 6], [1, 4, 3, 2, 7, 6, 5], [1, 4, 3, 5, 2, 6, 7], [1, 4, 3, 5, 2, 7, 6], [1, 4, 3, 5, 6, 2, 7], [1, 4, 3, 5, 6, 7, 2], [1, 4, 3, 5, 7, 2, 6], [1, 4, 3, 5, 7, 6, 2], [1, 4, 3, 6, 2, 5, 7], [1, 4, 3, 6, 2, 7, 5], [1, 4, 3, 6, 5, 2, 7], [1, 4, 3, 6, 5, 7, 2], [1, 4, 3, 6, 7, 2, 5], [1, 4, 3, 6, 7, 5, 2], [1, 4, 3, 7, 2, 5, 6], [1, 4, 3, 7, 2, 6, 5], [1, 4, 3, 7, 5, 2, 6], [1, 4, 3, 7, 5, 6, 2], [1, 4, 3, 7, 6, 2, 5], [1, 4, 3, 7, 6, 5, 2], [1, 4, 5, 2, 3, 6, 7], [1, 4, 5, 2, 3, 7, 6], [1, 4, 5, 2, 6, 3, 7], [1, 4, 5, 2, 6, 7, 3], [1, 4, 5, 2, 7, 3, 6], [1, 4, 5, 2, 7, 6, 3], [1, 4, 5, 3, 2, 6, 7], [1, 4, 5, 3, 2, 7, 6], [1, 4, 5, 3, 6, 2, 7], [1, 4, 5, 3, 6, 7, 2], [1, 4, 5, 3, 7, 2, 6], [1, 4, 5, 3, 7, 6, 2], [1, 4, 5, 6, 2, 3, 7], [1, 4, 5, 6, 2, 7, 3], [1, 4, 5, 6, 3, 2, 7], [1, 4, 5, 6, 3, 7, 2], [1, 4, 5, 6, 7, 2, 3], [1, 4, 5, 6, 7, 3, 2], [1, 4, 5, 7, 2, 3, 6], [1, 4, 5, 7, 2, 6, 3], [1, 4, 5, 7, 3, 2, 6], [1, 4, 5, 7, 3, 6, 2], [1, 4, 5, 7, 6, 2, 3], [1, 4, 5, 7, 6, 3, 2], [1, 4, 6, 2, 3, 5, 7], [1, 4, 6, 2, 3, 7, 5], [1, 4, 6, 2, 5, 3, 7], [1, 4, 6, 2, 5, 7, 3], [1, 4, 6, 2, 7, 3, 5], [1, 4, 6, 2, 7, 5, 3], [1, 4, 6, 3, 2, 5, 7], [1, 4, 6, 3, 2, 7, 5], [1, 4, 6, 3, 5, 2, 7], [1, 4, 6, 3, 5, 7, 2], [1, 4, 6, 3, 7, 2, 5], [1, 4, 6, 3, 7, 5, 2], [1, 4, 6, 5, 2, 3, 7], [1, 4, 6, 5, 2, 7, 3], [1, 4, 6, 5, 3, 2, 7], [1, 4, 6, 5, 3, 7, 2], [1, 4, 6, 5, 7, 2, 3], [1, 4, 6, 5, 7, 3, 2], [1, 4, 6, 7, 2, 3, 5], [1, 4, 6, 7, 2, 5, 3], [1, 4, 6, 7, 3, 2, 5], [1, 4, 6, 7, 3, 5, 2], [1, 4, 6, 7, 5, 2, 3], [1, 4, 6, 7, 5, 3, 2], [1, 4, 7, 2, 3, 5, 6], [1, 4, 7, 2, 3, 6, 5], [1, 4, 7, 2, 5, 3, 6], [1, 4, 7, 2, 5, 6, 3], [1, 4, 7, 2, 6, 3, 5], [1, 4, 7, 2, 6, 5, 3], [1, 4, 7, 3, 2, 5, 6], [1, 4, 7, 3, 2, 6, 5], [1, 4, 7, 3, 5, 2, 6], [1, 4, 7, 3,

30.10.12

Vorlesung1 -- Sage

5, 6, 2], [1, 4, 7, 3, 6, 2, 5], [1, 4, 7, 3, 6, 5, 2], [1, 4, 7, 5, 2, 3, 6], [1, 4, 7, 5, 2, 6, 3], [1, 4, 7, 5, 3, 2, 6], [1, 4, 7, 5, 3, 6, 2], [1, 4, 7, 5, 6, 2, 3], [1, 4, 7, 5, 6, 3, 2], [1, 4, 7, 6, 2, 3, 5], [1, 4, 7, 6, 2, 5, 3], [1, 4, 7, 6, 3, 2, 5], [1, 4, 7, 6, 3, 5, 2], [1, 4, 7, 6, 5, 2, 3], [1, 4, 7, 6, 5, 3, 2], [1, 5, 2, 3, 4, 6, 7], [1, 5, 2, 3, 4, 7, 6], [1, 5, 2, 3, 6, 4, 7], [1, 5, 2, 3, 6, 7, 4], [1, 5, 2, 3, 7, 4, 6], [1, 5, 2, 3, 7, 6, 4], [1, 5, 2, 4, 3, 6, 7], [1, 5, 2, 4, 3, 7, 6], [1, 5, 2, 4, 6, 3, 7], [1, 5, 2, 4, 6, 7, 3], [1, 5, 2, 4, 7, 3, 6], [1, 5, 2, 4, 7, 6, 3], [1, 5, 2, 6, 3, 4, 7], [1, 5, 2, 6, 3, 7, 4], [1, 5, 2, 6, 4, 3, 7], [1, 5, 2, 6, 4, 7, 3], [1, 5, 2, 6, 7, 3, 4], [1, 5, 2, 6, 7, 4, 3], [1, 5, 2, 7, 3, 4, 6], [1, 5, 2, 7, 3, 6, 4], [1, 5, 2, 7, 4, 3, 6], [1, 5, 2, 7, 4, 6, 3], [1, 5, 2, 7, 6, 3, 4], [1, 5, 2, 7, 6, 4, 3], [1, 5, 3, 2, 4, 6, 7], [1, 5, 3, 2, 4, 7, 6], [1, 5, 3, 2, 6, 4, 7], [1, 5, 3, 2, 6, 7, 4], [1, 5, 3, 2, 7, 4, 6], [1, 5, 3, 2, 7, 6, 4], [1, 5, 3, 4, 2, 6, 7], [1, 5, 3, 4, 2, 7, 6], [1, 5, 3, 4, 6, 2, 7], [1, 5, 3, 4, 6, 7, 2], [1, 5, 3, 4, 7, 2, 6], [1, 5, 3, 4, 7, 6, 2], [1, 5, 3, 6, 2, 4, 7], [1, 5, 3, 6, 2, 7, 4], [1, 5, 3, 6, 4, 2, 7], [1, 5, 3, 6, 4, 7, 2], [1, 5, 3, 6, 7, 2, 4], [1, 5, 3, 6, 7, 4, 2], [1, 5, 3, 7, 2, 4, 6], [1, 5, 3, 7, 2, 6, 4], [1, 5, 3, 7, 4, 2, 6], [1, 5, 3, 7, 4, 6, 2], [1, 5, 3, 7, 6, 2, 4], [1, 5, 3, 7, 6, 4, 2], [1, 5, 4, 2, 3, 6, 7], [1, 5, 4, 2, 3, 7, 6], [1, 5, 4, 2, 6, 3, 7], [1, 5, 4, 2, 6, 7, 3], [1, 5, 4, 2, 7, 3, 6], [1, 5, 4, 2, 7, 6, 3], [1, 5, 4, 3, 2, 6, 7], [1, 5, 4, 3, 2, 7, 6], [1, 5, 4, 3, 6, 2, 7], [1, 5, 4, 3, 6, 7, 2], [1, 5, 4, 3, 7, 2, 6], [1, 5, 4, 3, 7, 6, 2], [1, 5, 4, 6, 2, 3, 7], [1, 5, 4, 6, 2, 7, 3], [1, 5, 4, 6, 3, 2, 7], [1, 5, 4, 6, 3, 7, 2], [1, 5, 4, 6, 7, 2, 3], [1, 5, 4, 6, 7, 3, 2], [1, 5, 4, 7, 2, 3, 6], [1, 5, 4, 7, 2, 6, 3], [1, 5, 4, 7, 3, 2, 6], [1, 5, 4, 7, 3, 6, 2], [1, 5, 4, 7, 6, 2, 3], [1, 5, 4, 7, 6, 3, 2], [1, 5, 6, 2, 3, 4, 7], [1, 5, 6, 2, 3, 7, 4], [1, 5, 6, 2, 4, 3, 7], [1, 5, 6, 2, 4, 7, 3], [1, 5, 6, 2, 7, 3, 4], [1, 5, 6, 2, 7, 4, 3], [1, 5, 6, 3, 2, 4, 7], [1, 5, 6, 3, 2, 7, 4], [1, 5, 6, 3, 4, 2, 7], [1, 5, 6, 3, 4, 7, 2], [1, 5, 6, 3, 7, 2, 4], [1, 5, 6, 3, 7, 4, 2], [1, 5, 6, 4, 2, 3, 7], [1, 5, 6, 4, 2, 7, 3], [1, 5, 6, 4, 3, 2, 7], [1, 5, 6, 4, 3, 7, 2], [1, 5, 6, 4, 7, 2, 3], [1, 5, 6, 4, 7, 3, 2], [1, 5, 6, 7, 2, 3, 4], [1, 5, 6, 7, 2, 4, 3], [1, 5, 6, 7, 3, 2, 4], [1, 5, 6, 7, 3, 4, 2], [1, 5, 6, 7, 4, 2, 3], [1, 5, 6, 7, 4, 3, 2], [1, 5, 7, 2, 3, 4, 6], [1, 5, 7, 2, 3, 6, 4], [1, 5, 7, 2, 4, 3, 6], [1, 5, 7, 2, 4, 6, 3], [1, 5, 7, 2, 6, 3, 4], [1, 5, 7, 2, 6, 4, 3], [1, 5, 7, 3, 2, 4, 6], [1, 5, 7, 3, 2, 6, 4], [1, 5, 7, 3, 4, 2, 6], [1, 5, 7, 3, 4, 6, 2], [1, 5, 7, 3, 6, 2, 4], [1, 5, 7, 3, 6, 4, 2], [1, 5, 7, 4, 2, 3, 6], [1, 5, 7, 4, 2, 6, 3], [1, 5, 7, 4, 3, 2, 6], [1, 5, 7, 4, 3, 6, 2], [1, 5, 7, 4, 6, 2, 3], [1, 5, 7, 4, 6, 3, 2], [1, 5, 7, 6, 2, 3, 4], [1, 5, 7, 6, 2, 4, 3], [1, 5, 7, 6, 3, 2, 4], [1, 5, 7, 6, 3, 4, 2], [1, 5, 7, 6, 4, 2, 3], [1, 5, 7, 6, 4, 3, 2], [1, 6, 2, 3, 4, 5, 7], [1, 6, 2, 3, 4, 7, 5], [1, 6, 2, 3, 5, 4, 7], [1, 6, 2, 3, 5, 7, 4], [1, 6, 2, 3, 7, 4, 5], [1, 6, 2, 3, 7, 5, 4], [1, 6, 2, 4, 3, 5, 7], [1, 6, 2, 4, 3, 7, 5], [1, 6, 2, 4, 5, 3, 7], [1, 6, 2, 4, 5, 7, 3], [1, 6, 2, 4, 7, 3, 5], [1, 6, 2, 4, 7, 5, 3], [1, 6, 2, 5, 3, 4, 7], [1, 6, 2, 5, 3, 7, 4], [1, 6, 2, 5, 4, 3, 7], [1, 6, 2, 5, 4, 7, 3], [1, 6, 2, 5, 7, 3, 4], [1, 6, 2, 5, 7, 4, 3], [1, 6, 2, 7, 3, 4, 5], [1, 6, 2, 7, 3, 5, 4], [1, 6, 2, 7, 4, 3, 5], [1, 6, 2, 7, 4, 5, 3], [1, 6, 2, 7, 5, 3, 4], [1, 6, 2, 7, 5, 4, 3], [1, 6, 3, 2, 4, 5, 7], [1, 6, 3, 2, 4, 7, 5], [1, 6, 3, 2, 5, 4, 7], [1, 6, 3, 2, 5, 7, 4], [1, 6, 3, 2, 7, 4, 5], [1, 6, 3, 2, 7, 5, 4], [1, 6, 3, 4, 2, 5, 7], [1, 6, 3, 4, 2, 7, 5], [1, 6, 3, 4, 5, 2, 7], [1, 6, 3, 4, 5, 7, 2], [1, 6, 3, 4, 7, 2, 5], [1, 6, 3, 4, 7, 5, 2], [1, 6, 3, 5, 2, 4, 7], [1, 6, 3, 5, 2, 7, 4], [1, 6, 3, 5, 4, 2, 7], [1, 6, 3, 5, 4, 7, 2], [1, 6, 3, 5, 7, 2, 4], [1, 6, 3, 5, 7, 4, 2], [1, 6, 3, 7, 2, 4, 5], [1, 6, 3, 7, 2, 5, 4], [1, 6, 3, 7, 4, 2, 5], [1, 6, 3, 7, 4, 5, 2], [1, 6, 3, 7, 5, 2, 4], [1, 6, 3, 7, 5, 4, 2], [1, 6, 4, 2, 3, 5, 7], [1, 6, 4, 2, 3, 7, 5], [1, 6, 4, 2, 5, 3, 7], [1, 6, 4, 2, 5, 7, 3], [1, 6, 4, 2, 7, 3, 5], [1, 6, 4, 2, 7, 5, 3], [1, 6, 4, 3, 2, 5, 7], [1, 6, 4, 3, 2, 7, 5], [1, 6, 4, 3, 5, 2, 7], [1, 6, 4, 3, 5, 7, 2], [1, 6, 4, 3, 7, 2, 5], [1, 6, 4, 3, 7, 5, 2], [1, 6, 4, 5, 2, 3, 7], [1, 6, 4, 5, 2, 7, 3], [1, 6, 4, 5, 3, 2, 7], [1, 6, 4, 5, 3, 7, 2], [1, 6, 4, 5, 7, 2, 3], [1, 6, 4, 5, 7, 3, 2], [1, 6, 4, 7, 2, 3, 5], [1, 6, 4, 7, 2, 5, 3], [1, 6, 4, 7, 3, 2, 5], [1, 6, 4, 7, 3, 5, 2], [1, 6, 4, 7, 5, 2, 3], [1, 6, 4, 7, 5, 3, 2], [1, 6, 5, 2, 3, 4, 7], [1, 6, 5, 2, 3, 7, 4], [1, 6, 5, 2, 4, 3, 7], [1, 6, 5, 2, 4, 7, 3], [1, 6, 5, 2, 7, 3, 4], [1, 6, 5, 2, 7, 4, 3], [1, 6, 5, 3, 2, 4, 7], [1, 6, 5, 3, 2, 7, 4], [1, 6, 5, 3, 4, 2, 7], [1, 6, 5, 3, 4, 7, 2], [1, 6, 5, 3, 7, 2, 4], [1, 6, 5, 3, 7, 4, 2], [1, 6, 5, 4, 2, 3, 7], [1, 6, 5, 4, 2, 7, 3], [1, 6, 5, 4, 3, 2, 7], [1, 6, 5, 4, 3, 7, 2], [1, 6, 5, 4, 7, 2, 3], [1, 6, 5, 4, 7, 3, 2], [1, 6, 5, 7, 2, 3, 4], [1, 6, 5, 7, 2, 4, 3], [1, 6, 5, 7, 3, 2, 4], [1, 6, 5, 7, 3, 4, 2], [1, 6, 5, 7, 4, 2, 3], [1, 6, 5, 7, 4, 3, 2], [1, 6, 7, 2, 3, 4, 5], [1, 6, 7, 2, 3, 5, 4], [1, 6, 7, 2, 4, 3, 5], [1, 6, 7, 2, 4, 5, 3], [1, 6, 7, 2, 5, 3, 4], [1, 6, 7, 2, 5, 4, 3], [1, 6, 7, 3, 2, 4, 5], [1, 6, 7, 3, 2, 5, 4], [1, 6, 7, 3, 4, 2, 5], [1, 6, 7, 3, 4, 5, 2], [1, 6, 7, 3, 5, 2, 4], [1, 6, 7, 3, 5, 4, 2], [1, 6, 7, 4, 2, 3, 5], [1, 6, 7, 4, 2, 5, 3], [1, 6, 7, 4, 3, 2, 5], [1, 6, 7, 4, 3, 5, 2], [1, 6, 7, 4, 5, 2, 3], [1, 6, 7, 4, 5, 3, 2], [1, 6, 7, 5, 2, 3, 4], [1, 6, 7, 5, 2, 4, 3], [1, 6, 7, 5, 3, 2, 4], [1, 6, 7, 5, 3, 4, 2], [1, 6, 7, 5, 4, 2, 3], [1, 6, 7, 5, 4, 3, 2], [1, 7, 2, 3, 4, 5, 6], [1, 7, 2, 3, 4, 6, 5], [1, 7, 2, 3, 5, 4, 6], [1, 7, 2, 3, 5, 6, 4], [1, 7, 2, 3, 6, 4, 5], [1, 7, 2, 3, 6, 5, 4], [1, 7, 2, 4, 3, 5, 6], [1, 7, 2, 4, 3, 6, 5], [1, 7, 2, 4, 5, 3, 6], [1, 7, 2, 4, 5, 6, 3], [1, 7, 2, 4, 6, 3, 5], [1, 7, 2, 4, 6, 5, 3], [1, 7, 2, 5, 3, 4, 6], [1, 7, 2, 5, 3, 6, 4], [1, 7, 2, 5, 4, 3, 6], [1, 7, 2, 5, 4, 6, 3], [1, 7, 2, 5, 6, 3, 4], [1, 7, 2, 5, 6, 4, 3], [1, 7, 2, 6, 3, 4, 5], localhost:8000/home/admin/4/

15/20

30.10.12

Vorlesung1 -- Sage

[1, 7, 2, 6, 3, 5, 4], [1, 7, 2, 6, 4, 3, 5], [1, 7, 2, 6, 4, 5, 3], [1, 7, 2, 6, 5, 3, 4], [1, 7, 2, 6, 5, 4, 3], [1, 7, 3, 2, 4, 5, 6], [1, 7, 3, 2, 4, 6, 5], [1, 7, 3, 2, 5, 4, 6], [1, 7, 3, 2, 5, 6, 4], [1, 7, 3, 2, 6, 4, 5], [1, 7, 3, 2, 6, 5, 4], [1, 7, 3, 4, 2, 5, 6], [1, 7, 3, 4, 2, 6, 5], [1, 7, 3, 4, 5, 2, 6], [1, 7, 3, 4, 5, 6, 2], [1, 7, 3, 4, 6, 2, 5], [1, 7, 3, 4, 6, 5, 2], [1, 7, 3, 5, 2, 4, 6], [1, 7, 3, 5, 2, 6, 4], [1, 7, 3, 5, 4, 2, 6], [1, 7, 3, 5, 4, 6, 2], [1, 7, 3, 5, 6, 2, 4], [1, 7, 3, 5, 6, 4, 2], [1, 7, 3, 6, 2, 4, 5], [1, 7, 3, 6, 2, 5, 4], [1, 7, 3, 6, 4, 2, 5], [1, 7, 3, 6, 4, 5, 2], [1, 7, 3, 6, 5, 2, 4], [1, 7, 3, 6, 5, 4, 2], [1, 7, 4, 2, 3, 5, 6], [1, 7, 4, 2, 3, 6, 5], [1, 7, 4, 2, 5, 3, 6], [1, 7, 4, 2, 5, 6, 3], [1, 7, 4, 2, 6, 3, 5], [1, 7, 4, 2, 6, 5, 3], [1, 7, 4, 3, 2, 5, 6], [1, 7, 4, 3, 2, 6, 5], [1, 7, 4, 3, 5, 2, 6], [1, 7, 4, 3, 5, 6, 2], [1, 7, 4, 3, 6, 2, 5], [1, 7, 4, 3, 6, 5, 2], [1, 7, 4, 5, 2, 3, 6], [1, 7, 4, 5, 2, 6, 3], [1, 7, 4, 5, 3, 2, 6], [1, 7, 4, 5, 3, 6, 2], [1, 7, 4, 5, 6, 2, 3], [1, 7, 4, 5, 6, 3, 2], [1, 7, 4, 6, 2, 3, 5], [1, 7, 4, 6, 2, 5, 3], [1, 7, 4, 6, 3, 2, 5], [1, 7, 4, 6, 3, 5, 2], [1, 7, 4, 6, 5, 2, 3], [1, 7, 4, 6, 5, 3, 2], [1, 7, 5, 2, 3, 4, 6], [1, 7, 5, 2, 3, 6, 4], [1, 7, 5, 2, 4, 3, 6], [1, 7, 5, 2, 4, 6, 3], [1, 7, 5, 2, 6, 3, 4], [1, 7, 5, 2, 6, 4, 3], [1, 7, 5, 3, 2, 4, 6], [1, 7, 5, 3, 2, 6, 4], [1, 7, 5, 3, 4, 2, 6], [1, 7, 5, 3, 4, 6, 2], [1, 7, 5, 3, 6, 2, 4], [1, 7, 5, 3, 6, 4, 2], [1, 7, 5, 4, 2, 3, 6], [1, 7, 5, 4, 2, 6, 3], [1, 7, 5, 4, 3, 2, 6], [1, 7, 5, 4, 3, 6, 2], [1, 7, 5, 4, 6, 2, 3], [1, 7, 5, 4, 6, 3, 2], [1, 7, 5, 6, 2, 3, 4], [1, 7, 5, 6, 2, 4, 3], [1, 7, 5, 6, 3, 2, 4], [1, 7, 5, 6, 3, 4, 2], [1, 7, 5, 6, 4, 2, 3], [1, 7, 5, 6, ... 3, 2, 4, 5, 6], [7, 1, 3, 2, 4, 6, 5], [7, 1, 3, 2, 5, 4, 6], [7, 1, 3, 2, 5, 6, 4], [7, 1, 3, 2, 6, 4, 5], [7, 1, 3, 2, 6, 5, 4], [7, 1, 3, 4, 2, 5, 6], [7, 1, 3, 4, 2, 6, 5], [7, 1, 3, 4, 5, 2, 6], [7, 1, 3, 4, 5, 6, 2], [7, 1, 3, 4, 6, 2, 5], [7, 1, 3, 4, 6, 5, 2], [7, 1, 3, 5, 2, 4, 6], [7, 1, 3, 5, 2, 6, 4], [7, 1, 3, 5, 4, 2, 6], [7, 1, 3, 5, 4, 6, 2], [7, 1, 3, 5, 6, 2, 4], [7, 1, 3, 5, 6, 4, 2], [7, 1, 3, 6, 2, 4, 5], [7, 1, 3, 6, 2, 5, 4], [7, 1, 3, 6, 4, 2, 5], [7, 1, 3, 6, 4, 5, 2], [7, 1, 3, 6, 5, 2, 4], [7, 1, 3, 6, 5, 4, 2], [7, 1, 4, 2, 3, 5, 6], [7, 1, 4, 2, 3, 6, 5], [7, 1, 4, 2, 5, 3, 6], [7, 1, 4, 2, 5, 6, 3], [7, 1, 4, 2, 6, 3, 5], [7, 1, 4, 2, 6, 5, 3], [7, 1, 4, 3, 2, 5, 6], [7, 1, 4, 3, 2, 6, 5], [7, 1, 4, 3, 5, 2, 6], [7, 1, 4, 3, 5, 6, 2], [7, 1, 4, 3, 6, 2, 5], [7, 1, 4, 3, 6, 5, 2], [7, 1, 4, 5, 2, 3, 6], [7, 1, 4, 5, 2, 6, 3], [7, 1, 4, 5, 3, 2, 6], [7, 1, 4, 5, 3, 6, 2], [7, 1, 4, 5, 6, 2, 3], [7, 1, 4, 5, 6, 3, 2], [7, 1, 4, 6, 2, 3, 5], [7, 1, 4, 6, 2, 5, 3], [7, 1, 4, 6, 3, 2, 5], [7, 1, 4, 6, 3, 5, 2], [7, 1, 4, 6, 5, 2, 3], [7, 1, 4, 6, 5, 3, 2], [7, 1, 5, 2, 3, 4, 6], [7, 1, 5, 2, 3, 6, 4], [7, 1, 5, 2, 4, 3, 6], [7, 1, 5, 2, 4, 6, 3], [7, 1, 5, 2, 6, 3, 4], [7, 1, 5, 2, 6, 4, 3], [7, 1, 5, 3, 2, 4, 6], [7, 1, 5, 3, 2, 6, 4], [7, 1, 5, 3, 4, 2, 6], [7, 1, 5, 3, 4, 6, 2], [7, 1, 5, 3, 6, 2, 4], [7, 1, 5, 3, 6, 4, 2], [7, 1, 5, 4, 2, 3, 6], [7, 1, 5, 4, 2, 6, 3], [7, 1, 5, 4, 3, 2, 6], [7, 1, 5, 4, 3, 6, 2], [7, 1, 5, 4, 6, 2, 3], [7, 1, 5, 4, 6, 3, 2], [7, 1, 5, 6, 2, 3, 4], [7, 1, 5, 6, 2, 4, 3], [7, 1, 5, 6, 3, 2, 4], [7, 1, 5, 6, 3, 4, 2], [7, 1, 5, 6, 4, 2, 3], [7, 1, 5, 6, 4, 3, 2], [7, 1, 6, 2, 3, 4, 5], [7, 1, 6, 2, 3, 5, 4], [7, 1, 6, 2, 4, 3, 5], [7, 1, 6, 2, 4, 5, 3], [7, 1, 6, 2, 5, 3, 4], [7, 1, 6, 2, 5, 4, 3], [7, 1, 6, 3, 2, 4, 5], [7, 1, 6, 3, 2, 5, 4], [7, 1, 6, 3, 4, 2, 5], [7, 1, 6, 3, 4, 5, 2], [7, 1, 6, 3, 5, 2, 4], [7, 1, 6, 3, 5, 4, 2], [7, 1, 6, 4, 2, 3, 5], [7, 1, 6, 4, 2, 5, 3], [7, 1, 6, 4, 3, 2, 5], [7, 1, 6, 4, 3, 5, 2], [7, 1, 6, 4, 5, 2, 3], [7, 1, 6, 4, 5, 3, 2], [7, 1, 6, 5, 2, 3, 4], [7, 1, 6, 5, 2, 4, 3], [7, 1, 6, 5, 3, 2, 4], [7, 1, 6, 5, 3, 4, 2], [7, 1, 6, 5, 4, 2, 3], [7, 1, 6, 5, 4, 3, 2], [7, 2, 1, 3, 4, 5, 6], [7, 2, 1, 3, 4, 6, 5], [7, 2, 1, 3, 5, 4, 6], [7, 2, 1, 3, 5, 6, 4], [7, 2, 1, 3, 6, 4, 5], [7, 2, 1, 3, 6, 5, 4], [7, 2, 1, 4, 3, 5, 6], [7, 2, 1, 4, 3, 6, 5], [7, 2, 1, 4, 5, 3, 6], [7, 2, 1, 4, 5, 6, 3], [7, 2, 1, 4, 6, 3, 5], [7, 2, 1, 4, 6, 5, 3], [7, 2, 1, 5, 3, 4, 6], [7, 2, 1, 5, 3, 6, 4], [7, 2, 1, 5, 4, 3, 6], [7, 2, 1, 5, 4, 6, 3], [7, 2, 1, 5, 6, 3, 4], [7, 2, 1, 5, 6, 4, 3], [7, 2, 1, 6, 3, 4, 5], [7, 2, 1, 6, 3, 5, 4], [7, 2, 1, 6, 4, 3, 5], [7, 2, 1, 6, 4, 5, 3], [7, 2, 1, 6, 5, 3, 4], [7, 2, 1, 6, 5, 4, 3], [7, 2, 3, 1, 4, 5, 6], [7, 2, 3, 1, 4, 6, 5], [7, 2, 3, 1, 5, 4, 6], [7, 2, 3, 1, 5, 6, 4], [7, 2, 3, 1, 6, 4, 5], [7, 2, 3, 1, 6, 5, 4], [7, 2, 3, 4, 1, 5, 6], [7, 2, 3, 4, 1, 6, 5], [7, 2, 3, 4, 5, 1, 6], [7, 2, 3, 4, 5, 6, 1], [7, 2, 3, 4, 6, 1, 5], [7, 2, 3, 4, 6, 5, 1], [7, 2, 3, 5, 1, 4, 6], [7, 2, 3, 5, 1, 6, 4], [7, 2, 3, 5, 4, 1, 6], [7, 2, 3, 5, 4, 6, 1], [7, 2, 3, 5, 6, 1, 4], [7, 2, 3, 5, 6, 4, 1], [7, 2, 3, 6, 1, 4, 5], [7, 2, 3, 6, 1, 5, 4], [7, 2, 3, 6, 4, 1, 5], [7, 2, 3, 6, 4, 5, 1], [7, 2, 3, 6, 5, 1, 4], [7, 2, 3, 6, 5, 4, 1], [7, 2, 4, 1, 3, 5, 6], [7, 2, 4, 1, 3, 6, 5], [7, 2, 4, 1, 5, 3, 6], [7, 2, 4, 1, 5, 6, 3], [7, 2, 4, 1, 6, 3, 5], [7, 2, 4, 1, 6, 5, 3], [7, 2, 4, 3, 1, 5, 6], [7, 2, 4, 3, 1, 6, 5], [7, 2, 4, 3, 5, 1, 6], [7, 2, 4, 3, 5, 6, 1], [7, 2, 4, 3, 6, 1, 5], [7, 2, 4, 3, 6, 5, 1], [7, 2, 4, 5, 1, 3, 6], [7, 2, 4, 5, 1, 6, 3], [7, 2, 4, 5, 3, 1, 6], [7, 2, 4, 5, 3, 6, 1], [7, 2, 4, 5, 6, 1, 3], [7, 2, 4, 5, 6, 3, 1], [7, 2, 4, 6, 1, 3, 5], [7, 2, 4, 6, 1, 5, 3], [7, 2, 4, 6, 3, 1, 5], [7, 2, 4, 6, 3, 5, 1], [7, 2, 4, 6, 5, 1, 3], [7, 2, 4, 6, 5, 3, 1], [7, 2, 5, 1, 3, 4, 6], [7, 2, 5, 1, 3, 6, 4], [7, 2, 5, 1, 4, 3, 6], [7, 2, 5, 1, 4, 6, 3], [7, 2, 5, 1, 6, 3, 4], [7, 2, 5, 1, 6, 4, 3], [7, 2, 5, 3, 1, 4, 6], [7, 2, 5, 3, 1, 6, 4], [7, 2, 5, 3, 4, 1, 6], [7, 2, 5, 3, 4, 6, 1], [7, 2, 5, 3, 6, 1, 4], [7, 2, 5, 3, 6, 4, 1], [7, 2, 5, 4, 1, 3, 6], [7, 2, 5, 4, 1, 6, 3], [7, 2, 5, 4, 3, 1, 6], [7, 2, 5, 4, 3, 6, 1], [7, 2, 5, 4, 6, 1, 3], [7, 2, 5, 4, 6, localhost:8000/home/admin/4/

16/20

30.10.12

Vorlesung1 -- Sage

3, 1], [7, 2, 5, 6, 1, 3, 4], [7, 2, 5, 6, 1, 4, 3], [7, 2, 5, 6, 3, 1, 4], [7, 2, 5, 6, 3, 4, 1], [7, 2, 5, 6, 4, 1, 3], [7, 2, 5, 6, 4, 3, 1], [7, 2, 6, 1, 3, 4, 5], [7, 2, 6, 1, 3, 5, 4], [7, 2, 6, 1, 4, 3, 5], [7, 2, 6, 1, 4, 5, 3], [7, 2, 6, 1, 5, 3, 4], [7, 2, 6, 1, 5, 4, 3], [7, 2, 6, 3, 1, 4, 5], [7, 2, 6, 3, 1, 5, 4], [7, 2, 6, 3, 4, 1, 5], [7, 2, 6, 3, 4, 5, 1], [7, 2, 6, 3, 5, 1, 4], [7, 2, 6, 3, 5, 4, 1], [7, 2, 6, 4, 1, 3, 5], [7, 2, 6, 4, 1, 5, 3], [7, 2, 6, 4, 3, 1, 5], [7, 2, 6, 4, 3, 5, 1], [7, 2, 6, 4, 5, 1, 3], [7, 2, 6, 4, 5, 3, 1], [7, 2, 6, 5, 1, 3, 4], [7, 2, 6, 5, 1, 4, 3], [7, 2, 6, 5, 3, 1, 4], [7, 2, 6, 5, 3, 4, 1], [7, 2, 6, 5, 4, 1, 3], [7, 2, 6, 5, 4, 3, 1], [7, 3, 1, 2, 4, 5, 6], [7, 3, 1, 2, 4, 6, 5], [7, 3, 1, 2, 5, 4, 6], [7, 3, 1, 2, 5, 6, 4], [7, 3, 1, 2, 6, 4, 5], [7, 3, 1, 2, 6, 5, 4], [7, 3, 1, 4, 2, 5, 6], [7, 3, 1, 4, 2, 6, 5], [7, 3, 1, 4, 5, 2, 6], [7, 3, 1, 4, 5, 6, 2], [7, 3, 1, 4, 6, 2, 5], [7, 3, 1, 4, 6, 5, 2], [7, 3, 1, 5, 2, 4, 6], [7, 3, 1, 5, 2, 6, 4], [7, 3, 1, 5, 4, 2, 6], [7, 3, 1, 5, 4, 6, 2], [7, 3, 1, 5, 6, 2, 4], [7, 3, 1, 5, 6, 4, 2], [7, 3, 1, 6, 2, 4, 5], [7, 3, 1, 6, 2, 5, 4], [7, 3, 1, 6, 4, 2, 5], [7, 3, 1, 6, 4, 5, 2], [7, 3, 1, 6, 5, 2, 4], [7, 3, 1, 6, 5, 4, 2], [7, 3, 2, 1, 4, 5, 6], [7, 3, 2, 1, 4, 6, 5], [7, 3, 2, 1, 5, 4, 6], [7, 3, 2, 1, 5, 6, 4], [7, 3, 2, 1, 6, 4, 5], [7, 3, 2, 1, 6, 5, 4], [7, 3, 2, 4, 1, 5, 6], [7, 3, 2, 4, 1, 6, 5], [7, 3, 2, 4, 5, 1, 6], [7, 3, 2, 4, 5, 6, 1], [7, 3, 2, 4, 6, 1, 5], [7, 3, 2, 4, 6, 5, 1], [7, 3, 2, 5, 1, 4, 6], [7, 3, 2, 5, 1, 6, 4], [7, 3, 2, 5, 4, 1, 6], [7, 3, 2, 5, 4, 6, 1], [7, 3, 2, 5, 6, 1, 4], [7, 3, 2, 5, 6, 4, 1], [7, 3, 2, 6, 1, 4, 5], [7, 3, 2, 6, 1, 5, 4], [7, 3, 2, 6, 4, 1, 5], [7, 3, 2, 6, 4, 5, 1], [7, 3, 2, 6, 5, 1, 4], [7, 3, 2, 6, 5, 4, 1], [7, 3, 4, 1, 2, 5, 6], [7, 3, 4, 1, 2, 6, 5], [7, 3, 4, 1, 5, 2, 6], [7, 3, 4, 1, 5, 6, 2], [7, 3, 4, 1, 6, 2, 5], [7, 3, 4, 1, 6, 5, 2], [7, 3, 4, 2, 1, 5, 6], [7, 3, 4, 2, 1, 6, 5], [7, 3, 4, 2, 5, 1, 6], [7, 3, 4, 2, 5, 6, 1], [7, 3, 4, 2, 6, 1, 5], [7, 3, 4, 2, 6, 5, 1], [7, 3, 4, 5, 1, 2, 6], [7, 3, 4, 5, 1, 6, 2], [7, 3, 4, 5, 2, 1, 6], [7, 3, 4, 5, 2, 6, 1], [7, 3, 4, 5, 6, 1, 2], [7, 3, 4, 5, 6, 2, 1], [7, 3, 4, 6, 1, 2, 5], [7, 3, 4, 6, 1, 5, 2], [7, 3, 4, 6, 2, 1, 5], [7, 3, 4, 6, 2, 5, 1], [7, 3, 4, 6, 5, 1, 2], [7, 3, 4, 6, 5, 2, 1], [7, 3, 5, 1, 2, 4, 6], [7, 3, 5, 1, 2, 6, 4], [7, 3, 5, 1, 4, 2, 6], [7, 3, 5, 1, 4, 6, 2], [7, 3, 5, 1, 6, 2, 4], [7, 3, 5, 1, 6, 4, 2], [7, 3, 5, 2, 1, 4, 6], [7, 3, 5, 2, 1, 6, 4], [7, 3, 5, 2, 4, 1, 6], [7, 3, 5, 2, 4, 6, 1], [7, 3, 5, 2, 6, 1, 4], [7, 3, 5, 2, 6, 4, 1], [7, 3, 5, 4, 1, 2, 6], [7, 3, 5, 4, 1, 6, 2], [7, 3, 5, 4, 2, 1, 6], [7, 3, 5, 4, 2, 6, 1], [7, 3, 5, 4, 6, 1, 2], [7, 3, 5, 4, 6, 2, 1], [7, 3, 5, 6, 1, 2, 4], [7, 3, 5, 6, 1, 4, 2], [7, 3, 5, 6, 2, 1, 4], [7, 3, 5, 6, 2, 4, 1], [7, 3, 5, 6, 4, 1, 2], [7, 3, 5, 6, 4, 2, 1], [7, 3, 6, 1, 2, 4, 5], [7, 3, 6, 1, 2, 5, 4], [7, 3, 6, 1, 4, 2, 5], [7, 3, 6, 1, 4, 5, 2], [7, 3, 6, 1, 5, 2, 4], [7, 3, 6, 1, 5, 4, 2], [7, 3, 6, 2, 1, 4, 5], [7, 3, 6, 2, 1, 5, 4], [7, 3, 6, 2, 4, 1, 5], [7, 3, 6, 2, 4, 5, 1], [7, 3, 6, 2, 5, 1, 4], [7, 3, 6, 2, 5, 4, 1], [7, 3, 6, 4, 1, 2, 5], [7, 3, 6, 4, 1, 5, 2], [7, 3, 6, 4, 2, 1, 5], [7, 3, 6, 4, 2, 5, 1], [7, 3, 6, 4, 5, 1, 2], [7, 3, 6, 4, 5, 2, 1], [7, 3, 6, 5, 1, 2, 4], [7, 3, 6, 5, 1, 4, 2], [7, 3, 6, 5, 2, 1, 4], [7, 3, 6, 5, 2, 4, 1], [7, 3, 6, 5, 4, 1, 2], [7, 3, 6, 5, 4, 2, 1], [7, 4, 1, 2, 3, 5, 6], [7, 4, 1, 2, 3, 6, 5], [7, 4, 1, 2, 5, 3, 6], [7, 4, 1, 2, 5, 6, 3], [7, 4, 1, 2, 6, 3, 5], [7, 4, 1, 2, 6, 5, 3], [7, 4, 1, 3, 2, 5, 6], [7, 4, 1, 3, 2, 6, 5], [7, 4, 1, 3, 5, 2, 6], [7, 4, 1, 3, 5, 6, 2], [7, 4, 1, 3, 6, 2, 5], [7, 4, 1, 3, 6, 5, 2], [7, 4, 1, 5, 2, 3, 6], [7, 4, 1, 5, 2, 6, 3], [7, 4, 1, 5, 3, 2, 6], [7, 4, 1, 5, 3, 6, 2], [7, 4, 1, 5, 6, 2, 3], [7, 4, 1, 5, 6, 3, 2], [7, 4, 1, 6, 2, 3, 5], [7, 4, 1, 6, 2, 5, 3], [7, 4, 1, 6, 3, 2, 5], [7, 4, 1, 6, 3, 5, 2], [7, 4, 1, 6, 5, 2, 3], [7, 4, 1, 6, 5, 3, 2], [7, 4, 2, 1, 3, 5, 6], [7, 4, 2, 1, 3, 6, 5], [7, 4, 2, 1, 5, 3, 6], [7, 4, 2, 1, 5, 6, 3], [7, 4, 2, 1, 6, 3, 5], [7, 4, 2, 1, 6, 5, 3], [7, 4, 2, 3, 1, 5, 6], [7, 4, 2, 3, 1, 6, 5], [7, 4, 2, 3, 5, 1, 6], [7, 4, 2, 3, 5, 6, 1], [7, 4, 2, 3, 6, 1, 5], [7, 4, 2, 3, 6, 5, 1], [7, 4, 2, 5, 1, 3, 6], [7, 4, 2, 5, 1, 6, 3], [7, 4, 2, 5, 3, 1, 6], [7, 4, 2, 5, 3, 6, 1], [7, 4, 2, 5, 6, 1, 3], [7, 4, 2, 5, 6, 3, 1], [7, 4, 2, 6, 1, 3, 5], [7, 4, 2, 6, 1, 5, 3], [7, 4, 2, 6, 3, 1, 5], [7, 4, 2, 6, 3, 5, 1], [7, 4, 2, 6, 5, 1, 3], [7, 4, 2, 6, 5, 3, 1], [7, 4, 3, 1, 2, 5, 6], [7, 4, 3, 1, 2, 6, 5], [7, 4, 3, 1, 5, 2, 6], [7, 4, 3, 1, 5, 6, 2], [7, 4, 3, 1, 6, 2, 5], [7, 4, 3, 1, 6, 5, 2], [7, 4, 3, 2, 1, 5, 6], [7, 4, 3, 2, 1, 6, 5], [7, 4, 3, 2, 5, 1, 6], [7, 4, 3, 2, 5, 6, 1], [7, 4, 3, 2, 6, 1, 5], [7, 4, 3, 2, 6, 5, 1], [7, 4, 3, 5, 1, 2, 6], [7, 4, 3, 5, 1, 6, 2], [7, 4, 3, 5, 2, 1, 6], [7, 4, 3, 5, 2, 6, 1], [7, 4, 3, 5, 6, 1, 2], [7, 4, 3, 5, 6, 2, 1], [7, 4, 3, 6, 1, 2, 5], [7, 4, 3, 6, 1, 5, 2], [7, 4, 3, 6, 2, 1, 5], [7, 4, 3, 6, 2, 5, 1], [7, 4, 3, 6, 5, 1, 2], [7, 4, 3, 6, 5, 2, 1], [7, 4, 5, 1, 2, 3, 6], [7, 4, 5, 1, 2, 6, 3], [7, 4, 5, 1, 3, 2, 6], [7, 4, 5, 1, 3, 6, 2], [7, 4, 5, 1, 6, 2, 3], [7, 4, 5, 1, 6, 3, 2], [7, 4, 5, 2, 1, 3, 6], [7, 4, 5, 2, 1, 6, 3], [7, 4, 5, 2, 3, 1, 6], [7, 4, 5, 2, 3, 6, 1], [7, 4, 5, 2, 6, 1, 3], [7, 4, 5, 2, 6, 3, 1], [7, 4, 5, 3, 1, 2, 6], [7, 4, 5, 3, 1, 6, 2], [7, 4, 5, 3, 2, 1, 6], [7, 4, 5, 3, 2, 6, 1], [7, 4, 5, 3, 6, 1, 2], [7, 4, 5, 3, 6, 2, 1], [7, 4, 5, 6, 1, 2, 3], [7, 4, 5, 6, 1, 3, 2], [7, 4, 5, 6, 2, 1, 3], [7, 4, 5, 6, 2, 3, 1], [7, 4, 5, 6, 3, 1, 2], [7, 4, 5, 6, 3, 2, 1], [7, 4, 6, 1, 2, 3, 5], [7, 4, 6, 1, 2, 5, 3], [7, 4, 6, 1, 3, 2, 5], [7, 4, 6, 1, 3, 5, 2], [7, 4, 6, 1, 5, 2, 3], [7, 4, 6, 1, 5, 3, 2], [7, 4, 6, 2, 1, 3, 5], [7, 4, 6, 2, 1, 5, 3], [7, 4, 6, 2, 3, 1, 5], [7, 4, 6, 2, 3, 5, 1], [7, 4, 6, 2, 5, 1, 3], [7, 4, 6, 2, 5, 3, 1], [7, 4, 6, 3, 1, 2, 5], [7, 4, 6, 3, 1, 5, 2], [7, 4, 6, 3, 2, 1, 5], [7, 4, 6, 3, 2, 5, 1], [7, 4, 6, 3, 5, 1, 2], [7, 4, 6, 3, 5, 2, 1], [7, 4, 6, 5, 1, 2, 3], [7, 4, 6, 5, 1, 3, 2], [7, 4, 6, 5, 2, 1, 3], [7, 4, 6, 5, 2, 3, 1], [7, 4, 6, 5, 3, 1, 2], [7, 4, 6, 5, 3, 2, 1], [7, 5, 1, 2, 3, 4, 6], [7, 5, 1, 2, 3, 6, 4], [7, 5, 1, 2, 4, 3, 6], [7, localhost:8000/home/admin/4/

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30.10.12

Vorlesung1 -- Sage

5, 1, 2, 4, 6, 3], [7, 5, 1, 2, 6, 3, 4], [7, 5, 1, 2, 6, 4, 3], [7, 5, 1, 3, 2, 4, 6], [7, 5, 1, 3, 2, 6, 4], [7, 5, 1, 3, 4, 2, 6], [7, 5, 1, 3, 4, 6, 2], [7, 5, 1, 3, 6, 2, 4], [7, 5, 1, 3, 6, 4, 2], [7, 5, 1, 4, 2, 3, 6], [7, 5, 1, 4, 2, 6, 3], [7, 5, 1, 4, 3, 2, 6], [7, 5, 1, 4, 3, 6, 2], [7, 5, 1, 4, 6, 2, 3], [7, 5, 1, 4, 6, 3, 2], [7, 5, 1, 6, 2, 3, 4], [7, 5, 1, 6, 2, 4, 3], [7, 5, 1, 6, 3, 2, 4], [7, 5, 1, 6, 3, 4, 2], [7, 5, 1, 6, 4, 2, 3], [7, 5, 1, 6, 4, 3, 2], [7, 5, 2, 1, 3, 4, 6], [7, 5, 2, 1, 3, 6, 4], [7, 5, 2, 1, 4, 3, 6], [7, 5, 2, 1, 4, 6, 3], [7, 5, 2, 1, 6, 3, 4], [7, 5, 2, 1, 6, 4, 3], [7, 5, 2, 3, 1, 4, 6], [7, 5, 2, 3, 1, 6, 4], [7, 5, 2, 3, 4, 1, 6], [7, 5, 2, 3, 4, 6, 1], [7, 5, 2, 3, 6, 1, 4], [7, 5, 2, 3, 6, 4, 1], [7, 5, 2, 4, 1, 3, 6], [7, 5, 2, 4, 1, 6, 3], [7, 5, 2, 4, 3, 1, 6], [7, 5, 2, 4, 3, 6, 1], [7, 5, 2, 4, 6, 1, 3], [7, 5, 2, 4, 6, 3, 1], [7, 5, 2, 6, 1, 3, 4], [7, 5, 2, 6, 1, 4, 3], [7, 5, 2, 6, 3, 1, 4], [7, 5, 2, 6, 3, 4, 1], [7, 5, 2, 6, 4, 1, 3], [7, 5, 2, 6, 4, 3, 1], [7, 5, 3, 1, 2, 4, 6], [7, 5, 3, 1, 2, 6, 4], [7, 5, 3, 1, 4, 2, 6], [7, 5, 3, 1, 4, 6, 2], [7, 5, 3, 1, 6, 2, 4], [7, 5, 3, 1, 6, 4, 2], [7, 5, 3, 2, 1, 4, 6], [7, 5, 3, 2, 1, 6, 4], [7, 5, 3, 2, 4, 1, 6], [7, 5, 3, 2, 4, 6, 1], [7, 5, 3, 2, 6, 1, 4], [7, 5, 3, 2, 6, 4, 1], [7, 5, 3, 4, 1, 2, 6], [7, 5, 3, 4, 1, 6, 2], [7, 5, 3, 4, 2, 1, 6], [7, 5, 3, 4, 2, 6, 1], [7, 5, 3, 4, 6, 1, 2], [7, 5, 3, 4, 6, 2, 1], [7, 5, 3, 6, 1, 2, 4], [7, 5, 3, 6, 1, 4, 2], [7, 5, 3, 6, 2, 1, 4], [7, 5, 3, 6, 2, 4, 1], [7, 5, 3, 6, 4, 1, 2], [7, 5, 3, 6, 4, 2, 1], [7, 5, 4, 1, 2, 3, 6], [7, 5, 4, 1, 2, 6, 3], [7, 5, 4, 1, 3, 2, 6], [7, 5, 4, 1, 3, 6, 2], [7, 5, 4, 1, 6, 2, 3], [7, 5, 4, 1, 6, 3, 2], [7, 5, 4, 2, 1, 3, 6], [7, 5, 4, 2, 1, 6, 3], [7, 5, 4, 2, 3, 1, 6], [7, 5, 4, 2, 3, 6, 1], [7, 5, 4, 2, 6, 1, 3], [7, 5, 4, 2, 6, 3, 1], [7, 5, 4, 3, 1, 2, 6], [7, 5, 4, 3, 1, 6, 2], [7, 5, 4, 3, 2, 1, 6], [7, 5, 4, 3, 2, 6, 1], [7, 5, 4, 3, 6, 1, 2], [7, 5, 4, 3, 6, 2, 1], [7, 5, 4, 6, 1, 2, 3], [7, 5, 4, 6, 1, 3, 2], [7, 5, 4, 6, 2, 1, 3], [7, 5, 4, 6, 2, 3, 1], [7, 5, 4, 6, 3, 1, 2], [7, 5, 4, 6, 3, 2, 1], [7, 5, 6, 1, 2, 3, 4], [7, 5, 6, 1, 2, 4, 3], [7, 5, 6, 1, 3, 2, 4], [7, 5, 6, 1, 3, 4, 2], [7, 5, 6, 1, 4, 2, 3], [7, 5, 6, 1, 4, 3, 2], [7, 5, 6, 2, 1, 3, 4], [7, 5, 6, 2, 1, 4, 3], [7, 5, 6, 2, 3, 1, 4], [7, 5, 6, 2, 3, 4, 1], [7, 5, 6, 2, 4, 1, 3], [7, 5, 6, 2, 4, 3, 1], [7, 5, 6, 3, 1, 2, 4], [7, 5, 6, 3, 1, 4, 2], [7, 5, 6, 3, 2, 1, 4], [7, 5, 6, 3, 2, 4, 1], [7, 5, 6, 3, 4, 1, 2], [7, 5, 6, 3, 4, 2, 1], [7, 5, 6, 4, 1, 2, 3], [7, 5, 6, 4, 1, 3, 2], [7, 5, 6, 4, 2, 1, 3], [7, 5, 6, 4, 2, 3, 1], [7, 5, 6, 4, 3, 1, 2], [7, 5, 6, 4, 3, 2, 1], [7, 6, 1, 2, 3, 4, 5], [7, 6, 1, 2, 3, 5, 4], [7, 6, 1, 2, 4, 3, 5], [7, 6, 1, 2, 4, 5, 3], [7, 6, 1, 2, 5, 3, 4], [7, 6, 1, 2, 5, 4, 3], [7, 6, 1, 3, 2, 4, 5], [7, 6, 1, 3, 2, 5, 4], [7, 6, 1, 3, 4, 2, 5], [7, 6, 1, 3, 4, 5, 2], [7, 6, 1, 3, 5, 2, 4], [7, 6, 1, 3, 5, 4, 2], [7, 6, 1, 4, 2, 3, 5], [7, 6, 1, 4, 2, 5, 3], [7, 6, 1, 4, 3, 2, 5], [7, 6, 1, 4, 3, 5, 2], [7, 6, 1, 4, 5, 2, 3], [7, 6, 1, 4, 5, 3, 2], [7, 6, 1, 5, 2, 3, 4], [7, 6, 1, 5, 2, 4, 3], [7, 6, 1, 5, 3, 2, 4], [7, 6, 1, 5, 3, 4, 2], [7, 6, 1, 5, 4, 2, 3], [7, 6, 1, 5, 4, 3, 2], [7, 6, 2, 1, 3, 4, 5], [7, 6, 2, 1, 3, 5, 4], [7, 6, 2, 1, 4, 3, 5], [7, 6, 2, 1, 4, 5, 3], [7, 6, 2, 1, 5, 3, 4], [7, 6, 2, 1, 5, 4, 3], [7, 6, 2, 3, 1, 4, 5], [7, 6, 2, 3, 1, 5, 4], [7, 6, 2, 3, 4, 1, 5], [7, 6, 2, 3, 4, 5, 1], [7, 6, 2, 3, 5, 1, 4], [7, 6, 2, 3, 5, 4, 1], [7, 6, 2, 4, 1, 3, 5], [7, 6, 2, 4, 1, 5, 3], [7, 6, 2, 4, 3, 1, 5], [7, 6, 2, 4, 3, 5, 1], [7, 6, 2, 4, 5, 1, 3], [7, 6, 2, 4, 5, 3, 1], [7, 6, 2, 5, 1, 3, 4], [7, 6, 2, 5, 1, 4, 3], [7, 6, 2, 5, 3, 1, 4], [7, 6, 2, 5, 3, 4, 1], [7, 6, 2, 5, 4, 1, 3], [7, 6, 2, 5, 4, 3, 1], [7, 6, 3, 1, 2, 4, 5], [7, 6, 3, 1, 2, 5, 4], [7, 6, 3, 1, 4, 2, 5], [7, 6, 3, 1, 4, 5, 2], [7, 6, 3, 1, 5, 2, 4], [7, 6, 3, 1, 5, 4, 2], [7, 6, 3, 2, 1, 4, 5], [7, 6, 3, 2, 1, 5, 4], [7, 6, 3, 2, 4, 1, 5], [7, 6, 3, 2, 4, 5, 1], [7, 6, 3, 2, 5, 1, 4], [7, 6, 3, 2, 5, 4, 1], [7, 6, 3, 4, 1, 2, 5], [7, 6, 3, 4, 1, 5, 2], [7, 6, 3, 4, 2, 1, 5], [7, 6, 3, 4, 2, 5, 1], [7, 6, 3, 4, 5, 1, 2], [7, 6, 3, 4, 5, 2, 1], [7, 6, 3, 5, 1, 2, 4], [7, 6, 3, 5, 1, 4, 2], [7, 6, 3, 5, 2, 1, 4], [7, 6, 3, 5, 2, 4, 1], [7, 6, 3, 5, 4, 1, 2], [7, 6, 3, 5, 4, 2, 1], [7, 6, 4, 1, 2, 3, 5], [7, 6, 4, 1, 2, 5, 3], [7, 6, 4, 1, 3, 2, 5], [7, 6, 4, 1, 3, 5, 2], [7, 6, 4, 1, 5, 2, 3], [7, 6, 4, 1, 5, 3, 2], [7, 6, 4, 2, 1, 3, 5], [7, 6, 4, 2, 1, 5, 3], [7, 6, 4, 2, 3, 1, 5], [7, 6, 4, 2, 3, 5, 1], [7, 6, 4, 2, 5, 1, 3], [7, 6, 4, 2, 5, 3, 1], [7, 6, 4, 3, 1, 2, 5], [7, 6, 4, 3, 1, 5, 2], [7, 6, 4, 3, 2, 1, 5], [7, 6, 4, 3, 2, 5, 1], [7, 6, 4, 3, 5, 1, 2], [7, 6, 4, 3, 5, 2, 1], [7, 6, 4, 5, 1, 2, 3], [7, 6, 4, 5, 1, 3, 2], [7, 6, 4, 5, 2, 1, 3], [7, 6, 4, 5, 2, 3, 1], [7, 6, 4, 5, 3, 1, 2], [7, 6, 4, 5, 3, 2, 1], [7, 6, 5, 1, 2, 3, 4], [7, 6, 5, 1, 2, 4, 3], [7, 6, 5, 1, 3, 2, 4], [7, 6, 5, 1, 3, 4, 2], [7, 6, 5, 1, 4, 2, 3], [7, 6, 5, 1, 4, 3, 2], [7, 6, 5, 2, 1, 3, 4], [7, 6, 5, 2, 1, 4, 3], [7, 6, 5, 2, 3, 1, 4], [7, 6, 5, 2, 3, 4, 1], [7, 6, 5, 2, 4, 1, 3], [7, 6, 5, 2, 4, 3, 1], [7, 6, 5, 3, 1, 2, 4], [7, 6, 5, 3, 1, 4, 2], [7, 6, 5, 3, 2, 1, 4], [7, 6, 5, 3, 2, 4, 1], [7, 6, 5, 3, 4, 1, 2], [7, 6, 5, 3, 4, 2, 1], [7, 6, 5, 4, 1, 2, 3], [7, 6, 5, 4, 1, 3, 2], [7, 6, 5, 4, 2, 1, 3], [7, 6, 5, 4, 2, 3, 1], [7, 6, 5, 4, 3, 1, 2], [7, 6, 5, 4, 3, 2, 1]] full_output.txt

localhost:8000/home/admin/4/

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30.10.12

Vorlesung1 -- Sage

# LGS kösen

L=range(1,9)

L [1, 2, 3, 4, 5, 6, 7, 8] LL=Permutations(L)

LL Permutations of the set [1, 2, 3, 4, 5, 6, 7, 8] LL[20000] [4, 8, 6, 5, 2, 3, 1, 7] B=vector(LL[40000]).outer_product(vector(LL[20000]))

B [32 64 48 40 16 24 [28 56 42 35 14 21 [16 32 24 20 8 12 [ 8 16 12 10 4 6 [20 40 30 25 10 15 [24 48 36 30 12 18 [4 8 6 5 2 3 [12 24 18 15 6 9

8 56] 7 49] 4 28] 2 14] 5 35] 6 42] 1 7] 3 21]

y= vector([1488, 1302, 744, 372, 930, 1116, 186, 558])

x=B\y

x # Wie kommt das Ergebnis zustande (93/2, 0, 0, 0, 0, 0, 0, 0) rank(B) 1 M=matrix(ZZ,9,[5,0,0, 0,8,0, 0,4,9, 0,0,0, 5,0,0, 0,3,0, 0,6,7, 3,0,0, 0,0,1, 1,5,0, 0,0,0, 0,0,0, 0,0,0, 2,0,8, 0,0,0, 0,0,0, 0,0,0, 0,1,8, 7,0,0, 0,0,4, 1,5,0, 0,3,0, 0,0,2, 0,0,0, 4,9,0, 0,5,0, 0,0,3]) M [5 0 0 0 8 0 0 4 9] [0 0 0 5 0 0 0 3 0] [0 6 7 3 0 0 0 0 1] [1 5 0 0 0 0 0 0 0] [0 0 0 2 0 8 0 0 0] [0 0 0 0 0 0 0 1 8] [7 0 0 0 0 4 1 5 0] [0 3 0 0 0 2 0 0 0] [4 9 0 0 5 0 0 0 3] rank(M) 9 M=Matrix(ZZ,M)

y=vector([136, 46, 113, 44, 34, 71, 76, 30, 122])

localhost:8000/home/admin/4/

19/20

30.10.12

Vorlesung1 -- Sage

y (136, 46, 113, 44, 34, 71, 76, 30, 122) x=M\y

# Löse Mx=y

x (4, 8, 6, 5, 2, 3, 1, 7, 8)

localhost:8000/home/admin/4/

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