cuentafunciones.sagews

def cartesiano(A,n):
    fuf=[]
    for ii in range(n):
        fuf.append(A)
    return cartesian_product(fuf)    

A={'a','b','c'}
tutu=cartesiano(A,4)
('a','a','a','b') in tutu

True

tutu.cardinality()

81

B=range(3)
tuta=cartesiano(B,4)

(0,0,1,2) in tuta

def locus(pp):# exige que despues del cero vaya un 1
    kk=len(pp)
    for ii in range(kk-1):
        if pp[ii]==0:
            if not(pp[ii+1]==1):
                return False# hay un 0 cuyo siguiente no es 1
        if pp[kk-1]==0: #el último es cero
            return False
    return True       

True

# cuántas palabras de longitud 4 cumplen locus

ruta=[uu for uu in cartesiano(B,3) if locus(uu)]

print ruta

[(0, 1, 1), (0, 1, 2), (1, 0, 1), (1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 0, 1), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)]

len(ruta)

12 64

# cuántas palabras de longitud n cumplen locus

def vale(nn):
    ruta=[uu for uu in cartesiano(B,nn) if locus(uu)]
    return len(ruta) 

vale(3)

12

vale(4)

12

for ij in range(7):
    print vale(ij)

1 3 5 12 29 70 169

aa=matrix([[2,1],[1,0]])
var('n')
aa^n

n [ 1/2*(sqrt(2) + 1)^n*(sqrt(1/2) + 1) - 1/2*(-sqrt(2) + 1)^n*(sqrt(1/2) - 1) 1/2*sqrt(1/2)*(sqrt(2) + 1)^n - 1/2*sqrt(1/2)*(-sqrt(2) + 1)^n] [1/2*(sqrt(2) + 1)^n*(sqrt(2) - 1)*(sqrt(1/2) + 1) + 1/2*(-sqrt(2) + 1)^n*(sqrt(2) + 1)*(sqrt(1/2) - 1) 1/2*sqrt(1/2)*(-sqrt(2) + 1)^n*(sqrt(2) + 1) + 1/2*sqrt(1/2)*(sqrt(2) + 1)^n*(sqrt(2) - 1)]

def locis(pp):# Busca un cero seguido de un 1
    kk=len(pp)
    for ii in range(kk-1):
        if pp[ii]==0:#encontró el 0
            if pp[ii+1]==1:
                return True# y le sigue  1 EUREKA!
    return False       

locis([0,0,0,1,0])

True

locis([1,1, 2,1, 2,1,0])

False

def vile(nn):
    ruta=[uu for uu in cartesiano(B,nn) if locis(uu)]
    return len(ruta) 

vile(6)

352

for ik in range(10):
    vile(ik)

0 0 1 6 26 99 352 1200 3977 12918

# nos preguntamos por las funciones con 3 elementos en el reccorido que suman k
def suma(uu):    # suma los elementos de uu
    conta=0
    for mm in uu:
        conta+=mm
    return conta    


AA=range(3)
for ii in range(1,8):
    sss=cartesiano(AA,ii)
    rr=vector(ZZ,ii*3)
    for uu in sss:
        toy=suma(uu)
        rr[toy]+=1
    print rr   

(1, 1, 1) (1, 2, 3, 2, 1, 0) (1, 3, 6, 7, 6, 3, 1, 0, 0) (1, 4, 10, 16, 19, 16, 10, 4, 1, 0, 0, 0) (1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1, 0, 0, 0, 0) (1, 6, 21, 50, 90, 126, 141, 126, 90, 50, 21, 6, 1, 0, 0, 0, 0, 0) (1, 7, 28, 77, 161, 266, 357, 393, 357, 266, 161, 77, 28, 7, 1, 0, 0, 0, 0, 0, 0)

def imagenes(uu):
    ron=[]
    for xx in uu:
        ron.append(xx)
    return Set(ron)
imagenes((1,2,1,3,1,2,1))

{1, 2, 3}

def iny(uu):
    if imagenes(uu).cardinality()==len(uu):
        return True
    else:
        return False
    
def inyectivas(n,m):
    AA=range(m)
    Qu=cartesiano(AA,n)
    gorda=[uu for uu in Qu if iny(uu)]
    return gorda

inyectivas(3,2)

[]

inyectivas(2,3)

[(0, 1), (0, 2), (1, 0), (1, 2), (2, 0), (2, 1)]

inyectivas(2,4)

[(0, 1), (0, 2), (0, 3), (1, 0), (1, 2), (1, 3), (2, 0), (2, 1), (2, 3), (3, 0), (3, 1), (3, 2)]

def cuentinyec(n,m):
    return len(inyectivas(n,m))

cuentinyec(3,10)

720

# vamos a contar palabras de 4 dígitos sin repetición que contienen el 8
melba=inyectivas(4,10)
nico=[uu for uu in melba if 8 in imagenes(uu)]

len(nico)

2016

(3,2,4,5) in nico

False

(1,2,2,8) in nico

False

(2,6,4,8) in nico

True

AA=range(10)
mel=cartesiano(AA,4)
nic=[uu for uu in mel if 8 in imagenes(uu)]

len(nic)

3439

4*9^3+6*9^2+4*9+1

3439

naco=[uu for uu in mel if imagenes(uu).cardinality()==3]

len(naco)

4320

6*10*9*8

4320

naca=[uu for uu in mel if imagenes(uu).cardinality()==2]
len(naca)

630

3*10*9+90*4

630

def sobre(uu,mm):
    if imagenes(uu).cardinality()==mm:
        return True
    else:
        return False

def sobrevas(n,m):
    AA=range(m)
    Qu=cartesiano(AA,n)
    gorda=[uu for uu in Qu if sobre(uu,m)]
    return gorda

sobrevas(4,2)

[(0, 0, 0, 1), (0, 0, 1, 0), (0, 0, 1, 1), (0, 1, 0, 0), (0, 1, 0, 1), (0, 1, 1, 0), (0, 1, 1, 1), (1, 0, 0, 0), (1, 0, 0, 1), (1, 0, 1, 0), (1, 0, 1, 1), (1, 1, 0, 0), (1, 1, 0, 1), (1, 1, 1, 0)]

sobrevas(3,3)

[(0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0)]

len(sobrevas(4,3))

36

A=range(3)
mel=cartesiano(A,4)
nace=[uu for uu in mel if 1 in imagenes(uu)]

nace

[(0, 0, 0, 1), (0, 0, 1, 0), (0, 0, 1, 1), (0, 0, 1, 2), (0, 0, 2, 1), (0, 1, 0, 0), (0, 1, 0, 1), (0, 1, 0, 2), (0, 1, 1, 0), (0, 1, 1, 1), (0, 1, 1, 2), (0, 1, 2, 0), (0, 1, 2, 1), (0, 1, 2, 2), (0, 2, 0, 1), (0, 2, 1, 0), (0, 2, 1, 1), (0, 2, 1, 2), (0, 2, 2, 1), (1, 0, 0, 0), (1, 0, 0, 1), (1, 0, 0, 2), (1, 0, 1, 0), (1, 0, 1, 1), (1, 0, 1, 2), (1, 0, 2, 0), (1, 0, 2, 1), (1, 0, 2, 2), (1, 1, 0, 0), (1, 1, 0, 1), (1, 1, 0, 2), (1, 1, 1, 0), (1, 1, 1, 1), (1, 1, 1, 2), (1, 1, 2, 0), (1, 1, 2, 1), (1, 1, 2, 2), (1, 2, 0, 0), (1, 2, 0, 1), (1, 2, 0, 2), (1, 2, 1, 0), (1, 2, 1, 1), (1, 2, 1, 2), (1, 2, 2, 0), (1, 2, 2, 1), (1, 2, 2, 2), (2, 0, 0, 1), (2, 0, 1, 0), (2, 0, 1, 1), (2, 0, 1, 2), (2, 0, 2, 1), (2, 1, 0, 0), (2, 1, 0, 1), (2, 1, 0, 2), (2, 1, 1, 0), (2, 1, 1, 1), (2, 1, 1, 2), (2, 1, 2, 0), (2, 1, 2, 1), (2, 1, 2, 2), (2, 2, 0, 1), (2, 2, 1, 0), (2, 2, 1, 1), (2, 2, 1, 2), (2, 2, 2, 1)]

len(nace)

65

def cuentasobres(nn,mm):
    return len(sobrevas(nn,mm))

cuentasobres(8,4)

40824

4^8-4*(4-1)^8+binomial(4,2)*(4-2)^8-binomial(4,3)*(4-3)^8 

40824

def F(n,m,k):   #Yesid
    G = PotenciaProdCartesiano(range(m),n) #G:= [m]^[n]
    K = Set(range(k))
    F_salida = []
    for f in G:
        Conjunto = K.intersection(Set(set(f)))
        if Conjunto.cardinality()== k:
            F_salida.append(f)
    return F_salida

def PotenciaProdCartesiano(A, n):   #Yesid
    B =[]
    for i in range (n):
        B.append(A)
    return cartesian_product(B)

F(4,2,2)

[(0, 0, 0, 1), (0, 0, 1, 0), (0, 0, 1, 1), (0, 1, 0, 0), (0, 1, 0, 1), (0, 1, 1, 0), (0, 1, 1, 1), (1, 0, 0, 0), (1, 0, 0, 1), (1, 0, 1, 0), (1, 0, 1, 1), (1, 1, 0, 0), (1, 1, 0, 1), (1, 1, 1, 0)]

len(F(6,3,2))

602

def Stir2(n):
    RR=[[],[  1 ]]
    for ii in range(2,n):
        nota=[0,1]
        for jj in range(2,ii+1):
            qq=RR[ii-1][jj-1]+jj*RR[ii-1][jj]
            nota.append(qq)
        nota.append(0)    
        RR.append(nota)
    return RR

Stir2(6)

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "", line 6, in Stir2 IndexError: list index out of range

def Stirl2(n):
    RR=[[],[0,1,0]]
    for kk in range(2,n+1):
        SS=RR[kk-1]
        TT=[0]
        ll=len(SS)
        for ii in range(1,ll):
            qq=SS[ii-1]+ii*SS[ii]
            TT.append(qq)
        TT.append(0)
        RR.append(TT)
    return RR
   

Stirl2(4)

[[], [0, 1, 0], [0, 1, 1, 0], [0, 1, 3, 1, 0], [0, 1, 7, 6, 1, 0]]

for nono in range(10):
    print Stirl2(10)[nono]

[] [0, 1, 0] [0, 1, 1, 0] [0, 1, 3, 1, 0] [0, 1, 7, 6, 1, 0] [0, 1, 15, 25, 10, 1, 0] [0, 1, 31, 90, 65, 15, 1, 0] [0, 1, 63, 301, 350, 140, 21, 1, 0] [0, 1, 127, 966, 1701, 1050, 266, 28, 1, 0] [0, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1, 0]

 1*binomial(5,1)+ 63*binomial(5,2)*2+ 301*binomial(5,3)*6+ 350*binomial(5,4)*24+binomial(5,5)* 140*120

5^7

78125 78125

7^5

16807

utu=Permutations(5)

len(utu)

120

for uu in Permutations(4):
    print uu

[1, 2, 3, 4] [1, 2, 4, 3] [1, 3, 2, 4] [1, 3, 4, 2] [1, 4, 2, 3] [1, 4, 3, 2] [2, 1, 3, 4] [2, 1, 4, 3] [2, 3, 1, 4] [2, 3, 4, 1] [2, 4, 1, 3] [2, 4, 3, 1] [3, 1, 2, 4] [3, 1, 4, 2] [3, 2, 1, 4] [3, 2, 4, 1] [3, 4, 1, 2] [3, 4, 2, 1] [4, 1, 2, 3] [4, 1, 3, 2] [4, 2, 1, 3] [4, 2, 3, 1] [4, 3, 1, 2] [4, 3, 2, 1]

def escreci(uu):
    ti=len(uu)
    for ii in range(ti-1):
        if not uu[ii]<uu[ii+1]:
            return False
            break
    return True        


def nescre(n,m):
    A=range(m)
    rita=cartesiano(A,n)
    yeye=[]
    for uu in rita:
        if escreci(uu):
            yeye.append(uu)
    return yeye        

escreci([1 ,5, 4, 6])

False

nescre(3,2)

[]

nescre(2,3)

[(0, 1), (0, 2), (1, 2)]

nescre(2,4)

[(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]

for yoyo in range(4,12):
    print yoyo,len(nescre(4,yoyo))

4 1 5 5 6 15 7 35 8 70 9 126 10 210 11 330

    

def orbita(uu,jj):
    tt=[]
    while not jj in tt:
        tt.append(jj)
        yotas=uu[jj-1]
        jj=yotas
    return tt

orbita([2,3,5,4,1],1)

[1, 2, 3, 5]

for muchacha in Permutations(3):
    print muchacha,orbita(muchacha,2)

[1, 2, 3] [2] [1, 3, 2] [2, 3] [2, 1, 3] [2, 1] [2, 3, 1] [2, 3, 1] [3, 1, 2] [2, 1, 3] [3, 2, 1] [2]

def ciclos(uu):
    toma=[]
    for coco in uu:
        toma.append(Set(orbita(uu,coco)))
    return Set(toma)

ciclos([3,4,1,2,5])

{{1, 3}, {2, 4}, {5}}

Set([4,5,3,4,2,5])

{2, 3, 4, 5}

for muchacha in Permutations(4):
    print muchacha,ciclos(muchacha)

[1, 2, 3, 4] {{4}, {2}, {3}, {1}} [1, 2, 4, 3] {{3, 4}, {2}, {1}} [1, 3, 2, 4] {{4}, {2, 3}, {1}} [1, 3, 4, 2] {{2, 3, 4}, {1}} [1, 4, 2, 3] {{2, 3, 4}, {1}} [1, 4, 3, 2] {{2, 4}, {3}, {1}} [2, 1, 3, 4] {{1, 2}, {4}, {3}} [2, 1, 4, 3] {{1, 2}, {3, 4}} [2, 3, 1, 4] {{4}, {1, 2, 3}} [2, 3, 4, 1] {{1, 2, 3, 4}} [2, 4, 1, 3] {{1, 2, 3, 4}} [2, 4, 3, 1] {{3}, {1, 2, 4}} [3, 1, 2, 4] {{4}, {1, 2, 3}} [3, 1, 4, 2] {{1, 2, 3, 4}} [3, 2, 1, 4] {{1, 3}, {4}, {2}} [3, 2, 4, 1] {{1, 3, 4}, {2}} [3, 4, 1, 2] {{1, 3}, {2, 4}} [3, 4, 2, 1] {{1, 2, 3, 4}} [4, 1, 2, 3] {{1, 2, 3, 4}} [4, 1, 3, 2] {{3}, {1, 2, 4}} [4, 2, 1, 3] {{1, 3, 4}, {2}} [4, 2, 3, 1] {{1, 4}, {2}, {3}} [4, 3, 1, 2] {{1, 2, 3, 4}} [4, 3, 2, 1] {{2, 3}, {1, 4}}

def nciclos(nn,kk):  #cuantas pertunaciones hay de orden nn con kk ciclos
    rotica=[uu for uu in Permutations(nn)if ciclos(uu).cardinality()==kk]
    return len(rotica)

nciclos(5,3)

35

print 0,
for nn in range(5):
    print nciclos(5,nn+1),

0 24 50 35 10 1

print 0,
for nn in range(6):
    print nciclos(6,nn+1),

0 120 274 225 85 15 1

120+274+225+85+15+1==factorial(6)

True

for ii in range(1,8):
    print 0,
    for nn in range(ii):
        print nciclos(ii,nn+1),
    print ''    

0 1 0 1 1 0 2 3 1 0 6 11 6 1 0 24 50 35 10 1 0 120 274 225 85 15 1 0 720 1764 1624 735 175 21 1

def Stirl1(n):
    RR=[[],[0,1,0]]
    for kk in range(2,n+1):
        SS=RR[kk-1]
        TT=[0]
        ll=len(SS)
        for ii in range(1,ll):
            qq=SS[ii-1]+(kk-1)*SS[ii]
            TT.append(qq)
        TT.append(0)
        RR.append(TT)
    return RR

Stirl1(5)

[[], [0, 1, 0], [0, 1, 1, 0], [0, 2, 3, 1, 0], [0, 6, 11, 6, 1, 0], [0, 24, 50, 35, 10, 1, 0]]

def ImStirl1(n):
    RR=[[],[0,1,0]]
    for kk in range(2,n+1):
        SS=RR[kk-1]
        TT=[0]
        ll=len(SS)
        for ii in range(1,ll):
            qq=SS[ii-1]+(kk-1)*SS[ii]
            TT.append(qq)
        TT.append(0)
        RR.append(TT)
        print TT
        


ImStirl1(10)

[0, 1, 1, 0] [0, 2, 3, 1, 0] [0, 6, 11, 6, 1, 0] [0, 24, 50, 35, 10, 1, 0] [0, 120, 274, 225, 85, 15, 1, 0] [0, 720, 1764, 1624, 735, 175, 21, 1, 0] [0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 0] [0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 0] [0, 362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 1, 0]