def cardinality(self):
"""
Returns the number of Lyndon words with the evaluation e.
EXAMPLES::
sage: LyndonWords([]).cardinality()
0
sage: LyndonWords([2,2]).cardinality()
1
sage: LyndonWords([2,3,2]).cardinality()
30
Check to make sure that the count matches up with the number of
Lyndon words generated.
::
sage: comps = [[],[2,2],[3,2,7],[4,2]]+Compositions(4).list()
sage: lws = [ LyndonWords(comp) for comp in comps]
sage: all( [ lw.cardinality() == len(lw.list()) for lw in lws] )
True
"""
evaluation = self.e
le = __builtin__.list(evaluation)
if len(evaluation) == 0:
return 0
n = sum(evaluation)
return sum([moebius(j)*factorial(n/j) / prod([factorial(ni/j) for ni in evaluation]) for j in divisors(gcd(le))])/n
def cardinality(self):
"""
Returns the number of Lyndon words with the evaluation e.
EXAMPLES::
sage: LyndonWords([]).cardinality()
0
sage: LyndonWords([2,2]).cardinality()
1
sage: LyndonWords([2,3,2]).cardinality()
30
Check to make sure that the count matches up with the number of
Lyndon words generated.
::
sage: comps = [[],[2,2],[3,2,7],[4,2]]+Compositions(4).list()
sage: lws = [ LyndonWords(comp) for comp in comps]
sage: all( [ lw.cardinality() == len(lw.list()) for lw in lws] )
True
"""
evaluation = self.e
le = __builtin__.list(evaluation)
if len(evaluation) == 0:
return 0
n = sum(evaluation)
return sum([
moebius(j) * factorial(n / j) /
prod([factorial(ni / j) for ni in evaluation])
for j in divisors(gcd(le))
]) / n
def cardinality(self):
"""
EXAMPLES::
sage: [ ContreTableaux(n).cardinality() for n in range(0, 11)]
[1, 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700]
"""
return prod( [ factorial(3*k+1)/factorial(self.n+k) for k in range(self.n)] )
def cardinality(self):
"""
EXAMPLES::
sage: [ ContreTableaux(n).cardinality() for n in range(0, 11)]
[1, 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700]
"""
return prod( [ factorial(3*k+1)/factorial(self.n+k) for k in range(self.n)] )
def by_taylor_expansion(self, fs, k):
"""
We combine the theta decomposition and the heat operator as in [Sko].
This yields a bijections of Jacobi forms of weight `k` and
`M_k \times S_{k+2} \times .. \times S_{k+2m}`.
NOTE:
To make ``phi_divs`` integral we introduce an extra factor
`2^{\mathrm{index}} * \mathrm{factorial}(k + 2 * \mathrm{index} - 1)`.
"""
## we introduce an abbreviations
p = self.__p
PS = self.power_series_ring()
if not len(fs) == self.__precision.jacobi_index() + 1:
raise ValueError(
"fs must be a list of m + 1 elliptic modular forms or their fourier expansion"
)
qexp_prec = self._qexp_precision()
if qexp_prec is None: # there are no forms below the precision
return dict()
f_divs = dict()
for (i, f) in enumerate(fs):
f_divs[(i, 0)] = PS(f(qexp_prec), qexp_prec)
for i in xrange(self.__precision.jacobi_index() + 1):
for j in xrange(1, self.__precision.jacobi_index() - i + 1):
f_divs[(i, j)] = f_divs[(i, j - 1)].derivative().shift(1)
phi_divs = list()
for i in xrange(self.__precision.jacobi_index() + 1):
## This is the formula in Skoruppas thesis. He uses d/ d tau instead of d / dz which yields
## a factor 4 m
phi_divs.append(
sum(f_divs[(j, i - j)] *
(4 * self.__precision.jacobi_index())**i * binomial(i, j) *
(2**self.index() // 2**i) * prod(2 * (i - l) + 1
for l in xrange(1, i)) *
(factorial(k + 2 * self.index() - 1) //
factorial(i + k + j - 1)) *
factorial(2 * self.__precision.jacobi_index() + k - 1)
for j in xrange(i + 1)))
phi_coeffs = dict()
for r in xrange(self.index() + 1):
series = sum(map(operator.mul, self._theta_factors()[r], phi_divs))
series = self._eta_factor() * series
for n in xrange(qexp_prec):
phi_coeffs[(n, r)] = int(series[n].lift()) % p
return phi_coeffs
def cardinality(self):
r"""
Return the cardinality of ``self``.
EXAMPLES::
sage: [AlternatingSignMatrices(n).cardinality() for n in range(0, 11)]
[1, 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700]
"""
return prod( [ factorial(3*k+1)/factorial(self._n+k)
for k in range(self._n)] )
def cardinality(self):
r"""
Cardinality of ``self``.
EXAMPLES::
sage: M = MonotoneTriangles(4)
sage: M.cardinality()
42
"""
return prod( [ factorial(3*k+1)/factorial(self._n+k)
for k in range(self._n)] )
def cardinality(self):
r"""
Return the cardinality of ``self``.
EXAMPLES::
sage: [AlternatingSignMatrices(n).cardinality() for n in range(0, 11)]
[1, 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700]
"""
return prod([
factorial(3 * k + 1) / factorial(self._n + k)
for k in range(self._n)
])
def cardinality(self):
"""
TESTS::
sage: [ AlternatingSignMatrices(n).cardinality() for n in range(0, 11)]
[1, 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700]
::
sage: asms = [ AlternatingSignMatrices(n) for n in range(6) ]
sage: all( [ asm.cardinality() == len(asm.list()) for asm in asms] )
True
"""
return prod( [ factorial(3*k+1)/factorial(self.n+k) for k in range(self.n)] )
def by_taylor_expansion(self, fs, k) :
"""
We combine the theta decomposition and the heat operator as in [Sko].
This yields a bijections of Jacobi forms of weight `k` and
`M_k \times S_{k+2} \times .. \times S_{k+2m}`.
NOTE:
To make ``phi_divs`` integral we introduce an extra factor
`2^{\mathrm{index}} * \mathrm{factorial}(k + 2 * \mathrm{index} - 1)`.
"""
## we introduce an abbreviations
p = self.__p
PS = self.power_series_ring()
if not len(fs) == self.__precision.jacobi_index() + 1 :
raise ValueError("fs must be a list of m + 1 elliptic modular forms or their fourier expansion")
qexp_prec = self._qexp_precision()
if qexp_prec is None : # there are no forms below the precision
return dict()
f_divs = dict()
for (i, f) in enumerate(fs) :
f_divs[(i, 0)] = PS(f(qexp_prec), qexp_prec)
for i in xrange(self.__precision.jacobi_index() + 1) :
for j in xrange(1, self.__precision.jacobi_index() - i + 1) :
f_divs[(i,j)] = f_divs[(i, j - 1)].derivative().shift(1)
phi_divs = list()
for i in xrange(self.__precision.jacobi_index() + 1) :
## This is the formula in Skoruppas thesis. He uses d/ d tau instead of d / dz which yields
## a factor 4 m
phi_divs.append( sum( f_divs[(j, i - j)] * (4 * self.__precision.jacobi_index())**i
* binomial(i,j) * ( 2**self.index() // 2**i)
* prod(2*(i - l) + 1 for l in xrange(1, i))
* (factorial(k + 2*self.index() - 1) // factorial(i + k + j - 1))
* factorial(2*self.__precision.jacobi_index() + k - 1)
for j in xrange(i + 1) ) )
phi_coeffs = dict()
for r in xrange(self.index() + 1) :
series = sum( map(operator.mul, self._theta_factors()[r], phi_divs) )
series = self._eta_factor() * series
for n in xrange(qexp_prec) :
phi_coeffs[(n, r)] = int(series[n].lift()) % p
return phi_coeffs
def cardinality(self):
r"""
Cardinality of ``self``.
EXAMPLES::
sage: M = MonotoneTriangles(4)
sage: M.cardinality()
42
"""
return prod([
factorial(3 * k + 1) / factorial(self._n + k)
for k in range(self._n)
])
def cpi(self, p):
"""
Returns the centrally primitive idempotent for the symmetric group
of order n for the irreducible corresponding indexed by the
partition p.
EXAMPLES::
sage: QS3 = SymmetricGroupAlgebra(QQ,3)
sage: QS3.cpi([2,1])
2/3*[1, 2, 3] - 1/3*[2, 3, 1] - 1/3*[3, 1, 2]
sage: QS3.cpi([3])
1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1]
sage: QS3.cpi([1,1,1])
1/6*[1, 2, 3] - 1/6*[1, 3, 2] - 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] - 1/6*[3, 2, 1]
"""
if p not in partition.Partitions_n(self.n):
raise TypeError, "p must be a partition of %s"%self.n
character_table = eval(gap.eval("Display(Irr(SymmetricGroup(%d)));"%self.n))
cpi = self(0)
np = partition.Partitions_n(self.n).list()
np.reverse()
p_index = np.index(p)
big_coeff = character_table[p_index][0]/factorial(self.n)
for g in permutation.StandardPermutations_n(self.n):
cpi += big_coeff * character_table[p_index][np.index(g.inverse().cycle_type())] * self(g)
return cpi
def cardinality(self):
r"""
Return the cardinality of ``self``.
The number of `n \times n` alternating sign matrices is equal to
.. MATH::
\prod_{k=0}^{n-1} \frac{(3k+1)!}{(n+k)!} = \frac{1! 4! 7! 10!
\cdots (3n-2)!}{n! (n+1)! (n+2)! (n+3)! \cdots (2n-1)!}
EXAMPLES::
sage: [AlternatingSignMatrices(n).cardinality() for n in range(0, 11)]
[1, 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700]
"""
return Integer(prod([factorial(3 * k + 1) / factorial(self._n + k) for k in range(self._n)]))
def cardinality(self):
r"""
Return the cardinality of ``self``.
The number of `n \times n` alternating sign matrices is equal to
.. MATH::
\prod_{k=0}^{n-1} \frac{(3k+1)!}{(n+k)!} = \frac{1! 4! 7! 10!
\cdots (3n-2)!}{n! (n+1)! (n+2)! (n+3)! \cdots (2n-1)!}
EXAMPLES::
sage: [AlternatingSignMatrices(n).cardinality() for n in range(0, 11)]
[1, 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700]
"""
return Integer(prod( [ factorial(3*k+1)/factorial(self._n+k)
for k in range(self._n)] ))
def cardinality(self):
r"""
Cardinality of ``self``.
The number of monotone triangles with `n` rows is equal to
.. MATH::
\prod_{k=0}^{n-1} \frac{(3k+1)!}{(n+k)!} = \frac{1! 4! 7! 10!
\cdots (3n-2)!}{n! (n+1)! (n+2)! (n+3)! \cdots (2n-1)!}
EXAMPLES::
sage: M = MonotoneTriangles(4)
sage: M.cardinality()
42
"""
return Integer(prod([factorial(3 * k + 1) / factorial(self._n + k) for k in range(self._n)]))
def cardinality(self):
r"""
Cardinality of ``self``.
The number of monotone triangles with `n` rows is equal to
.. MATH::
\prod_{k=0}^{n-1} \frac{(3k+1)!}{(n+k)!} = \frac{1! 4! 7! 10!
\cdots (3n-2)!}{n! (n+1)! (n+2)! (n+3)! \cdots (2n-1)!}
EXAMPLES::
sage: M = MonotoneTriangles(4)
sage: M.cardinality()
42
"""
return Integer(prod( [ factorial(3*k+1)/factorial(self._n+k)
for k in range(self._n)] ))
def cpi(self, p):
"""
Return the centrally primitive idempotent for the symmetric group
of order `n` corresponding to the irreducible representation
indexed by the partition ``p``.
EXAMPLES::
sage: QS3 = SymmetricGroupAlgebra(QQ,3)
sage: QS3.cpi([2,1])
2/3*[1, 2, 3] - 1/3*[2, 3, 1] - 1/3*[3, 1, 2]
sage: QS3.cpi([3])
1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1]
sage: QS3.cpi([1,1,1])
1/6*[1, 2, 3] - 1/6*[1, 3, 2] - 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] - 1/6*[3, 2, 1]
sage: QS0 = SymmetricGroupAlgebra(QQ, 0)
sage: QS0.cpi(Partition([]))
[]
TESTS::
sage: QS3.cpi([2,2])
Traceback (most recent call last):
...
TypeError: p (= [2, 2]) must be a partition of n (= 3)
"""
if p not in partition.Partitions_n(self.n):
raise TypeError(
"p (= {p}) must be a partition of n (= {n})".format(p=p,
n=self.n))
character_table = eval(
gap.eval("Display(Irr(SymmetricGroup(%d)));" % self.n))
cpi = self.zero()
np = partition.Partitions_n(self.n).list()
np.reverse()
p_index = np.index(p)
big_coeff = character_table[p_index][0] / factorial(self.n)
character_row = character_table[p_index]
dct = {
g: big_coeff * character_row[np.index(g.cycle_type())]
for g in permutation.StandardPermutations_n(self.n)
}
return self._from_dict(dct)
def cpi(self, p):
"""
Return the centrally primitive idempotent for the symmetric group
of order `n` corresponding to the irreducible representation
indexed by the partition ``p``.
EXAMPLES::
sage: QS3 = SymmetricGroupAlgebra(QQ,3)
sage: QS3.cpi([2,1])
2/3*[1, 2, 3] - 1/3*[2, 3, 1] - 1/3*[3, 1, 2]
sage: QS3.cpi([3])
1/6*[1, 2, 3] + 1/6*[1, 3, 2] + 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] + 1/6*[3, 2, 1]
sage: QS3.cpi([1,1,1])
1/6*[1, 2, 3] - 1/6*[1, 3, 2] - 1/6*[2, 1, 3] + 1/6*[2, 3, 1] + 1/6*[3, 1, 2] - 1/6*[3, 2, 1]
sage: QS0 = SymmetricGroupAlgebra(QQ, 0)
sage: QS0.cpi(Partition([]))
[]
TESTS::
sage: QS3.cpi([2,2])
Traceback (most recent call last):
...
TypeError: p (= [2, 2]) must be a partition of n (= 3)
"""
if p not in partition.Partitions_n(self.n):
raise TypeError("p (= {p}) must be a partition of n (= {n})".format(p=p, n=self.n))
character_table = eval(gap.eval("Display(Irr(SymmetricGroup(%d)));"%self.n))
cpi = self.zero()
np = partition.Partitions_n(self.n).list()
np.reverse()
p_index = np.index(p)
big_coeff = character_table[p_index][0] / factorial(self.n)
character_row = character_table[p_index]
dct = { g : big_coeff * character_row[np.index(g.cycle_type())]
for g in permutation.StandardPermutations_n(self.n) }
return self._from_dict(dct)
def kappa(alpha):
r"""
Returns `\kappa_\alpha` which is n! divided by the number
of standard tableaux of shape `\alpha`.
EXAMPLES::
sage: from sage.combinat.symmetric_group_algebra import kappa
sage: kappa(Partition([2,1]))
3
sage: kappa([2,1])
3
"""
try:
n = alpha.size()
except AttributeError:
n = sum(alpha)
return factorial(n)/StandardTableaux(alpha).cardinality()
def by_taylor_expansion(self, fs, k):
r"""
We combine the theta decomposition and the heat operator as in the
thesis of Nils Skoruppa. This yields a bijections of the space of weak
Jacobi forms of weight `k` and index `m` with the product of spaces
of elliptic modular forms `M_k \times S_{k+2} \times .. \times S_{k+2m}`.
INPUT:
- ``fs`` -- A list of functions that given an integer `p` return the
q-expansion of a modular form with rational coefficients
up to precision `p`. These modular forms correspond to
the components of the above product.
- `k` -- An integer. The weight of the weak Jacobi form to be computed.
NOTE:
In order to make ``phi_divs`` integral we introduce an extra factor
`2^{\mathrm{index}} * \mathrm{factorial}(k + 2 * \mathrm{index} - 1)`.
"""
## we introduce an abbreviations
p = self.__p
PS = self.power_series_ring()
if not len(fs) == self.__precision.jacobi_index() + 1:
raise ValueError( "fs (which has length {0}) must be a list of {1} Fourier expansions" \
.format(len(fs), self.__precision.jacobi_index() + 1) )
qexp_prec = self._qexp_precision()
if qexp_prec is None: # there are no forms below the precision
return dict()
f_divs = dict()
for (i, f) in enumerate(fs):
f_divs[(i, 0)] = PS(f(qexp_prec), qexp_prec)
for i in xrange(self.__precision.jacobi_index() + 1):
for j in xrange(1, self.__precision.jacobi_index() - i + 1):
f_divs[(i, j)] = f_divs[(i, j - 1)].derivative().shift(1)
phi_divs = list()
for i in xrange(self.__precision.jacobi_index() + 1):
## This is the formula in Skoruppas thesis. He uses d/ d tau instead of d / dz which yields
## a factor 4 m
phi_divs.append(
sum(f_divs[(j, i - j)] *
(4 * self.__precision.jacobi_index())**i * binomial(i, j) *
(2**self.index() // 2**i) * prod(2 * (i - l) + 1
for l in xrange(1, i)) *
(factorial(k + 2 * self.index() - 1) //
factorial(i + k + j - 1)) *
factorial(2 * self.__precision.jacobi_index() + k - 1)
for j in xrange(i + 1)))
phi_coeffs = dict()
for r in xrange(self.index() + 1):
series = sum(map(operator.mul, self._theta_factors()[r], phi_divs))
series = self._eta_factor() * series
for n in xrange(qexp_prec):
phi_coeffs[(n, r)] = int(series[n].lift()) % p
return phi_coeffs
def by_taylor_expansion(self, fs, k) :
r"""
We combine the theta decomposition and the heat operator as in the
thesis of Nils Skoruppa. This yields a bijections of the space of weak
Jacobi forms of weight `k` and index `m` with the product of spaces
of elliptic modular forms `M_k \times S_{k+2} \times .. \times S_{k+2m}`.
INPUT:
- ``fs`` -- A list of functions that given an integer `p` return the
q-expansion of a modular form with rational coefficients
up to precision `p`. These modular forms correspond to
the components of the above product.
- `k` -- An integer. The weight of the weak Jacobi form to be computed.
NOTE:
In order to make ``phi_divs`` integral we introduce an extra factor
`2^{\mathrm{index}} * \mathrm{factorial}(k + 2 * \mathrm{index} - 1)`.
"""
## we introduce an abbreviations
p = self.__p
PS = self.power_series_ring()
if not len(fs) == self.__precision.jacobi_index() + 1 :
raise ValueError( "fs (which has length {0}) must be a list of {1} Fourier expansions" \
.format(len(fs), self.__precision.jacobi_index() + 1) )
qexp_prec = self._qexp_precision()
if qexp_prec is None : # there are no forms below the precision
return dict()
f_divs = dict()
for (i, f) in enumerate(fs) :
f_divs[(i, 0)] = PS(f(qexp_prec), qexp_prec)
for i in xrange(self.__precision.jacobi_index() + 1) :
for j in xrange(1, self.__precision.jacobi_index() - i + 1) :
f_divs[(i,j)] = f_divs[(i, j - 1)].derivative().shift(1)
phi_divs = list()
for i in xrange(self.__precision.jacobi_index() + 1) :
## This is the formula in Skoruppas thesis. He uses d/ d tau instead of d / dz which yields
## a factor 4 m
phi_divs.append( sum( f_divs[(j, i - j)] * (4 * self.__precision.jacobi_index())**i
* binomial(i,j) * ( 2**self.index() // 2**i)
* prod(2*(i - l) + 1 for l in xrange(1, i))
* (factorial(k + 2*self.index() - 1) // factorial(i + k + j - 1))
* factorial(2*self.__precision.jacobi_index() + k - 1)
for j in xrange(i + 1) ) )
phi_coeffs = dict()
for r in xrange(self.index() + 1) :
series = sum( map(operator.mul, self._theta_factors()[r], phi_divs) )
series = self._eta_factor() * series
for n in xrange(qexp_prec) :
phi_coeffs[(n, r)] = int(series[n].lift()) % p
return phi_coeffs
