Merge branch 'public/combinat/invol-nsym-2' of git://trac.sagemath.org/sage into nsym

Author: darijgr

src/sage/combinat/ncsf_qsym/generic_basis_code.py

@@ -1201,9 +1201,9 @@ def internal_product(self, other):
 
             Explicit formulas can be given for internal products of
             elements of the complete and the Psi bases. First, the formula
-            for the complete basis ([NCSF1]_ Proposition 5.1): If `I` and
+            for the Complete basis ([NCSF1]_ Proposition 5.1): If `I` and
             `J` are two compositions of lengths `p` and `q`, respectively,
-            then the corresponding complete homogeneous non-commutative
+            then the corresponding Complete homogeneous non-commutative
             symmetric functions `S^I` and `S^J` have internal product
 
             .. MATH::
@@ -1225,7 +1225,7 @@ def internal_product(self, other):
             The formula on the Psi basis ([NCSF2]_ Lemma 3.10) is more
             complicated. Let `I` and `J` be two compositions of lengths
             `p` and `q`, respectively, having the same size `|I| = |J|`.
-            We denote by `\Psi^K` the element of the Psi-basis
+            We denote by `\Psi^K` the element of the Psi basis
             corresponding to any composition `K`.
 
             - If `p > q`, then `\Psi^I * \Psi^J` is plainly `0`.
@@ -1242,11 +1242,11 @@ def internal_product(self, other):
               composition `K = (k_1, k_2, \ldots, k_s)`, denote by
               `\Gamma_K` the non-commutative symmetric function
               `k_1 [\ldots [[\Psi_{k_1}, \Psi_{k_2}], \Psi_{k_3}],
-              \ldots \Psi_{k_s}]`. For any subset `S` of
-              `\{ 1, 2, \ldots, q \}`, let `J_S` be the composition
+              \ldots \Psi_{k_s}]`. For any subset `A` of
+              `\{ 1, 2, \ldots, q \}`, let `J_A` be the composition
               obtained from `J` by removing the `r`-th parts for all
-              `r \notin S` (while keeping the `r`-th parts for all
-              `r \in S` in order). Then, `\Psi^I * \Psi^J` equals the
+              `r \notin A` (while keeping the `r`-th parts for all
+              `r \in A` in order). Then, `\Psi^I * \Psi^J` equals the
               sum of `\Gamma_{J_{K_1}} \Gamma_{J_{K_2}} \cdots
               \Gamma_{J_{K_p}}` over all ordered set partitions
               `(K_1, K_2, \ldots, K_p)` of `\{ 1, 2, \ldots, q \}`

src/sage/combinat/ncsf_qsym/ncsf.py

@@ -968,7 +968,7 @@ def internal_coproduct(self):
                     sage: def int_copr_on_F_via_M(I):
                     ....:     result = tensor([F.zero(), F.zero()])
                     ....:     w = M(F(I)).internal_coproduct()
-                    ....:     for lam, a in w.monomial_coefficients().items():
+                    ....:     for lam, a in w:
                     ....:         (U, V) = lam
                     ....:         result += a * tensor([F(M(U)), F(M(V))])
                     ....:     return result
@@ -987,7 +987,7 @@ def internal_coproduct(self):
                     sage: def int_copr_of_e_in_M(mu):
                     ....:     result = tensor([M.zero(), M.zero()])
                     ....:     w = e(mu).internal_coproduct()
-                    ....:     for lam, a in w.monomial_coefficients().items():
+                    ....:     for lam, a in w:
                     ....:         (nu, kappa) = lam
                     ....:         result += a * tensor([M(e(nu)), M(e(kappa))])
                     ....:     return result
@@ -1011,7 +1011,7 @@ def internal_coproduct(self):
                 F = parent.realization_of().F()
                 from sage.categories.tensor import tensor
                 result = tensor([parent.zero(), parent.zero()])
-                for lam, a in F(self).internal_coproduct().monomial_coefficients().items():
+                for lam, a in F(self).internal_coproduct():
                     (I, J) = lam
                     result += a * tensor([parent(F(I)), parent(F(J))])
                 return result
@@ -1116,7 +1116,7 @@ def frobenius(self, n):
                 M = parent.realization_of().M()
                 C = parent._basis_keys
                 dct = {C(map(lambda i: n * i, I)): coeff
-                       for (I, coeff) in M(self).monomial_coefficients().items()}
+                       for (I, coeff) in M(self)}
                 result_in_M_basis = M._from_dict(dct)
                 return parent(result_in_M_basis)
 
@@ -1193,8 +1193,7 @@ def star_involution(self):
                 # involution componentwise, then convert back.
                 parent = self.parent()
                 M = parent.realization_of().M()
-                dct = {I.reversed(): coeff
-                       for (I, coeff) in M(self).monomial_coefficients().items()}
+                dct = {I.reversed(): coeff for (I, coeff) in M(self)}
                 return parent(M._from_dict(dct))
 
             def omega_involution(self):
@@ -1348,8 +1347,7 @@ def psi_involution(self):
                 # involution componentwise, then convert back.
                 parent = self.parent()
                 F = parent.realization_of().F()
-                dct = {I.complement(): coeff
-                       for (I, coeff) in F(self).monomial_coefficients().items()}
+                dct = {I.complement(): coeff for (I, coeff) in F(self)}
                 return parent(F._from_dict(dct))
 
             def expand(self, n, alphabet='x'):
@@ -1707,8 +1705,7 @@ def lambda_of_monomial(self, I, n):
                                                for k in lam])
             QQ_result *= (-1) ** n
             # QQ_result is now \lambda^n(M_I) over QQ.
-            result = self.sum_of_terms([(J, ZZ(coeff)) for (J, coeff) in
-                                        QQ_result.monomial_coefficients().items()],
+            result = self.sum_of_terms([(J, ZZ(coeff)) for (J, coeff) in QQ_result],
                                         distinct = True)
             return result
 
@@ -2293,7 +2290,7 @@ def internal_coproduct(self):
                 result = F2.zero()
                 from sage.categories.tensor import tensor
                 from sage.combinat.permutation import Permutation
-                for I, a in self.monomial_coefficients().items():
+                for I, a in self:
                     # We must add a * \Delta^\times(F_I) to result.
                     from sage.combinat.permutation import descents_composition_last
                     pi = descents_composition_last(I)
@@ -2364,8 +2361,7 @@ def star_involution(self):
                     0
                 """
                 parent = self.parent()
-                dct = {I.reversed(): coeff
-                       for (I, coeff) in self.monomial_coefficients().items()}
+                dct = {I.reversed(): coeff for (I, coeff) in self}
                 return parent._from_dict(dct)
 
     F = Fundamental
@@ -2558,7 +2554,7 @@ def _from_fundamental_on_basis(self, comp):
                 QS[1, 2, 1, 1] + QS[1, 3, 1] - QS[2, 2, 1]
             """
             comp = Composition(comp)
-            if comp == []:
+            if not comp._list:
                 return self.one()
             comps = compositions_order(comp.size())
             T = self._from_fundamental_transition_matrix(comp.size())
@@ -2625,7 +2621,7 @@ def _to_Monomial_on_basis(self, J):
             4*M[1, 1, 1, 1, 1] + 3*M[1, 1, 1, 2] + 2*M[1, 1, 2, 1] + M[1, 1, 3] + M[1, 2, 1, 1] + M[1, 2, 2] + M[2, 1, 1, 1] + M[2, 1, 2]
             """
             M = self.realization_of().Monomial()
-            if J == []:
+            if not J._list:
                 return M([])
             C = Compositions()
             C_size = Compositions(J.size())

src/sage/combinat/ncsf_qsym/qsym.py

@@ -1782,27 +1782,27 @@ def t_completion(self, t):
 
         If `\lambda = (\lambda_1, \lambda_2, \lambda_3, \ldots)` is a
         partition and `t` is an integer greater or equal to
-        `\left| \lambda \right| + \lambda_1`, then the *`t`-completion of
-        `\lambda`* is defined as the partition
-        `(t - \left| \lambda \right|, \lambda_1, \lambda_2, \lambda_3,
-        \ldots)` of `t`. This partition is denoted by `\lambda[t]` in
-        [BOR09]_, by `\lambda_{[t]}` in [BdVO12]_ and by `\lambda(t)` in
-        [CO10]_.
+        `\left\lvert \lambda \right\rvert + \lambda_1`, then the
+        `t`-*completion of* `\lambda` is defined as the partition
+        `(t - \left\lvert \lambda \right\rvert, \lambda_1, \lambda_2,
+        \lambda_3, \ldots)` of `t`. This partition is denoted by `\lambda[t]`
+        in [BOR09]_, by `\lambda_{[t]}` in [BdVO12]_, and by `\lambda(t)`
+        in[CO10]_.
 
         REFERENCES:
 
-        .. [BOR09] Emmanuel Briand, Rosa Orellana, Mercedes Rosas,
+        .. [BOR09] Emmanuel Briand, Rosa Orellana, Mercedes Rosas.
            *The stability of the Kronecker products of Schur
            functions*.
            :arxiv:`0907.4652v2`.
 
-        .. [CO10] Jonathan Comes, Viktor Ostrik,
-           *On blocks of Deligne's category
-           `\underline{\mathrm{Rep}}(S_t)`*.
+        .. [CO10] Jonathan Comes, Viktor Ostrik.
+           *On blocks of Deligne's category*
+           `\underline{\mathrm{Rep}}(S_t)`.
            :arxiv:`0910.5695v2`,
            http://pages.uoregon.edu/jcomes/blocks.pdf
 
-        .. [BdVO12] Christopher Bowman, Maud De Visscher, Rosa Orellana,
+        .. [BdVO12] Christopher Bowman, Maud De Visscher, Rosa Orellana.
            *The partition algebra and the Kronecker coefficients*.
            :arXiv:`1210.5579v6`.
 
@@ -1827,12 +1827,14 @@ def t_completion(self, t):
             sage: Partition([4, 2, 2, 1]).t_completion(10)
             Traceback (most recent call last):
             ...
-            ValueError: [1, 4, 2, 2, 1] is not an element of Partitions
+            ValueError: 5-completion is not defined
             sage: Partition([4, 2, 2, 1]).t_completion(5)
             Traceback (most recent call last):
             ...
-            ValueError: [-4, 4, 2, 2, 1] is not an element of Partitions
+            ValueError: 5-completion is not defined
         """
+        if not self._list and t <= self.size() + self._list[0]:
+            raise ValueError("{}-completion is not defined".format(t))
         return Partition([t - self.size()] + self._list)
 
     def larger_lex(self, rhs):
@@ -2002,17 +2004,17 @@ def to_dyck_word(self, n=None):
             True
         """
         from sage.combinat.dyck_word import DyckWord
-        if self == []:
+        if not self._list:
             if n is None:
                 return DyckWord([])
-            return DyckWord([1 for _ in range(n)] + [0 for _ in range(n)])
+            return DyckWord([1]*n + [0]*n)
         list_of_word = []
         if n is None:
             n = max(i + l + 1 for (i, l) in enumerate(self))
             # This n is also max(i+j for (i,j) in self.cells()) + 2.
         list_of_word.extend([1]*(n-self.length()))
         copy_part = list(self)
-        while copy_part != []:
+        while copy_part:
             c = copy_part.pop()
             list_of_word.extend([0]*c)
             for i in range(len(copy_part)):

src/sage/combinat/partition.py

@@ -650,26 +650,27 @@ def omega(self):
             *omega involution*. It sends the power-sum symmetric function
             `p_k` to `(-1)^{k-1} p_k` for every positive integer `k`.
 
-            The images of some bases under the omega automorphism are given
-            by
+            The images of some bases under the omega automorphism are given by
 
             .. MATH::
 
                 \omega(e_{\lambda}) = h_{\lambda}, \qquad
                 \omega(h_{\lambda}) = e_{\lambda}, \qquad
-                \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)} p_{\lambda}, \qquad
+                \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)}
+                p_{\lambda}, \qquad
                 \omega(s_{\lambda}) = s_{\lambda^{\prime}},
 
             where `\lambda` is any partition, where `\ell(\lambda)` denotes
             the length (:meth:`~sage.combinat.partition.Partition.length`)
-            of the partition `\lambda`, where `\lambda^{\prime}' denotes the
+            of the partition `\lambda`, where `\lambda^{\prime}` denotes the
             conjugate partition
             (:meth:`~sage.combinat.partition.Partition.conjugate`) of
             `\lambda`, and where the usual notations for bases are used
             (`e` = elementary, `h` = complete homogeneous, `p` = powersum,
             `s` = Schur).
 
-            ``omega_involution`` is a synonym for the ``omega`` method.
+            :meth:`omega_involution()` is a synonym for the :meth`omega()`
+            method.
 
             OUTPUT:
 

src/sage/combinat/sf/dual.py

@@ -108,26 +108,27 @@ def omega(self):
             *omega involution*. It sends the power-sum symmetric function
             `p_k` to `(-1)^{k-1} p_k` for every positive integer `k`.
 
-            The images of some bases under the omega automorphism are given
-            by
+            The images of some bases under the omega automorphism are given by
 
             .. MATH::
 
                 \omega(e_{\lambda}) = h_{\lambda}, \qquad
                 \omega(h_{\lambda}) = e_{\lambda}, \qquad
-                \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)} p_{\lambda}, \qquad
+                \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)}
+                p_{\lambda}, \qquad
                 \omega(s_{\lambda}) = s_{\lambda^{\prime}},
 
             where `\lambda` is any partition, where `\ell(\lambda)` denotes
             the length (:meth:`~sage.combinat.partition.Partition.length`)
-            of the partition `\lambda`, where `\lambda^{\prime}' denotes the
+            of the partition `\lambda`, where `\lambda^{\prime}` denotes the
             conjugate partition
             (:meth:`~sage.combinat.partition.Partition.conjugate`) of
             `\lambda`, and where the usual notations for bases are used
             (`e` = elementary, `h` = complete homogeneous, `p` = powersum,
             `s` = Schur).
 
-            ``omega_involution`` is a synonym for the ``omega`` method.
+            :meth:`omega_involution()` is a synonym for the :meth`omega()`
+            method.
 
             EXAMPLES::
 

src/sage/combinat/sf/elementary.py

@@ -129,26 +129,27 @@ def omega(self):
             *omega involution*. It sends the power-sum symmetric function
             `p_k` to `(-1)^{k-1} p_k` for every positive integer `k`.
 
-            The images of some bases under the omega automorphism are given
-            by
+            The images of some bases under the omega automorphism are given by
 
             .. MATH::
 
                 \omega(e_{\lambda}) = h_{\lambda}, \qquad
                 \omega(h_{\lambda}) = e_{\lambda}, \qquad
-                \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)} p_{\lambda}, \qquad
+                \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)}
+                p_{\lambda}, \qquad
                 \omega(s_{\lambda}) = s_{\lambda^{\prime}},
 
             where `\lambda` is any partition, where `\ell(\lambda)` denotes
             the length (:meth:`~sage.combinat.partition.Partition.length`)
-            of the partition `\lambda`, where `\lambda^{\prime}' denotes the
+            of the partition `\lambda`, where `\lambda^{\prime}` denotes the
             conjugate partition
             (:meth:`~sage.combinat.partition.Partition.conjugate`) of
             `\lambda`, and where the usual notations for bases are used
             (`e` = elementary, `h` = complete homogeneous, `p` = powersum,
             `s` = Schur).
 
-            ``omega_involution`` is a synonym for the ``omega`` method.
+            :meth:`omega_involution()` is a synonym for the :meth:`omega()`
+            method.
 
             OUTPUT:
 

src/sage/combinat/sf/homogeneous.py

@@ -179,26 +179,27 @@ def omega(self):
             *omega involution*. It sends the power-sum symmetric function
             `p_k` to `(-1)^{k-1} p_k` for every positive integer `k`.
 
-            The images of some bases under the omega automorphism are given
-            by
+            The images of some bases under the omega automorphism are given by
 
             .. MATH::
 
                 \omega(e_{\lambda}) = h_{\lambda}, \qquad
                 \omega(h_{\lambda}) = e_{\lambda}, \qquad
-                \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)} p_{\lambda}, \qquad
+                \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)}
+                p_{\lambda}, \qquad
                 \omega(s_{\lambda}) = s_{\lambda^{\prime}},
 
             where `\lambda` is any partition, where `\ell(\lambda)` denotes
             the length (:meth:`~sage.combinat.partition.Partition.length`)
-            of the partition `\lambda`, where `\lambda^{\prime}' denotes the
+            of the partition `\lambda`, where `\lambda^{\prime}` denotes the
             conjugate partition
             (:meth:`~sage.combinat.partition.Partition.conjugate`) of
             `\lambda`, and where the usual notations for bases are used
             (`e` = elementary, `h` = complete homogeneous, `p` = powersum,
             `s` = Schur).
 
-            ``omega_involution`` is a synonym for the ``omega`` method.
+            :meth:`omega_involution()` is a synonym for the :meth:`omega()`
+            method.
 
             OUTPUT:
 

src/sage/combinat/sf/powersum.py

@@ -209,30 +209,31 @@ def omega(self):
             *omega involution*. It sends the power-sum symmetric function
             `p_k` to `(-1)^{k-1} p_k` for every positive integer `k`.
 
-            The images of some bases under the omega automorphism are given
-            by
+            The images of some bases under the omega automorphism are given by
 
             .. MATH::
 
                 \omega(e_{\lambda}) = h_{\lambda}, \qquad
                 \omega(h_{\lambda}) = e_{\lambda}, \qquad
-                \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)} p_{\lambda}, \qquad
+                \omega(p_{\lambda}) = (-1)^{|\lambda| - \ell(\lambda)}
+                p_{\lambda}, \qquad
                 \omega(s_{\lambda}) = s_{\lambda^{\prime}},
 
             where `\lambda` is any partition, where `\ell(\lambda)` denotes
             the length (:meth:`~sage.combinat.partition.Partition.length`)
-            of the partition `\lambda`, where `\lambda^{\prime}' denotes the
+            of the partition `\lambda`, where `\lambda^{\prime}` denotes the
             conjugate partition
             (:meth:`~sage.combinat.partition.Partition.conjugate`) of
             `\lambda`, and where the usual notations for bases are used
             (`e` = elementary, `h` = complete homogeneous, `p` = powersum,
             `s` = Schur).
 
-            ``omega_involution`` is a synonym for the ``omega`` method.
+            :meth:`omega_involution()` is a synonym for the :meth:`omega()`
+            method.
 
             OUTPUT:
 
-            - the image of ``self`` under omega as an element of the Schur basis
+            - the image of ``self`` under the omega automorphism
 
             EXAMPLES::