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authorDarij Grinberg <darijgrinberg@gmail.com>2016-05-11 05:14:28 +0200
committerDarij Grinberg <darijgrinberg@gmail.com>2016-05-11 05:14:28 +0200
commitbb0b6985540afd1297a6272d50fdb6292e03ce6f (patch)
treef3db2b572fef81313fa3899c9e9bf3425226c58f
parentdocumentation & comments (diff)
example from Hopf notes
-rw-r--r--src/sage/combinat/ncsf_qsym/qsym.py4
-rw-r--r--src/sage/combinat/sf/sfa.py6
-rw-r--r--src/sage/combinat/sf/witt.py2
-rw-r--r--src/sage/matroids/matroid.pyx26
4 files changed, 32 insertions, 6 deletions
diff --git a/src/sage/combinat/ncsf_qsym/qsym.py b/src/sage/combinat/ncsf_qsym/qsym.py
index 07598d5..f6ec21d 100644
--- a/src/sage/combinat/ncsf_qsym/qsym.py
+++ b/src/sage/combinat/ncsf_qsym/qsym.py
@@ -13,7 +13,7 @@ REFERENCES:
.. [GriRei2014] Darij Grinberg, Victor Reiner,
*Hopf algebras in combinatorics*,
- 30 September 2014. :arxiv:`1409.8356v1`.
+ 30 September 2014. :arxiv:`1409.8356v3`.
.. [Mal1993] Claudia Malvenuto, *Produits et coproduits des fonctions
quasi-symetriques et de l'algebre des descentes*,
@@ -2876,7 +2876,7 @@ class QuasiSymmetricFunctions(UniqueRepresentation, Parent):
the `n`-fold concatenation of this Lyndon word with
itself, occurring `n!` times in that shuffle power. But this
can be deduced from Section 2 of [Rad1979]_. See also
- Chapter 6 of [GriRei2014]_, specifically Theorem 6.99, for a
+ Chapter 6 of [GriRei2014]_, specifically Theorem 6.107, for a
complete proof.) More precisely, he showed that
`\mathrm{QSym}` is generated, as a free commutative
`\mathbf{k}`-algebra, by the elements `\lambda^n(M_I)`, where
diff --git a/src/sage/combinat/sf/sfa.py b/src/sage/combinat/sf/sfa.py
index ed4ade0..7a50838 100644
--- a/src/sage/combinat/sf/sfa.py
+++ b/src/sage/combinat/sf/sfa.py
@@ -1008,7 +1008,7 @@ class SymmetricFunctionsBases(Category_realization_of_parent):
basis ``self``. These functions are defined below.
The Carlitz-Shareshian-Wachs symmetric functions have been
- introduced in [GriRei2014]_, Exercise 2.84, as
+ introduced in [GriRei2014]_, Exercise 2.87, as
refinements of a certain particular case of chromatic
quasisymmetric functions defined by Shareshian and Wachs.
Their definitions are as follows:
@@ -1030,7 +1030,7 @@ class SymmetricFunctionsBases(Category_realization_of_parent):
X_{n, d, s} = \sum_{w \in W(n, d, s)} x_w .
This is a symmetric function (according to
- [GriRei2014]_, Exercise 2.84(b)), and for `s = 0` equals
+ [GriRei2014]_, Exercise 2.87(b)), and for `s = 0` equals
the `t^d`-coefficient of the descent enumerator of Smirnov
words of length `n` (an example of a chromatic
quasisymmetric function which happens to be symmetric --
@@ -1049,7 +1049,7 @@ class SymmetricFunctionsBases(Category_realization_of_parent):
`w = (w_1, w_2, \ldots, w_n) \in W(n, d, s)`. These
three power series `U_{n, d, s}`, `V_{n, d, s}` and
`W_{n, d, s}` are symmetric functions as well
- ([GriRei2014]_, Exercise 2.84(c)). Their sum is
+ ([GriRei2014]_, Exercise 2.87(c)). Their sum is
`X_{n, d, s}`.
REFERENCES:
diff --git a/src/sage/combinat/sf/witt.py b/src/sage/combinat/sf/witt.py
index 7af7a2f..e550bca 100644
--- a/src/sage/combinat/sf/witt.py
+++ b/src/sage/combinat/sf/witt.py
@@ -28,7 +28,7 @@ class SymmetricFunctionAlgebra_witt(multiplicative.SymmetricFunctionAlgebra_mult
denoted by `(x_{\lambda})` in [HazWitt1]_, section 9.63, and by
`(q_{\lambda})` in [DoranIV1996]_. We will denote this basis by
`(w_{\lambda})` (which is precisely how it is denoted in
- [GriRei2014]_, Exercise 2.76(d)). It is a multiplicative basis
+ [GriRei2014]_, Exercise 2.79(d)). It is a multiplicative basis
(meaning that `w_{\emptyset} = 1` and that every partition
`\lambda` satisfies
`w_{\lambda} = w_{\lambda_1} w_{\lambda_2} w_{\lambda_3} \cdots`,
diff --git a/src/sage/matroids/matroid.pyx b/src/sage/matroids/matroid.pyx
index 51f9ec2..a6aa0e9 100644
--- a/src/sage/matroids/matroid.pyx
+++ b/src/sage/matroids/matroid.pyx
@@ -6459,6 +6459,19 @@ cdef class Matroid(SageObject):
True
sage: M.is_max_weight_independent_generic()
False
+
+ Here is an example from [GriRei2014]_ (Example 7.56 in v3)::
+
+ sage: A = Matrix(QQ, [[ 1, 1, 0, 0],
+ ....: [-1, 0, 1, 1],
+ ....: [ 0, -1, -1, -1]])
+ sage: M = Matroid(A)
+ sage: M.is_max_weight_independent_generic()
+ False
+ sage: M.is_max_weight_independent_generic(weights={0: 1, 1: 3, 2: 3, 3: 2})
+ True
+ sage: M.is_max_weight_independent_generic(weights={0: 1, 1: 3, 2: 2, 3: 2})
+ False
"""
res = []
r = 0
@@ -6588,6 +6601,19 @@ cdef class Matroid(SageObject):
True
sage: M.is_max_weight_coindependent_generic()
False
+
+ Here is an example from [GriRei2014]_ (Example 7.56 in v3)::
+
+ sage: A = Matrix(QQ, [[ 1, 1, 0, 0],
+ ....: [-1, 0, 1, 1],
+ ....: [ 0, -1, -1, -1]])
+ sage: M = Matroid(A)
+ sage: M.is_max_weight_coindependent_generic()
+ False
+ sage: M.is_max_weight_coindependent_generic(weights={0: 1, 1: 3, 2: 3, 3: 2})
+ True
+ sage: M.is_max_weight_coindependent_generic(weights={0: 1, 1: 3, 2: 2, 3: 2})
+ False
"""
res = []
r = 0