--- sage/combinat/partition.py 2009-07-17 23:38:48.000000000 +0200 +++ ../sage-combinat/sage/combinat/partition.py 2009-07-17 23:25:31.000000000 +0200 @@ -20,9 +20,9 @@ - Dan Drake (2009-03-28): deprecate RestrictedPartitions and implement Partitions_parts_in -EXAMPLES: There are 5 partitions of the integer 4. +EXAMPLES: -:: +There are 5 partitions of the integer 4:: sage: Partitions(4).cardinality() 5 @@ -30,17 +30,13 @@ [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]] We can use the method .first() to get the 'first' partition of a -number. - -:: +number:: sage: Partitions(4).first() [4] Using the method .next(), we can calculate the 'next' partition. -When we are at the last partition, None will be returned. - -:: +When we are at the last partition, None will be returned:: sage: Partitions(4).next([4]) [3, 1] @@ -48,10 +44,8 @@ True We can use ``iter`` to get an object which iterates over the partitions one -by one to save memory. Note that when we do something like -``for part in Partitions(4)`` this iterator is used in the background. - -:: +by one to save memory. Note that when we do something like +``for part in Partitions(4)`` this iterator is used in the background:: sage: g = iter(Partitions(4)) @@ -95,9 +89,7 @@ The min_part options is complementary to max_part and selects partitions having only 'large' parts. Here is the list of all -partitions of 4 with each part at least 2. - -:: +partitions of 4 with each part at least 2:: sage: Partitions(4, min_part=2).list() [[4], [2, 2]] @@ -119,9 +111,7 @@ Finally, here are the partitions of 4 with [1,1,1] as an inner bound. Note that inner sets min_length to the length of its -argument. - -:: +argument:: sage: Partitions(4, inner=[1,1,1]).list() [[2, 1, 1], [1, 1, 1, 1]] @@ -143,9 +133,7 @@ [[7, 4], [6, 5], [6, 4, 1], [6, 3, 2], [5, 4, 2], [5, 3, 2, 1]] Partition objects can also be created individually with the -Partition function. - -:: +Partition function:: sage: Partition([2,1]) [2, 1] @@ -154,9 +142,7 @@ methods that we can use. For example, we can get the conjugate of a partition. Geometrically, the conjugate of a partition is the reflection of that partition through its main diagonal. Of course, -this operation is an involution. - -:: +this operation is an involution:: sage: Partition([4,1]).conjugate() [2, 1, 1, 1] @@ -165,9 +151,7 @@ We can go back and forth between the exponential notations of a partition. The exponential notation can be padded with extra -zeros. - -:: +zeros:: sage: Partition([6,4,4,2,1]).to_exp() [1, 1, 0, 2, 0, 1] @@ -180,17 +164,13 @@ sage: Partition([6,4,4,2,1]).to_exp(10) [1, 1, 0, 2, 0, 1, 0, 0, 0, 0] -We can get the coordinates of the corners of a partition. - -:: +We can get the coordinates of the corners of a partition:: sage: Partition([4,3,1]).corners() [[0, 3], [1, 2], [2, 0]] We can compute the core and quotient of a partition and build -the partition back up from them. - -:: +the partition back up from them:: sage: Partition([6,3,2,2]).core(3) [2, 1, 1] @@ -208,20 +188,12 @@ # Copyright (C) 2007 Mike Hansen , # # Distributed under the terms of the GNU General Public License (GPL) -# -# This code is distributed in the hope that it will be useful, -# but WITHOUT ANY WARRANTY; without even the implied warranty of -# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU -# General Public License for more details. -# -# The full text of the GPL is available at: -# # http://www.gnu.org/licenses/ #***************************************************************************** from sage.interfaces.all import gap, gp from sage.rings.all import QQ, ZZ, infinity, factorial, gcd -from sage.misc.all import prod, sage_eval +from sage.misc.all import prod from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing import sage.combinat.misc as misc import sage.combinat.skew_partition @@ -239,29 +211,31 @@ from integer_list import IntegerListsLex from sage.functions.other import ceil -def Partition(mu=None, **key_word): - """ - A partition is a weakly decreasing ordered sequence of non-negative - integers. This function returns a Sage partition object which can - be specified in one of the following ways:: - * a list (the default) - * using exponential notation - * by beta numbers (TODO) - * specifying the core and the quotient - * specifying the core and the canonical quotient (TODO) - See the examples below. - - Sage follows the usual python conventions when dealing with partitions, - so that the first part of the partition ``mu=Partition([4,3,2,2])`` is - ``mu[0]``, the second part is ``mu[1]`` and so on. As is usual, Sage ignores - trailing zeros at the end of partitions. - - EXAMPLES:: - - sage: Partition([3,2,1]) - [3, 2, 1] - sage: Partition([3,2,1,0]) - [3, 2, 1] +def Partition(mu=None, **keyword): + """ + A partition is a weakly decreasing ordered sequence of non-negative + integers. This function returns a Sage partition object which can + be specified in one of the following ways:: + + * a list (the default) + * using exponential notation + * by beta numbers (TODO) + * specifying the core and the quotient + * specifying the core and the canonical quotient (TODO) + + See the examples below. + + Sage follows the usual python conventions when dealing with partitions, + so that the first part of the partition ``mu=Partition([4,3,2,2])`` is + ``mu[0]``, the second part is ``mu[1]`` and so on. As is usual, Sage ignores + trailing zeros at the end of partitions. + + EXAMPLES:: + + sage: Partition([3,2,1]) + [3, 2, 1] + sage: Partition([3,2,1,0]) + [3, 2, 1] sage: Partition(exp=[2,1,1]) [3, 2, 1, 1] sage: Partition(core=[2,1], quotient=[[2,1],[3],[1,1,1]]) @@ -275,24 +249,24 @@ ... ValueError: [1, 2, 3] is not a valid partition """ - if mu is not None and len(key_word)==0: + if mu is not None and len(keyword)==0: mu = [i for i in mu if i != 0] if mu in Partitions_all(): return Partition_class(mu) else: raise ValueError, "%s is not a valid partition"%mu - elif 'beta_numbers' in key_word and len(key_word)==1: + elif 'beta_numbers' in keyword and len(keyword)==1: raise NotImplementedError - elif 'exp' in key_word and len(key_word)==1: - return from_exp(key_word['exp']) - elif 'core' in key_word and 'quotient' in key_word and len(key_word)==2: - return from_core_and_quotient(key_word['core'], key_word['quotient']) - elif 'core' in key_word and 'canonical_quotient' in key_word and len(key_word)==2: + elif 'exp' in keyword and len(keyword)==1: + return from_exp(keyword['exp']) + elif 'core' in keyword and 'quotient' in keyword and len(keyword)==2: + return from_core_and_quotient(keyword['core'], keyword['quotient']) + elif 'core' in keyword and 'canonical_quotient' in keyword and len(keyword)==2: raise NotImplementedError - elif 'core_and_quotient' in key_word and len(key_word)==1: + elif 'core_and_quotient' in keyword and len(keyword)==1: from sage.misc.misc import deprecation deprecation('"core_and_quotient=(*)" is deprecated. Use "core=[*], quotient=[*]" instead.') - return from_core_and_quotient(*key_word['core_and_quotient']) + return from_core_and_quotient(*keyword['core_and_quotient']) else: raise ValueError, 'incorrect syntax for Partition()' @@ -304,9 +278,9 @@ This function is for internal use only; use Partition(exp=*) instead. - + EXAMPLES:: - + sage: Partition(exp=[1,2,1]) [3, 2, 2, 1] """ @@ -318,7 +292,7 @@ def from_core_and_quotient(core, quotient): """ ** This function is being deprecated - use Partition(core=*, quotient=*) instead ** - + Returns a partition from its core and quotient. Algorithm from mupad-combinat. @@ -327,9 +301,9 @@ This function is for internal use only; use Partition(core=*, quotient=*) instead. - + EXAMPLES:: - + sage: Partition(core=[2,1], quotient=[[2,1],[3],[1,1,1]]) [11, 5, 5, 3, 2, 2, 2] """ @@ -397,11 +371,11 @@ return sage.combinat.skew_partition.SkewPartition([self[:], p]) - def power(self,k): + def power(self, k): """ - partition_power( pi, k ) returns the partition corresponding to - the `k`-th power of a permutation with cycle structure pi - (thus describes the powermap of symmetric groups). + Returns the cycle type of the `k`-th power of any permutation + with cycle type ``self`` (thus describes the powermap of + symmetric groups). Wraps GAP's PowerPartition. @@ -514,7 +488,7 @@ def sign(self): r""" - Returns the sign of a permutation with cycle type of the partition. + Returns the sign of any permutation with cycle type ``self``. This function corresponds to a homomorphism from the symmetric group `S_n` into the cyclic group of order 2, whose kernel @@ -1267,7 +1241,7 @@ def centralizer_size(self, t=0, q=0): """ Returns the size of the centralizer of any permutation of cycle type - p. If m_i is the multiplicity of i as a part of p, this is given + ``self``. If m_i is the multiplicity of i as a part of p, this is given by `\prod_i (i^m[i])*(m[i]!)`. Including the optional parameters t and q gives the q-t analog which is the former product times `\prod_{i=1}^{length(p)} (1 - q^{p[i]}) / (1 - t^{p[i]}).` @@ -1294,7 +1268,7 @@ def aut(self): r""" Returns a factor for the number of permutations with cycle type - self. self.aut() returns + ``self``. self.aut() returns `1^{j_1}j_1! \cdots n^{j_n}j_n!` where `j_k` is the number of parts in self equal to k. @@ -1417,30 +1391,44 @@ return res def r_core(self, length): - """ *** deprecate *** """ - from sage.misc.misc import deprecation - deprecation('r_core is deprecated. Use core instead.') - return self.core(self, length) + """ + This function is deprecated. + + EXAMPLES:: + + sage: Partition([6,3,2,2]).r_core(3) + doctest:1: DeprecationWarning: r_core is deprecated. Please use core instead. + [2, 1, 1] + + Please use :meth:`core` instead:: + + sage: Partition([6,3,2,2]).core(3) + [2, 1, 1] + + """ + from sage.misc.misc import deprecation + deprecation('r_core is deprecated. Please use core instead.') + return self.core(length) def core(self, length): """ Returns the core of the partition -- in the literature the core is - commonly referred to as the k-core, p-core, r-core, ... . The + commonly referred to as the `k`-core, `p`-core, `r`-core, ... . The construction of the core can be visualized by repeatedly removing - border strips of size r from p until this is no longer possible. + border strips of size `r` from ``self`` until this is no longer possible. The remaining partition is the core. - + EXAMPLES:: - + sage: Partition([6,3,2,2]).core(3) [2, 1, 1] sage: Partition([]).core(3) [] sage: Partition([8,7,7,4,1,1,1,1,1]).core(3) [2, 1, 1] - + TESTS:: - + sage: Partition([3,3,3,2,1]).core(3) [] sage: Partition([10,8,7,7]).core(4) @@ -1471,23 +1459,35 @@ return filter(lambda x: x != 0, part) def r_quotient(self, length): - """ *** deprecate *** """ - from sage.misc.misc import deprecation - deprecation('r_quotient is deprecated. Use quotient instead.') - return self.quotient(self,length) + """ + This function is deprecated. + + EXAMPLES:: + + sage: Partition([6,3,2,2]).r_quotient(3) + doctest:1: DeprecationWarning: r_quotient is deprecated. Please use quotient instead. + [[], [], [2, 1]] + Please use :meth:`quotient` instead:: + + sage: Partition([6,3,2,2]).quotient(3) + [[], [], [2, 1]] + """ + from sage.misc.misc import deprecation + deprecation('r_quotient is deprecated. Please use quotient instead.') + return self.quotient(length) def quotient(self, length): """ Returns the quotient of the partition -- in the literature the - core is commonly referred to as the k-core, p-core, r-core, ... . The - quotient is a list of r partitions, constructed in the following - way. Label each cell in p with its content, modulo r. Let R_i be - the set of rows ending in a box labelled i, and C_i be the set of - columns ending in a box labelled i. Then the jth component of the - quotient of p is the partition defined by intersecting R_j with - C_j+1. - + core is commonly referred to as the `k`-core, `p`-core, `r`-core, ... . The + quotient is a list of `r` partitions, constructed in the following + way. Label each cell in `p` with its content, modulo `r`. Let `R_i` be + the set of rows ending in a box labelled `i`, and `C_i` be the set of + columns ending in a box labelled `i`. Then the `j`-th component of the + quotient of `p` is the partition defined by intersecting `R_j` with + `C_j+1`. + EXAMPLES:: sage: Partition([7,7,5,3,3,3,1]).quotient(3) @@ -4251,7 +4251,7 @@ def partition_power(pi,k): """ partition_power( pi, k ) returns the partition corresponding to - the `k`-th power of a permutation with cycle structure pi + the `k`-th power of a permutation with cycle structure ``pi`` (thus describes the powermap of symmetric groups). Wraps GAP's PowerPartition. @@ -4295,11 +4295,11 @@ is exactly the alternating group `A_n`. Partitions of sign `1` are called even partitions while partitions of sign `-1` are called odd. - + This function is deprecated: use Partition( pi ).sign() instead. EXAMPLES:: - + sage: Partition([5,3]).sign() 1 sage: Partition([5,2]).sign() @@ -4390,4 +4390,4 @@ from sage.misc.misc import deprecation deprecation('"partition_associated deprecated. Use Partition(pi).conjugte() instead') return Partition(pi).conjugate() - +