# HG changeset patch # User Jason Bandlow # Date 1245094287 14400 # Node ID 8c7e07da06802ea104ec774369cec32cf96acf90 # Parent 6c7354ace3b6e0bf4c33ddb730531bd23a9fb92c imported patch partition-update-AM.patch diff -r 6c7354ace3b6 -r 8c7e07da0680 sage/combinat/partition.py --- a/sage/combinat/partition.py Sat Jun 06 09:55:28 2009 -0700 +++ b/sage/combinat/partition.py Mon Jun 15 15:31:27 2009 -0400 @@ -187,21 +187,21 @@ sage: Partition([4,3,1]).corners() [[0, 3], [1, 2], [2, 0]] -We can compute the r-core and r-quotient of a partition and build +We can compute the core and quotient of a partition and build the partition back up from them. :: - sage: Partition([6,3,2,2]).r_core(3) + sage: Partition([6,3,2,2]).core(3) [2, 1, 1] - sage: Partition([7,7,5,3,3,3,1]).r_quotient(3) + sage: Partition([7,7,5,3,3,3,1]).quotient(3) [[2], [1], [2, 2, 2]] sage: p = Partition([11,5,5,3,2,2,2]) - sage: p.r_core(3) + sage: p.core(3) [] - sage: p.r_quotient(3) + sage: p.quotient(3) [[2, 1], [4], [1, 1, 1]] - sage: Partition(core_and_quotient=([],[[2, 1], [4], [1, 1, 1]])) + sage: Partition(core=[],quotient=[[2, 1], [4], [1, 1, 1]]) [11, 5, 5, 3, 2, 2, 2] """ #***************************************************************************** @@ -237,96 +237,135 @@ from integer_vector import IntegerVectors from cartesian_product import CartesianProduct from integer_list import IntegerListsLex - -def Partition(l=None, exp=None, core_and_quotient=None): - """ - Returns a partition object. - - Note that Sage uses the English convention for partitions and - tableaux. - - EXAMPLES:: - +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] sage: Partition(exp=[2,1,1]) [3, 2, 1, 1] - sage: Partition(core_and_quotient=([2,1], [[2,1],[3],[1,1,1]])) + sage: Partition(core=[2,1], quotient=[[2,1],[3],[1,1,1]]) [11, 5, 5, 3, 2, 2, 2] - sage: Partition([3,2,1]) - [3, 2, 1] sage: [2,1] in Partitions() True sage: [2,1,0] in Partitions() False - sage: Partition([2,1,0]) - [2, 1] sage: Partition([1,2,3]) Traceback (most recent call last): ... ValueError: [1, 2, 3] is not a valid partition """ - number_of_arguments = 0 - for arg in ['l', 'exp', 'core_and_quotient']: - if locals()[arg] is not None: - number_of_arguments += 1 - - if number_of_arguments != 1: - raise ValueError, "you must specify exactly one argument" - - if l is not None: - l = [i for i in l if i != 0] - if l in Partitions_all(): - return Partition_class(l) - else: - raise ValueError, "%s is not a valid partition"%l - elif exp is not None: - return from_exp(exp) - else: - return from_core_and_quotient(*core_and_quotient) - + if mu is not None and len(key_word)==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: + raise NotImplementedError + elif 'exp' in key_word and len(key_word)==1: + return Partition_class([]).from_exp(key_word['exp']) + elif 'core' in key_word and 'quotient' in key_word and len(key_word)==2: + return Partition_class([]).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: + raise NotImplementedError + elif 'core_and_quotient' in key_word and len(key_word)==1: + from sage.misc.misc import deprecation + deprecation('"core_and_quotient=(*)" is deprecated. Use "core=[*], quotient=[*]" instead.') + return Partition_class([]).from_core_and_quotient(*key_word['core_and_quotient']) + else: + raise ValueError, 'incorrect syntax for Partition()' def from_exp(a): """ + ** This function is being deprecated - use Partition(exp=*) instead ** Returns a partition from its list of multiplicities. EXAMPLES:: - sage: partition.from_exp([1,2,1]) + sage: Partition(exp=[1,2,1]) [3, 2, 2, 1] """ - - p = [] - for i in reversed(range(len(a))): - p += [i+1]*a[i] - - return Partition(p) - + from sage.misc.misc import deprecation + deprecation('"from_exp" is deprecated. Use "Partition(exp=[*])" instead.') + return Partition(exp=a) def from_core_and_quotient(core, quotient): """ - Returns a partition from its r-core and r-quotient. + ** This function is being deprecated - use Partition(core=*, quotient=*) instead ** + + Returns a partition from its core and quotient. Algorithm from mupad-combinat. EXAMPLES:: - sage: partition.from_core_and_quotient([2,1], [[2,1],[3],[1,1,1]]) + sage: Partition(core=[2,1], quotient=[[2,1],[3],[1,1,1]]) [11, 5, 5, 3, 2, 2, 2] """ - from sage.functions.all import ceil - length = len(quotient) - k = length*max( [ceil(len(core)/length),len(core)] + [len(q) for q in quotient] ) + length - v = [ core[i]-(i+1)+1 for i in range(len(core))] + [ -i + 1 for i in range(len(core)+1,k+1) ] - w = [ filter(lambda x: (x-i) % length == 0, v) for i in range(1, length+1) ] - new_w = [] - for i in range(length): - new_w += [ w[i][j] + length*quotient[i][j] for j in range(len(quotient[i]))] - new_w += [ w[i][j] for j in range(len(quotient[i]), len(w[i])) ] - new_w.sort() - new_w.reverse() - return filter(lambda x: x != 0, [new_w[i-1]+i-1 for i in range(1, len(new_w)+1)]) - + from sage.misc.misc import deprecation + deprecation('"from_core_and_quotient" is deprecated. Use "Partition(core=[*],quotient=[*])" instead.') + return Partition(core=core,quotient=quotient) class Partition_class(CombinatorialObject): + def from_exp(self,exp): + """ + Returns a partition from its list of multiplicities. + + EXAMPLES:: + + sage: Partition(exp=[1,2,1]) + [3, 2, 2, 1] + """ + p = [] + for i in reversed(range(len(exp))): + p += [i+1]*exp[i] + return Partition(p) + + def from_core_and_quotient(self, core, quotient): + """ + Returns a partition from its core and quotient. + + Algorithm from mupad-combinat. + + EXAMPLES:: + + sage: Partition(core=[2,1], quotient=[[2,1],[3],[1,1,1]]) + [11, 5, 5, 3, 2, 2, 2] + """ + length = len(quotient) + k = length*max( [ceil(len(core)/length),len(core)] + [len(q) for q in quotient] ) + length + v = [ core[i]-(i+1)+1 for i in range(len(core))] + [ -i + 1 for i in range(len(core)+1,k+1) ] + w = [ filter(lambda x: (x-i) % length == 0, v) for i in range(1, length+1) ] + new_w = [] + for i in range(length): + new_w += [ w[i][j] + length*quotient[i][j] for j in range(len(quotient[i]))] + new_w += [ w[i][j] for j in range(len(quotient[i]), len(w[i])) ] + new_w.sort() + new_w.reverse() + return Partition([new_w[i-1]+i-1 for i in range(1, len(new_w)+1)]) + + def ferrers_diagram(self): """ Return the Ferrers diagram of pi. @@ -344,18 +383,6 @@ ***** ** * - sage: pi = Partitions(10).list()[11] ## [6,1,1,1,1] - sage: print pi.ferrers_diagram() - ****** - * - * - * - * - sage: pi = Partitions(10).list()[8] ## [6, 3, 1] - sage: print pi.ferrers_diagram() - ****** - *** - * """ return '\n'.join(['*'*p for p in self]) @@ -507,8 +534,7 @@ def sign(self): r""" - partition_sign( pi ) returns the sign of a permutation with cycle - structure given by the partition pi. + Returns the sign of a permutation with cycle type of the partition. This function corresponds to a homomorphism from the symmetric group `S_n` into the cyclic group of order 2, whose kernel @@ -516,8 +542,6 @@ `1` are called even partitions while partitions of sign `-1` are called odd. - Wraps GAP's SignPartition. - EXAMPLES:: sage: Partition([5,3]).sign() @@ -577,8 +601,7 @@ - http://en.wikipedia.org/wiki/Zolotarev's_lemma """ - ans=gap.eval("SignPartition(%s)"%(self)) - return sage_eval(ans) + return (-1)**(self.size()-self.length()) def up(self): r""" @@ -791,7 +814,6 @@ sage: Partition([5,4,2,1,1,1]).associated().associated() [5, 4, 2, 1, 1, 1] """ - return self.conjugate() def arm(self, i, j): @@ -1415,28 +1437,35 @@ return res def r_core(self, length): - """ - Returns the r-core of the partition p. The construction of the - r-core can be visualized by repeatedly removing border strips of - size r from p until this is no longer possible. The remaining - partition is the r-core. - - EXAMPLES:: - - sage: Partition([6,3,2,2]).r_core(3) + """ *** deprecate *** """ + from sage.misc.misc import deprecation + deprecation('r_core is deprecated. Use core instead.') + return self.core(self, 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 + construction of the core can be visualized by repeatedly removing + border strips of size r from p 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([]).r_core(3) - [] - sage: Partition([8,7,7,4,1,1,1,1,1]).r_core(3) + 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]).r_core(3) - [] - sage: Partition([10,8,7,7]).r_core(4) - [] - sage: Partition([21,15,15,9,6,6,6,3,3]).r_core(3) + sage: Partition([3,3,3,2,1]).core(3) + [] + sage: Partition([10,8,7,7]).core(4) + [] + sage: Partition([21,15,15,9,6,6,6,3,3]).core(3) [] """ p = self @@ -1462,36 +1491,45 @@ return filter(lambda x: x != 0, part) def r_quotient(self, length): - """ - Returns the r-quotient of the partition p. The r-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 r-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]).r_quotient(3) + """ *** deprecate *** """ + from sage.misc.misc import deprecation + deprecation('r_quotient is deprecated. Use quotient instead.') + return self.quotient(self,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. + + EXAMPLES:: + + sage: Partition([7,7,5,3,3,3,1]).quotient(3) [[2], [1], [2, 2, 2]] TESTS:: - sage: Partition([8,7,7,4,1,1,1,1,1]).r_quotient(3) + sage: Partition([8,7,7,4,1,1,1,1,1]).quotient(3) [[2, 1], [2, 2], [2]] - sage: Partition([10,8,7,7]).r_quotient(4) + sage: Partition([10,8,7,7]).quotient(4) [[2], [3], [2], [1]] - sage: Partition([6,3,3]).r_quotient(3) + sage: Partition([6,3,3]).quotient(3) [[1], [1], [2]] - sage: Partition([3,3,3,2,1]).r_quotient(3) + sage: Partition([3,3,3,2,1]).quotient(3) [[1], [1, 1], [1]] - sage: Partition([6,6,6,3,3,3]).r_quotient(3) + sage: Partition([6,6,6,3,3,3]).quotient(3) [[2, 1], [2, 1], [2, 1]] - sage: Partition([21,15,15,9,6,6,6,3,3]).r_quotient(3) + sage: Partition([21,15,15,9,6,6,6,3,3]).quotient(3) [[5, 2, 1], [5, 2, 1], [7, 3, 2]] - sage: Partition([21,15,15,9,6,6,3,3]).r_quotient(3) + sage: Partition([21,15,15,9,6,6,3,3]).quotient(3) [[5, 2], [5, 2, 1], [7, 3, 1]] - sage: Partition([14,12,11,10,10,10,10,9,6,4,3,3,2,1]).r_quotient(5) + sage: Partition([14,12,11,10,10,10,10,9,6,4,3,3,2,1]).quotient(5) [[3, 3], [2, 2, 1], [], [3, 3, 3], [1]] """ p = self @@ -3984,25 +4022,15 @@ EXAMPLES:: - sage: print ferrers_diagram([5,5,2,1]) + sage: print Partition([5,5,2,1]).ferrers_diagram() ***** ***** ** * - sage: pi = partitions_list(10)[30] ## [6,1,1,1,1] - sage: print ferrers_diagram(pi) - ****** - * - * - * - * - sage: pi = partitions_list(10)[33] ## [6, 3, 1] - sage: print ferrers_diagram(pi) - ****** - *** - * - """ - return '\n'.join(['*'*p for p in pi]) + """ + from sage.misc.misc import deprecation + deprecation('"ferrers_diagram deprecated. Use Partition(pi).ferrers_diagram() instead') + return Partition(pi).ferrers_diagram() def ordered_partitions(n,k=None): @@ -4277,7 +4305,9 @@ def partition_sign(pi): r""" - partition_sign( pi ) returns the sign of a permutation with cycle + ** This function is being deprecated -- use Partition(*).sign() instead. ** + + Partition( pi ).sign() returns the sign of a permutation with cycle structure given by the partition pi. This function corresponds to a homomorphism from the symmetric @@ -4285,14 +4315,14 @@ is exactly the alternating group `A_n`. Partitions of sign `1` are called even partitions while partitions of sign `-1` are called odd. - - Wraps GAP's SignPartition. - - EXAMPLES:: - - sage: partition_sign([5,3]) + + This function is deprecated: use Partition( pi ).sign() instead. + + EXAMPLES:: + + sage: Partition([5,3]).sign() 1 - sage: partition_sign([5,2]) + sage: Partition([5,2]).sign() -1 Zolotarev's lemma states that the Legendre symbol @@ -4322,7 +4352,7 @@ sage: p = PermutationGroupElement('(1, 2, 4, 8, 5, 10, 9, 7, 3, 6)') sage: p.sign() -1 - sage: partition_sign([10]) + sage: Partition([10]).sign() -1 sage: kronecker_symbol(11,2) -1 @@ -4338,7 +4368,7 @@ 1 sage: kronecker_symbol(3,11) 1 - sage: partition_sign([5,1,1,1,1,1]) + sage: Partition([5,1,1,1,1,1]).sign() 1 In both cases, Zolotarev holds. @@ -4347,26 +4377,29 @@ - http://en.wikipedia.org/wiki/Zolotarev's_lemma """ - ans=gap.eval("SignPartition(%s)"%(pi)) - return sage_eval(ans) + from sage.misc.misc import deprecation + deprecation('"partition_sign deprecated. Use Partition(pi).sign() instead') + return Partition(pi).sign() def partition_associated(pi): """ + ** This function is being deprecated -- use Partition(*).conjugate() instead. ** + partition_associated( pi ) returns the "associated" (also called "conjugate" in the literature) partition of the partition pi which is obtained by transposing the corresponding Ferrers diagram. EXAMPLES:: - sage: partition_associated([2,2]) + sage: Partition([2,2]).conjugate() [2, 2] - sage: partition_associated([6,3,1]) + sage: Partition([6,3,1]).conjugate() [3, 2, 2, 1, 1, 1] - sage: print ferrers_diagram([6,3,1]) + sage: print Partition([6,3,1]).ferrers_diagram() ****** *** * - sage: print ferrers_diagram([3,2,2,1,1,1]) + sage: print Partition([6,3,1]).conjugate().ferrers_diagram() *** ** ** @@ -4374,5 +4407,7 @@ * * """ - return list(Partition(pi).conjugate()) - + from sage.misc.misc import deprecation + deprecation('"partition_associated deprecated. Use Partition(pi).conjugte() instead') + return Partition(pi).conjugate() + diff -r 6c7354ace3b6 -r 8c7e07da0680 sage/combinat/ribbon_tableau.py --- a/sage/combinat/ribbon_tableau.py Sat Jun 06 09:55:28 2009 -0700 +++ b/sage/combinat/ribbon_tableau.py Mon Jun 15 15:31:27 2009 -0400 @@ -142,7 +142,7 @@ """ if shape in partition.Partitions(): shape = partition.Partition(shape) - shape = skew_partition.SkewPartition([shape, shape.r_core(length)]) + shape = skew_partition.SkewPartition([shape, shape.core(length)]) else: shape = skew_partition.SkewPartition(shape) @@ -880,7 +880,7 @@ sage: a = SkewPartition([[8,7,6,5,1,1],[2,1,1]]) sage: weight = [3,3,2] sage: k = 3 - sage: s = SemistandardMultiSkewTableaux(a.r_quotient(k),weight) + sage: s = SemistandardMultiSkewTableaux(a.quotient(k),weight) sage: len(s.list()) 34 sage: RibbonTableaux(a,weight,k).cardinality() diff -r 6c7354ace3b6 -r 8c7e07da0680 sage/combinat/sf/llt.py --- a/sage/combinat/sf/llt.py Sat Jun 06 09:55:28 2009 -0700 +++ b/sage/combinat/sf/llt.py Mon Jun 15 15:31:27 2009 -0400 @@ -146,14 +146,14 @@ elif isinstance(skp, list) and skp[0] in sage.combinat.skew_partition.SkewPartitions(): #skp is a list of skew partitions - skp = [sage.combinat.partition.Partition(core_and_quotient=([], skp[i][0])) for i in range(len(skp))] - skp += [sage.combinat.partition.Partition(core_and_quotient=([], skp[i][1])) for i in range(len(skp))] + skp = [sage.combinat.partition.Partition(core=[], quotient=skp[i][0]) for i in range(len(skp))] + skp += [sage.combinat.partition.Partition(core=[], quotient=skp[i][1]) for i in range(len(skp))] mu = sage.combinat.partition.Partitions(ZZ(sum( [ s.size() for s in skp] ) / self.level())) elif isinstance(skp, list) and skp[0] in sage.combinat.partition.Partitions(): #skp is a list of partitions - skp = sage.combinat.partition.Partition(core_and_quotient=([], skp)) + skp = sage.combinat.partition.Partition(core=[], quotient=skp) mu = sage.combinat.partition.Partitions( ZZ(sum(skp) / self.level() )) else: raise ValueError, "LLT polynomials not defined for %s"%skp diff -r 6c7354ace3b6 -r 8c7e07da0680 sage/combinat/skew_partition.py --- a/sage/combinat/skew_partition.py Sat Jun 06 09:55:28 2009 -0700 +++ b/sage/combinat/skew_partition.py Mon Jun 15 15:31:27 2009 -0400 @@ -452,22 +452,28 @@ return G - def r_quotient(self, k): + 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) + + def quotient(self, k): """ The quotient map extended to skew partitions. EXAMPLES:: - sage: SkewPartition([[3, 3, 2, 1], [2, 1]]).r_quotient(2) + sage: SkewPartition([[3, 3, 2, 1], [2, 1]]).quotient(2) [[[3], []], [[], []]] """ ## k-th element is the skew partition built using the k-th partition of the ## k-quotient of the outer and the inner partition. ## This bijection is only defined if the inner and the outer partition ## have the same core - if self.inner().r_core(k) == self.outer().r_core(k): - rqinner = self.inner().r_quotient(k) - rqouter = self.outer().r_quotient(k) + if self.inner().core(k) == self.outer().core(k): + rqinner = self.inner().quotient(k) + rqouter = self.outer().quotient(k) return [ SkewPartition_class([rqouter[i],rqinner[i]]) for i in range(k) ] else: raise ValueError, "quotient map is only defined for skew partitions with inner and outer partitions having the same core"