***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information 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. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St000046 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. ----------------------------------------------------------------------------- References: [1] Reyes, J.-C. U. Random walk, semi-direct products, and card shuffling [[MathSciNet:2703300]] ----------------------------------------------------------------------------- Code: class RandomToRandom: def __init__(self, n): self._n = n @cached_method def W(self): return SymmetricGroup(self._n) def cycle(self, i,j): """ EXAMPLES:: sage: RandomToRandom(5).cycle(2,4) (2,3,4) """ return self.W()([tuple(range(i,j+1))]) def r2r(self, i,j): """ EXAMPLES:: sage: R = RandomToRandom(4) sage: R.r2r(1,3,4) (1,2,4) sage: R.r2r(3,1,4) (1,4,2) sage: r2r(1,2,4) (1,4) sage: RandomToRandom(3).r2r(1,2,3) (1,3) """ return self.cycle(i,self._n) * (~self.cycle(j,self._n)) def operator(self, representation): """ EXAMPLES:: sage: R = RandomToRandom(3) sage: representation = attrcall("matrix") # emulates the natural representation sage: R.operator(representation) [5 1 3] [1 5 3] [3 3 3] sage: representation = R.W().algebra(QQ).monomial # emulates the regular representation sage: R.operator(representation) 3*B[()] + 2*B[(2,3)] + B[(1,2,3)] + B[(1,3,2)] + 2*B[(1,3)] sage: R.operator(Partition([2,1])) [2 2] [2 2] """ if isinstance(representation, Partition): assert representation.size() == self._n representation = SymmetricGroupRepresentation(representation) E = self.W().domain() return sum(representation(self.r2r(i,j)) for i in E for j in E) def max_eigenvalue_on_simple_representation(self, p): return max(self.operator(p).eigenvalues()) def max_eigenvalue_on_simple_representations(self): """ EXAMPLES:: sage: R = RandomToRandom(4) sage: dict(R.max_eigenvalue_on_simple_representations()) {[1, 1, 1, 1]: 0, [2, 1, 1]: 6, [2, 2]: 4, [3, 1]: 10, [4]: 16} sage: R = RandomToRandom(5) sage: dict(R.max_eigenvalue_on_simple_representations()) {[1, 1, 1, 1]: 0, [2, 1, 1]: 6, [2, 2]: 4, [3, 1]: 10, [4]: 16} """ from sage.sets.family import Family return Family(Partitions(self._n), self.max_eigenvalue_on_simple_representation) @cached_function def getRandomToRandom(n): return RandomToRandom(n) def statistic(L): n = sum(L) R = getRandomToRandom(n) d = R.max_eigenvalue_on_simple_representations() return d[L] ----------------------------------------------------------------------------- Statistic values: [1] => 1 [2] => 4 [1,1] => 0 [3] => 9 [2,1] => 4 [1,1,1] => 1 [4] => 16 [3,1] => 10 [2,2] => 4 [2,1,1] => 6 [1,1,1,1] => 0 [5] => 25 [4,1] => 18 [3,2] => 11 [3,1,1] => 13 [2,2,1] => 7 [2,1,1,1] => 6 [1,1,1,1,1] => 1 [6] => 36 [5,1] => 28 [4,2] => 20 [4,1,1] => 22 [3,3] => 12 [3,2,1] => 15 [3,1,1,1] => 14 [2,2,2] => 8 [2,2,1,1] => 8 [2,1,1,1,1] => 8 [1,1,1,1,1,1] => 0 [7] => 49 [6,1] => 40 [5,2] => 31 [5,1,1] => 33 [4,3] => 22 [4,2,1] => 25 [4,1,1,1] => 24 [3,3,1] => 17 [3,2,2] => 17 [3,2,1,1] => 17 [3,1,1,1,1] => 17 [2,2,2,1] => 9 [2,2,1,1,1] => 9 [2,1,1,1,1,1] => 8 [1,1,1,1,1,1,1] => 1 [8] => 64 [7,1] => 54 [6,2] => 44 [6,1,1] => 46 [5,3] => 34 [5,2,1] => 37 [5,1,1,1] => 36 [4,4] => 24 [4,3,1] => 28 [4,2,2] => 28 [4,2,1,1] => 28 [4,1,1,1,1] => 28 [3,3,2] => 19 [3,3,1,1] => 19 [3,2,2,1] => 19 [3,2,1,1,1] => 19 [3,1,1,1,1,1] => 18 [2,2,2,2] => 10 [2,2,2,1,1] => 10 [2,2,1,1,1,1] => 10 [2,1,1,1,1,1,1] => 10 [1,1,1,1,1,1,1,1] => 0 [9] => 81 [8,1] => 70 [7,2] => 59 [7,1,1] => 61 [6,3] => 48 [6,2,1] => 51 [6,1,1,1] => 50 [5,4] => 37 [5,3,1] => 41 [5,2,2] => 41 ----------------------------------------------------------------------------- Created: Mar 22, 2013 at 22:05 by Nicolas M. ThiĆ©ry ----------------------------------------------------------------------------- Last Updated: Dec 29, 2016 at 09:25 by Christian Stump