***************************************************************************** * 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: St000749 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields $$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3. This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2]. ----------------------------------------------------------------------------- References: [1] Irie, Y. p-Saturations of Welter's Game and the Irreducible Representations of Symmetric Groups [[arXiv:1604.07214]] ----------------------------------------------------------------------------- Code: def branching_symmetric_group(la, p): """ Return a dictionary from partitions to multiplicities. """ la = Partition(la) l = {la: 1} for i in range(la.size()-p): l_new = dict() for mu in l: for r, _ in mu.removable_cells(): nu = mu.remove_cell(r) l_new[nu] = l_new.get(nu, 0) + l[mu] l = l_new return l def statistic(la): """Return the largest number such that the restriction of the irreducible representation corresponding to la has a component relative prime to 2. """ la = Partition(la) for m in range(la.size(), 0, -1): if any(gcd(StandardTableaux(mu).cardinality(), 2) == 1 for mu in branching_symmetric_group(la, m)): return la.size()-m ----------------------------------------------------------------------------- Statistic values: [1] => 0 [2] => 0 [1,1] => 0 [3] => 0 [2,1] => 1 [1,1,1] => 0 [4] => 0 [3,1] => 0 [2,2] => 2 [2,1,1] => 0 [1,1,1,1] => 0 [5] => 0 [4,1] => 1 [3,2] => 0 [3,1,1] => 1 [2,2,1] => 0 [2,1,1,1] => 1 [1,1,1,1,1] => 0 [6] => 0 [5,1] => 0 [4,2] => 0 [4,1,1] => 2 [3,3] => 0 [3,2,1] => 1 [3,1,1,1] => 2 [2,2,2] => 0 [2,2,1,1] => 0 [2,1,1,1,1] => 0 [1,1,1,1,1,1] => 0 [7] => 0 [6,1] => 1 [5,2] => 1 [5,1,1] => 0 [4,3] => 1 [4,2,1] => 0 [4,1,1,1] => 3 [3,3,1] => 0 [3,2,2] => 0 [3,2,1,1] => 0 [3,1,1,1,1] => 0 [2,2,2,1] => 1 [2,2,1,1,1] => 1 [2,1,1,1,1,1] => 1 [1,1,1,1,1,1,1] => 0 [8] => 0 [7,1] => 0 [6,2] => 2 [6,1,1] => 0 [5,3] => 2 [5,2,1] => 1 [5,1,1,1] => 0 [4,4] => 2 [4,3,1] => 1 [4,2,2] => 1 [4,2,1,1] => 1 [4,1,1,1,1] => 0 [3,3,2] => 1 [3,3,1,1] => 1 [3,2,2,1] => 1 [3,2,1,1,1] => 1 [3,1,1,1,1,1] => 0 [2,2,2,2] => 2 [2,2,2,1,1] => 2 [2,2,1,1,1,1] => 2 [2,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1] => 0 [9] => 0 [8,1] => 1 [7,2] => 0 [7,1,1] => 1 [6,3] => 3 [6,2,1] => 0 [6,1,1,1] => 1 [5,4] => 3 [5,3,1] => 2 [5,2,2] => 2 [5,2,1,1] => 0 [5,1,1,1,1] => 1 [4,4,1] => 2 [4,3,2] => 2 [4,3,1,1] => 2 [4,2,2,1] => 2 [4,2,1,1,1] => 0 [4,1,1,1,1,1] => 1 [3,3,3] => 2 [3,3,2,1] => 2 [3,3,1,1,1] => 2 [3,2,2,2] => 2 [3,2,2,1,1] => 2 [3,2,1,1,1,1] => 0 [3,1,1,1,1,1,1] => 1 [2,2,2,2,1] => 3 [2,2,2,1,1,1] => 3 [2,2,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1] => 0 [10] => 0 [9,1] => 0 [8,2] => 0 [8,1,1] => 2 [7,3] => 0 [7,2,1] => 1 [7,1,1,1] => 2 [6,4] => 4 [6,3,1] => 0 [6,2,2] => 0 [6,2,1,1] => 1 [6,1,1,1,1] => 2 [5,5] => 4 [5,4,1] => 3 [5,3,2] => 3 [5,3,1,1] => 0 [5,2,2,1] => 0 [5,2,1,1,1] => 1 [5,1,1,1,1,1] => 2 [4,4,2] => 3 [4,4,1,1] => 3 [4,3,3] => 3 [4,3,2,1] => 3 [4,3,1,1,1] => 0 [4,2,2,2] => 3 [4,2,2,1,1] => 0 [4,2,1,1,1,1] => 1 [4,1,1,1,1,1,1] => 2 [3,3,3,1] => 3 [3,3,2,2] => 3 [3,3,2,1,1] => 3 [3,3,1,1,1,1] => 0 [3,2,2,2,1] => 3 [3,2,2,1,1,1] => 0 [3,2,1,1,1,1,1] => 1 [3,1,1,1,1,1,1,1] => 2 [2,2,2,2,2] => 4 [2,2,2,2,1,1] => 4 [2,2,2,1,1,1,1] => 0 [2,2,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1] => 0 [11] => 0 [10,1] => 1 [9,2] => 1 [9,1,1] => 0 [8,3] => 1 [8,2,1] => 0 [8,1,1,1] => 3 [7,4] => 0 [7,3,1] => 1 [7,2,2] => 0 [7,2,1,1] => 2 [7,1,1,1,1] => 3 [6,5] => 5 [6,4,1] => 0 [6,3,2] => 1 [6,3,1,1] => 1 [6,2,2,1] => 1 [6,2,1,1,1] => 2 [6,1,1,1,1,1] => 3 [5,5,1] => 4 [5,4,2] => 4 [5,4,1,1] => 0 [5,3,3] => 4 [5,3,2,1] => 1 [5,3,1,1,1] => 1 [5,2,2,2] => 0 [5,2,2,1,1] => 1 [5,2,1,1,1,1] => 2 [5,1,1,1,1,1,1] => 3 [4,4,3] => 4 [4,4,2,1] => 4 [4,4,1,1,1] => 0 [4,3,3,1] => 4 [4,3,2,2] => 4 [4,3,2,1,1] => 1 [4,3,1,1,1,1] => 1 [4,2,2,2,1] => 0 [4,2,2,1,1,1] => 1 [4,2,1,1,1,1,1] => 2 [4,1,1,1,1,1,1,1] => 3 [3,3,3,2] => 4 [3,3,3,1,1] => 4 [3,3,2,2,1] => 4 [3,3,2,1,1,1] => 1 [3,3,1,1,1,1,1] => 0 [3,2,2,2,2] => 4 [3,2,2,2,1,1] => 0 [3,2,2,1,1,1,1] => 1 [3,2,1,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1,1] => 0 [2,2,2,2,2,1] => 5 [2,2,2,2,1,1,1] => 0 [2,2,2,1,1,1,1,1] => 1 [2,2,1,1,1,1,1,1,1] => 1 [2,1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1,1] => 0 [12] => 0 [11,1] => 0 [10,2] => 2 [10,1,1] => 0 [9,3] => 2 [9,2,1] => 1 [9,1,1,1] => 0 [8,4] => 0 [8,3,1] => 0 [8,2,2] => 1 [8,2,1,1] => 0 [8,1,1,1,1] => 4 [7,5] => 0 [7,4,1] => 1 [7,3,2] => 0 [7,3,1,1] => 2 [7,2,2,1] => 0 [7,2,1,1,1] => 3 [7,1,1,1,1,1] => 4 [6,6] => 6 [6,5,1] => 0 [6,4,2] => 0 [6,4,1,1] => 1 [6,3,3] => 2 [6,3,2,1] => 2 [6,3,1,1,1] => 2 [6,2,2,2] => 0 [6,2,2,1,1] => 2 [6,2,1,1,1,1] => 3 [6,1,1,1,1,1,1] => 4 [5,5,2] => 5 [5,5,1,1] => 0 [5,4,3] => 5 [5,4,2,1] => 0 [5,4,1,1,1] => 1 [5,3,3,1] => 2 [5,3,2,2] => 0 [5,3,2,1,1] => 2 [5,3,1,1,1,1] => 2 [5,2,2,2,1] => 1 [5,2,2,1,1,1] => 2 [5,2,1,1,1,1,1] => 3 [5,1,1,1,1,1,1,1] => 4 [4,4,4] => 5 [4,4,3,1] => 5 [4,4,2,2] => 5 [4,4,2,1,1] => 0 [4,4,1,1,1,1] => 0 [4,3,3,2] => 5 [4,3,3,1,1] => 2 [4,3,2,2,1] => 0 [4,3,2,1,1,1] => 2 [4,3,1,1,1,1,1] => 0 [4,2,2,2,2] => 0 [4,2,2,2,1,1] => 1 [4,2,2,1,1,1,1] => 2 [4,2,1,1,1,1,1,1] => 0 [4,1,1,1,1,1,1,1,1] => 0 [3,3,3,3] => 5 [3,3,3,2,1] => 5 [3,3,3,1,1,1] => 2 [3,3,2,2,2] => 5 [3,3,2,2,1,1] => 0 [3,3,2,1,1,1,1] => 0 [3,3,1,1,1,1,1,1] => 1 [3,2,2,2,2,1] => 0 [3,2,2,2,1,1,1] => 1 [3,2,2,1,1,1,1,1] => 0 [3,2,1,1,1,1,1,1,1] => 1 [3,1,1,1,1,1,1,1,1,1] => 0 [2,2,2,2,2,2] => 6 [2,2,2,2,2,1,1] => 0 [2,2,2,2,1,1,1,1] => 0 [2,2,2,1,1,1,1,1,1] => 2 [2,2,1,1,1,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1,1] => 0 [5,4,3,1] => 0 [5,4,2,2] => 1 [5,4,2,1,1] => 1 [5,3,3,2] => 0 [5,3,3,1,1] => 3 [5,3,2,2,1] => 1 [4,4,3,2] => 6 [4,4,3,1,1] => 0 [4,4,2,2,1] => 1 [4,3,3,2,1] => 0 [5,4,3,2] => 1 [5,4,3,1,1] => 1 [5,4,2,2,1] => 2 [5,3,3,2,1] => 1 [4,4,3,2,1] => 1 [5,4,3,2,1] => 2 ----------------------------------------------------------------------------- Created: Apr 05, 2017 at 11:28 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Sep 14, 2018 at 18:56 by Martin Rubey