***************************************************************************** * 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: St000184 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The size of the centralizer of any permutation of given cycle type. The centralizer (or commutant, equivalently normalizer) of an element $g$ of a group $G$ is the set of elements of $G$ that commute with $g$: $$C_g = \{h \in G : hgh^{-1} = g\}.$$ Its size thus depends only on the conjugacy class of $g$. The conjugacy classes of a permutation is determined by its cycle type, and the size of the centralizer of a permutation with cycle type $\lambda = (1^{a_1},2^{a_2},\dots)$ is $$|C| = \Pi j^{a_j} a_j!$$ For example, for any permutation with cycle type $\lambda = (3,2,2,1)$, $$|C| = (3^1 \cdot 1!)(2^2 \cdot 2!)(1^1 \cdot 1!) = 24.$$ There is exactly one permutation of the empty set, the identity, so the statistic on the empty partition is $1$. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(p): return p.centralizer_size() ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 2 [1,1] => 2 [3] => 3 [2,1] => 2 [1,1,1] => 6 [4] => 4 [3,1] => 3 [2,2] => 8 [2,1,1] => 4 [1,1,1,1] => 24 [5] => 5 [4,1] => 4 [3,2] => 6 [3,1,1] => 6 [2,2,1] => 8 [2,1,1,1] => 12 [1,1,1,1,1] => 120 [6] => 6 [5,1] => 5 [4,2] => 8 [4,1,1] => 8 [3,3] => 18 [3,2,1] => 6 [3,1,1,1] => 18 [2,2,2] => 48 [2,2,1,1] => 16 [2,1,1,1,1] => 48 [1,1,1,1,1,1] => 720 [7] => 7 [6,1] => 6 [5,2] => 10 [5,1,1] => 10 [4,3] => 12 [4,2,1] => 8 [4,1,1,1] => 24 [3,3,1] => 18 [3,2,2] => 24 [3,2,1,1] => 12 [3,1,1,1,1] => 72 [2,2,2,1] => 48 [2,2,1,1,1] => 48 [2,1,1,1,1,1] => 240 [1,1,1,1,1,1,1] => 5040 [8] => 8 [7,1] => 7 [6,2] => 12 [6,1,1] => 12 [5,3] => 15 [5,2,1] => 10 [5,1,1,1] => 30 [4,4] => 32 [4,3,1] => 12 [4,2,2] => 32 [4,2,1,1] => 16 [4,1,1,1,1] => 96 [3,3,2] => 36 [3,3,1,1] => 36 [3,2,2,1] => 24 [3,2,1,1,1] => 36 [3,1,1,1,1,1] => 360 [2,2,2,2] => 384 [2,2,2,1,1] => 96 [2,2,1,1,1,1] => 192 [2,1,1,1,1,1,1] => 1440 [1,1,1,1,1,1,1,1] => 40320 [9] => 9 [8,1] => 8 [7,2] => 14 [7,1,1] => 14 [6,3] => 18 [6,2,1] => 12 [6,1,1,1] => 36 [5,4] => 20 [5,3,1] => 15 [5,2,2] => 40 [5,2,1,1] => 20 [5,1,1,1,1] => 120 [4,4,1] => 32 [4,3,2] => 24 [4,3,1,1] => 24 [4,2,2,1] => 32 [4,2,1,1,1] => 48 [4,1,1,1,1,1] => 480 [3,3,3] => 162 [3,3,2,1] => 36 [3,3,1,1,1] => 108 [3,2,2,2] => 144 [3,2,2,1,1] => 48 [3,2,1,1,1,1] => 144 [3,1,1,1,1,1,1] => 2160 [2,2,2,2,1] => 384 [2,2,2,1,1,1] => 288 [2,2,1,1,1,1,1] => 960 [2,1,1,1,1,1,1,1] => 10080 [1,1,1,1,1,1,1,1,1] => 362880 [10] => 10 [9,1] => 9 [8,2] => 16 [8,1,1] => 16 [7,3] => 21 [7,2,1] => 14 [7,1,1,1] => 42 [6,4] => 24 [6,3,1] => 18 [6,2,2] => 48 [6,2,1,1] => 24 [6,1,1,1,1] => 144 [5,5] => 50 [5,4,1] => 20 [5,3,2] => 30 [5,3,1,1] => 30 [5,2,2,1] => 40 [5,2,1,1,1] => 60 [5,1,1,1,1,1] => 600 [4,4,2] => 64 [4,4,1,1] => 64 [4,3,3] => 72 [4,3,2,1] => 24 [4,3,1,1,1] => 72 [4,2,2,2] => 192 [4,2,2,1,1] => 64 [4,2,1,1,1,1] => 192 [4,1,1,1,1,1,1] => 2880 [3,3,3,1] => 162 [3,3,2,2] => 144 [3,3,2,1,1] => 72 [3,3,1,1,1,1] => 432 [3,2,2,2,1] => 144 [3,2,2,1,1,1] => 144 [3,2,1,1,1,1,1] => 720 [3,1,1,1,1,1,1,1] => 15120 [2,2,2,2,2] => 3840 [2,2,2,2,1,1] => 768 [2,2,2,1,1,1,1] => 1152 [2,2,1,1,1,1,1,1] => 5760 [2,1,1,1,1,1,1,1,1] => 80640 [1,1,1,1,1,1,1,1,1,1] => 3628800 ----------------------------------------------------------------------------- Created: May 04, 2014 at 23:41 by Lahiru Kariyawasam ----------------------------------------------------------------------------- Last Updated: Oct 29, 2017 at 16:33 by Martin Rubey