***************************************************************************** * 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: St000474 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: Dyson's crank of a partition. Let $\lambda$ be a partition and let $o(\lambda)$ be the number of parts that are equal to 1 ([[St000475]]), and let $\mu(\lambda)$ be the number of parts that are strictly larger than $o(\lambda)$ ([[St000473]]). Dyson's crank is then defined as $$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$$ ----------------------------------------------------------------------------- References: [1] Andrews, G. E., Garvan, F. G. Dyson's crank of a partition [[MathSciNet:0929094]] ----------------------------------------------------------------------------- Code: def statistic(L): ones = list(L).count(1) if ones == 0: return L[0] else: return sum(1 for part in L if part > ones) - ones ----------------------------------------------------------------------------- Statistic values: [1] => -1 [2] => 2 [1,1] => -2 [3] => 3 [2,1] => 0 [1,1,1] => -3 [4] => 4 [3,1] => 0 [2,2] => 2 [2,1,1] => -2 [1,1,1,1] => -4 [5] => 5 [4,1] => 0 [3,2] => 3 [3,1,1] => -1 [2,2,1] => 1 [2,1,1,1] => -3 [1,1,1,1,1] => -5 [6] => 6 [5,1] => 0 [4,2] => 4 [4,1,1] => -1 [3,3] => 3 [3,2,1] => 1 [3,1,1,1] => -3 [2,2,2] => 2 [2,2,1,1] => -2 [2,1,1,1,1] => -4 [1,1,1,1,1,1] => -6 [7] => 7 [6,1] => 0 [5,2] => 5 [5,1,1] => -1 [4,3] => 4 [4,2,1] => 1 [4,1,1,1] => -2 [3,3,1] => 1 [3,2,2] => 3 [3,2,1,1] => -1 [3,1,1,1,1] => -4 [2,2,2,1] => 2 [2,2,1,1,1] => -3 [2,1,1,1,1,1] => -5 [1,1,1,1,1,1,1] => -7 [8] => 8 [7,1] => 0 [6,2] => 6 [6,1,1] => -1 [5,3] => 5 [5,2,1] => 1 [5,1,1,1] => -2 [4,4] => 4 [4,3,1] => 1 [4,2,2] => 4 [4,2,1,1] => -1 [4,1,1,1,1] => -4 [3,3,2] => 3 [3,3,1,1] => 0 [3,2,2,1] => 2 [3,2,1,1,1] => -3 [3,1,1,1,1,1] => -5 [2,2,2,2] => 2 [2,2,2,1,1] => -2 [2,2,1,1,1,1] => -4 [2,1,1,1,1,1,1] => -6 [1,1,1,1,1,1,1,1] => -8 [9] => 9 [8,1] => 0 [7,2] => 7 [7,1,1] => -1 [6,3] => 6 [6,2,1] => 1 [6,1,1,1] => -2 [5,4] => 5 [5,3,1] => 1 [5,2,2] => 5 [5,2,1,1] => -1 [5,1,1,1,1] => -3 [4,4,1] => 1 [4,3,2] => 4 [4,3,1,1] => 0 [4,2,2,1] => 2 [4,2,1,1,1] => -2 [4,1,1,1,1,1] => -5 [3,3,3] => 3 [3,3,2,1] => 2 [3,3,1,1,1] => -3 [3,2,2,2] => 3 [3,2,2,1,1] => -1 [3,2,1,1,1,1] => -4 [3,1,1,1,1,1,1] => -6 [2,2,2,2,1] => 3 [2,2,2,1,1,1] => -3 [2,2,1,1,1,1,1] => -5 [2,1,1,1,1,1,1,1] => -7 [1,1,1,1,1,1,1,1,1] => -9 [10] => 10 [9,1] => 0 [8,2] => 8 [8,1,1] => -1 [7,3] => 7 [7,2,1] => 1 [7,1,1,1] => -2 [6,4] => 6 [6,3,1] => 1 [6,2,2] => 6 [6,2,1,1] => -1 [6,1,1,1,1] => -3 [5,5] => 5 [5,4,1] => 1 [5,3,2] => 5 [5,3,1,1] => 0 [5,2,2,1] => 2 [5,2,1,1,1] => -2 [5,1,1,1,1,1] => -5 [4,4,2] => 4 [4,4,1,1] => 0 [4,3,3] => 4 [4,3,2,1] => 2 [4,3,1,1,1] => -2 [4,2,2,2] => 4 [4,2,2,1,1] => -1 [4,2,1,1,1,1] => -4 [4,1,1,1,1,1,1] => -6 [3,3,3,1] => 2 [3,3,2,2] => 3 [3,3,2,1,1] => 0 [3,3,1,1,1,1] => -4 [3,2,2,2,1] => 3 [3,2,2,1,1,1] => -3 [3,2,1,1,1,1,1] => -5 [3,1,1,1,1,1,1,1] => -7 [2,2,2,2,2] => 2 [2,2,2,2,1,1] => -2 [2,2,2,1,1,1,1] => -4 [2,2,1,1,1,1,1,1] => -6 [2,1,1,1,1,1,1,1,1] => -8 [1,1,1,1,1,1,1,1,1,1] => -10 [11] => 11 [10,1] => 0 [9,2] => 9 [9,1,1] => -1 [8,3] => 8 [8,2,1] => 1 [8,1,1,1] => -2 [7,4] => 7 [7,3,1] => 1 [7,2,2] => 7 [7,2,1,1] => -1 [7,1,1,1,1] => -3 [6,5] => 6 [6,4,1] => 1 [6,3,2] => 6 [6,3,1,1] => 0 [6,2,2,1] => 2 [6,2,1,1,1] => -2 [6,1,1,1,1,1] => -4 [5,5,1] => 1 [5,4,2] => 5 [5,4,1,1] => 0 [5,3,3] => 5 [5,3,2,1] => 2 [5,3,1,1,1] => -2 [5,2,2,2] => 5 [5,2,2,1,1] => -1 [5,2,1,1,1,1] => -3 [5,1,1,1,1,1,1] => -6 [4,4,3] => 4 [4,4,2,1] => 2 [4,4,1,1,1] => -1 [4,3,3,1] => 2 [4,3,2,2] => 4 [4,3,2,1,1] => 0 [4,3,1,1,1,1] => -4 [4,2,2,2,1] => 3 [4,2,2,1,1,1] => -2 [4,2,1,1,1,1,1] => -5 [4,1,1,1,1,1,1,1] => -7 [3,3,3,2] => 3 [3,3,3,1,1] => 1 [3,3,2,2,1] => 3 [3,3,2,1,1,1] => -3 [3,3,1,1,1,1,1] => -5 [3,2,2,2,2] => 3 [3,2,2,2,1,1] => -1 [3,2,2,1,1,1,1] => -4 [3,2,1,1,1,1,1,1] => -6 [3,1,1,1,1,1,1,1,1] => -8 [2,2,2,2,2,1] => 4 [2,2,2,2,1,1,1] => -3 [2,2,2,1,1,1,1,1] => -5 [2,2,1,1,1,1,1,1,1] => -7 [2,1,1,1,1,1,1,1,1,1] => -9 [1,1,1,1,1,1,1,1,1,1,1] => -11 [12] => 12 [11,1] => 0 [10,2] => 10 [10,1,1] => -1 [9,3] => 9 [9,2,1] => 1 [9,1,1,1] => -2 [8,4] => 8 [8,3,1] => 1 [8,2,2] => 8 [8,2,1,1] => -1 [8,1,1,1,1] => -3 [7,5] => 7 [7,4,1] => 1 [7,3,2] => 7 [7,3,1,1] => 0 [7,2,2,1] => 2 [7,2,1,1,1] => -2 [7,1,1,1,1,1] => -4 [6,6] => 6 [6,5,1] => 1 [6,4,2] => 6 [6,4,1,1] => 0 [6,3,3] => 6 [6,3,2,1] => 2 [6,3,1,1,1] => -2 [6,2,2,2] => 6 [6,2,2,1,1] => -1 [6,2,1,1,1,1] => -3 [6,1,1,1,1,1,1] => -6 [5,5,2] => 5 [5,5,1,1] => 0 [5,4,3] => 5 [5,4,2,1] => 2 [5,4,1,1,1] => -1 [5,3,3,1] => 2 [5,3,2,2] => 5 [5,3,2,1,1] => 0 [5,3,1,1,1,1] => -3 [5,2,2,2,1] => 3 [5,2,2,1,1,1] => -2 [5,2,1,1,1,1,1] => -5 [5,1,1,1,1,1,1,1] => -7 [4,4,4] => 4 [4,4,3,1] => 2 [4,4,2,2] => 4 [4,4,2,1,1] => 0 [4,4,1,1,1,1] => -4 [4,3,3,2] => 4 [4,3,3,1,1] => 1 [4,3,2,2,1] => 3 [4,3,2,1,1,1] => -2 [4,3,1,1,1,1,1] => -5 [4,2,2,2,2] => 4 [4,2,2,2,1,1] => -1 [4,2,2,1,1,1,1] => -4 [4,2,1,1,1,1,1,1] => -6 [4,1,1,1,1,1,1,1,1] => -8 [3,3,3,3] => 3 [3,3,3,2,1] => 3 [3,3,3,1,1,1] => -3 [3,3,2,2,2] => 3 [3,3,2,2,1,1] => 0 [3,3,2,1,1,1,1] => -4 [3,3,1,1,1,1,1,1] => -6 [3,2,2,2,2,1] => 4 [3,2,2,2,1,1,1] => -3 [3,2,2,1,1,1,1,1] => -5 [3,2,1,1,1,1,1,1,1] => -7 [3,1,1,1,1,1,1,1,1,1] => -9 [2,2,2,2,2,2] => 2 [2,2,2,2,2,1,1] => -2 [2,2,2,2,1,1,1,1] => -4 [2,2,2,1,1,1,1,1,1] => -6 [2,2,1,1,1,1,1,1,1,1] => -8 [2,1,1,1,1,1,1,1,1,1,1] => -10 [1,1,1,1,1,1,1,1,1,1,1,1] => -12 ----------------------------------------------------------------------------- Created: Apr 19, 2016 at 10:47 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Oct 29, 2017 at 21:22 by Martin Rubey