***************************************************************************** * 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: St000146 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The Andrews-Garvan crank of a partition. If $\pi$ is a partition, let $l(\pi)$ be its length (number of parts), $\omega(\pi)$ be the number of parts equal to 1, and $\mu(\pi)$ be the number of parts larger than $\omega(\pi)$. The crank is then defined by $$ c(\pi) = \begin{cases} l(\pi) &\text{if \(\omega(\pi)=0\)}\\ \mu(\pi) - \omega(\pi) &\text{otherwise}. \end{cases} $$ This statistic was defined in [1] to explain Ramanujan's partition congruence $$p(11n+6) \equiv 0 \pmod{11}$$ in the same way as the Dyson rank ([[St000145]]) explains the congruences $$p(5n+4) \equiv 0 \pmod{5}$$ and $$p(7n+5) \equiv 0 \pmod{7}.$$ ----------------------------------------------------------------------------- References: [1] Andrews, G. E., Garvan, F. G. Dyson's crank of a partition [[MathSciNet:0929094]] [2] [[wikipedia:Ramanujan's congruences]] ----------------------------------------------------------------------------- Code: def statistic(p): nb_ones = p.to_list().count(1) if nb_ones == 0: return len(p) else: return len([i for i in p if i > nb_ones]) - nb_ones ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => -1 [2] => 1 [1,1] => -2 [3] => 1 [2,1] => 0 [1,1,1] => -3 [4] => 1 [3,1] => 0 [2,2] => 2 [2,1,1] => -2 [1,1,1,1] => -4 [5] => 1 [4,1] => 0 [3,2] => 2 [3,1,1] => -1 [2,2,1] => 1 [2,1,1,1] => -3 [1,1,1,1,1] => -5 [6] => 1 [5,1] => 0 [4,2] => 2 [4,1,1] => -1 [3,3] => 2 [3,2,1] => 1 [3,1,1,1] => -3 [2,2,2] => 3 [2,2,1,1] => -2 [2,1,1,1,1] => -4 [1,1,1,1,1,1] => -6 [7] => 1 [6,1] => 0 [5,2] => 2 [5,1,1] => -1 [4,3] => 2 [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] => 1 [7,1] => 0 [6,2] => 2 [6,1,1] => -1 [5,3] => 2 [5,2,1] => 1 [5,1,1,1] => -2 [4,4] => 2 [4,3,1] => 1 [4,2,2] => 3 [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] => 4 [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] => 1 [8,1] => 0 [7,2] => 2 [7,1,1] => -1 [6,3] => 2 [6,2,1] => 1 [6,1,1,1] => -2 [5,4] => 2 [5,3,1] => 1 [5,2,2] => 3 [5,2,1,1] => -1 [5,1,1,1,1] => -3 [4,4,1] => 1 [4,3,2] => 3 [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] => 4 [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] => 1 [9,1] => 0 [8,2] => 2 [8,1,1] => -1 [7,3] => 2 [7,2,1] => 1 [7,1,1,1] => -2 [6,4] => 2 [6,3,1] => 1 [6,2,2] => 3 [6,2,1,1] => -1 [6,1,1,1,1] => -3 [5,5] => 2 [5,4,1] => 1 [5,3,2] => 3 [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] => 3 [4,4,1,1] => 0 [4,3,3] => 3 [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] => 4 [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] => 5 [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 [8,3] => 2 [7,4] => 2 [6,5] => 2 [6,4,1] => 1 [6,1,1,1,1,1] => -4 [5,5,1] => 1 [5,4,2] => 3 [5,4,1,1] => 0 [5,3,3] => 3 [5,3,2,1] => 2 [5,3,1,1,1] => -2 [5,2,2,2] => 4 [5,2,2,1,1] => -1 [5,2,1,1,1,1] => -3 [4,4,3] => 3 [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 [4,2,2,2,1] => 3 [3,3,3,2] => 4 [3,3,3,1,1] => 1 [3,3,2,2,1] => 3 [3,2,2,2,2] => 5 [2,2,2,2,2,1] => 4 [7,5] => 2 [7,4,1] => 1 [6,6] => 2 [6,4,2] => 3 [5,5,2] => 3 [5,4,3] => 3 [5,4,2,1] => 2 [5,4,1,1,1] => -1 [5,3,3,1] => 2 [5,3,2,2] => 4 [5,3,2,1,1] => 0 [5,2,2,2,1] => 3 [5,2,2,1,1,1] => -2 [4,4,4] => 3 [4,4,3,1] => 2 [4,4,2,2] => 4 [4,4,2,1,1] => 0 [4,3,3,2] => 4 [4,3,3,1,1] => 1 [4,3,2,2,1] => 3 [3,3,3,3] => 4 [3,3,3,2,1] => 3 [3,3,2,2,2] => 5 [3,3,2,2,1,1] => 0 [3,2,2,2,2,1] => 4 [2,2,2,2,2,2] => 6 [8,5] => 2 [7,5,1] => 1 [7,4,2] => 3 [5,5,3] => 3 [5,4,4] => 3 [5,4,3,1] => 2 [5,4,2,2] => 4 [5,4,2,1,1] => 0 [5,4,1,1,1,1] => -3 [5,3,3,2] => 4 [5,3,3,1,1] => 1 [5,3,2,2,1] => 3 [5,3,2,1,1,1] => -2 [4,4,4,1] => 2 [4,4,3,2] => 4 [4,4,3,1,1] => 1 [4,4,2,2,1] => 3 [4,3,3,3] => 4 [4,3,3,2,1] => 3 [3,3,3,3,1] => 3 [3,3,3,2,2] => 5 [3,3,2,2,2,1] => 4 [9,5] => 2 [8,5,1] => 1 [7,5,2] => 3 [7,4,3] => 3 [6,4,4] => 3 [6,2,2,2,2] => 5 [5,5,4] => 3 [5,5,1,1,1,1] => -2 [5,4,3,2] => 4 [5,4,3,1,1] => 1 [5,4,2,2,1] => 3 [5,4,2,1,1,1] => -1 [5,3,3,2,1] => 3 [5,3,2,2,2] => 5 [5,2,2,2,2,1] => 4 [4,4,4,2] => 4 [4,4,3,3] => 4 [4,4,3,2,1] => 3 [4,3,2,2,2,1] => 4 [3,3,3,3,2] => 5 [3,3,3,3,1,1] => 2 [9,5,1] => 1 [8,5,2] => 3 [7,5,3] => 3 [6,5,4] => 3 [6,5,1,1,1,1] => -2 [6,3,3,3] => 4 [6,2,2,2,2,1] => 4 [5,5,5] => 3 [5,4,3,2,1] => 3 [5,4,3,1,1,1] => -1 [5,3,2,2,2,1] => 4 [4,4,4,3] => 4 [4,4,4,1,1,1] => 0 [3,3,3,3,3] => 5 [3,3,3,3,2,1] => 4 [8,5,3] => 3 [7,5,3,1] => 2 [5,5,3,3] => 4 [5,5,2,2,2] => 5 [5,4,3,2,1,1] => 1 [5,4,2,2,2,1] => 4 [4,4,4,4] => 4 [4,4,4,2,2] => 5 [4,3,3,3,2,1] => 4 [8,6,3] => 3 [6,5,3,3] => 4 [6,5,2,2,2] => 5 [6,4,4,3] => 4 [6,4,4,1,1,1] => 0 [6,3,3,3,2] => 5 [6,3,3,3,1,1] => 2 [5,5,4,3] => 4 [5,5,4,1,1,1] => 0 [5,5,2,2,2,1] => 4 [5,4,3,2,2,1] => 4 [5,3,3,3,2,1] => 4 [4,4,4,3,2] => 5 [4,4,4,3,1,1] => 2 [4,4,4,2,2,1] => 4 [4,4,4,3,2,1] => 4 [5,4,3,3,2,1] => 4 [6,3,3,3,2,1] => 4 [6,5,2,2,2,1] => 4 [5,5,3,3,1,1] => 2 [6,5,4,1,1,1] => 0 [5,5,3,3,2] => 5 [5,5,4,2,2] => 5 [6,4,4,2,2] => 5 [6,5,4,3] => 4 [9,6,3] => 3 [8,6,4] => 3 [5,4,4,3,2,1] => 4 [5,5,3,3,2,1] => 4 [5,5,4,2,2,1] => 4 [6,4,4,2,2,1] => 4 [5,5,4,3,1,1] => 2 [6,4,4,3,1,1] => 2 [6,5,3,3,1,1] => 2 [5,5,4,3,2] => 5 [6,4,4,3,2] => 5 [6,5,3,3,2] => 5 [6,5,4,2,2] => 5 [6,5,4,3,1] => 3 [6,5,4,1,1,1,1] => -2 [9,6,4] => 3 [8,5,4,2] => 4 [8,5,5,1] => 2 [5,5,4,3,2,1] => 4 [6,4,4,3,2,1] => 4 [6,5,3,3,2,1] => 4 [6,5,4,2,2,1] => 4 [6,5,4,3,1,1] => 2 [6,5,4,3,2] => 5 [6,5,2,2,2,2,1] => 5 [6,5,4,2,1,1,1] => 0 [7,5,4,3,1] => 3 [8,6,4,2] => 4 [10,6,4] => 3 [10,7,3] => 3 [9,7,4] => 3 [9,5,5,1] => 2 [6,5,4,3,2,1] => 4 [6,3,3,3,3,2,1] => 5 [6,5,3,2,2,2,1] => 5 [6,5,4,3,1,1,1] => 0 [11,7,3] => 3 [4,4,4,4,3,2,1] => 5 [6,4,3,3,3,2,1] => 5 [6,5,4,2,2,2,1] => 5 [6,5,4,3,2,1,1] => 2 [9,6,4,3] => 4 [5,4,4,4,3,2,1] => 5 [6,5,3,3,3,2,1] => 5 [6,5,4,3,2,2,1] => 5 [9,6,5,3] => 4 [8,6,5,3,1] => 3 [6,4,4,4,3,2,1] => 5 [6,5,4,3,3,2,1] => 5 [11,7,5,1] => 2 [9,7,5,3] => 4 [5,5,5,4,3,2,1] => 5 [6,5,4,4,3,2,1] => 5 [9,7,5,3,1] => 3 [10,7,5,3] => 4 [6,5,5,4,3,2,1] => 5 [9,7,5,4,1] => 3 [6,6,5,4,3,2,1] => 5 [7,6,5,4,3,2] => 6 [7,6,5,4,3,2,1] => 5 [7,6,5,4,3,1,1,1] => 1 [10,7,6,4,1] => 3 [9,7,6,4,2] => 5 [10,8,5,4,1] => 3 [7,6,5,4,3,2,1,1] => 3 [7,6,5,4,2,2,2,1] => 6 [10,8,6,4,1] => 3 [9,7,5,5,3,1] => 4 [7,6,5,4,3,2,2,1] => 6 [7,6,5,3,3,3,2,1] => 6 [11,8,6,4,1] => 3 [10,8,6,4,2] => 5 [7,6,5,4,3,3,2,1] => 6 [7,6,4,4,4,3,2,1] => 6 [11,8,6,5,1] => 3 [7,6,5,4,4,3,2,1] => 6 [7,5,5,5,4,3,2,1] => 6 [7,6,5,5,4,3,2,1] => 6 [6,6,6,5,4,3,2,1] => 6 [7,6,6,5,4,3,2,1] => 6 [12,9,7,5,1] => 3 [7,7,6,5,4,3,2,1] => 6 [13,9,7,5,1] => 3 [11,9,7,5,3,1] => 4 [11,8,7,5,4,1] => 4 [8,7,6,5,4,3,2,1] => 6 [8,7,6,5,4,3,2,1,1] => 4 [8,7,6,5,4,3,2,2,1] => 7 [8,7,6,5,4,3,3,2,1] => 7 [8,7,6,5,4,4,3,2,1] => 7 [11,9,7,5,5,3] => 6 [8,7,6,5,5,4,3,2,1] => 7 [8,7,6,6,5,4,3,2,1] => 7 [8,7,7,6,5,4,3,2,1] => 7 [8,8,7,6,5,4,3,2,1] => 7 [9,8,7,6,5,4,3,2,1] => 7 [11,9,7,7,5,3,3] => 7 ----------------------------------------------------------------------------- Created: Jul 05, 2013 at 14:36 by Olivier Mallet ----------------------------------------------------------------------------- Last Updated: Nov 29, 2021 at 10:47 by Martin Rubey