***************************************************************************** * 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: St000378 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3]. ----------------------------------------------------------------------------- References: [1] Lee, K., Li, L., Loehr, N. A. A Combinatorial Approach to the Symmetry of $q,t$-Catalan Numbers [[arXiv:1602.01126]] [2] Haglund, J. The $q$,$t$-Catalan numbers and the space of diagonal harmonics [[MathSciNet:2371044]] [3] Robin Why are the dinv-statistic and the partition length equidistributed? [[MathOverflow:131484]] ----------------------------------------------------------------------------- Code: def statistic(P): return sum(1 for c in P.cells() if P.arm_length(*c) - P.leg_length(*c) in [0,1]) ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 1 [2] => 2 [1,1] => 1 [3] => 2 [2,1] => 3 [1,1,1] => 1 [4] => 2 [3,1] => 4 [2,2] => 3 [2,1,1] => 2 [1,1,1,1] => 1 [5] => 2 [4,1] => 3 [3,2] => 5 [3,1,1] => 4 [2,2,1] => 3 [2,1,1,1] => 2 [1,1,1,1,1] => 1 [6] => 2 [5,1] => 3 [4,2] => 5 [4,1,1] => 4 [3,3] => 4 [3,2,1] => 6 [3,1,1,1] => 3 [2,2,2] => 3 [2,2,1,1] => 2 [2,1,1,1,1] => 2 [1,1,1,1,1,1] => 1 [7] => 2 [6,1] => 3 [5,2] => 4 [5,1,1] => 3 [4,3] => 5 [4,2,1] => 7 [4,1,1,1] => 4 [3,3,1] => 6 [3,2,2] => 5 [3,2,1,1] => 4 [3,1,1,1,1] => 3 [2,2,2,1] => 3 [2,2,1,1,1] => 2 [2,1,1,1,1,1] => 2 [1,1,1,1,1,1,1] => 1 [8] => 2 [7,1] => 3 [6,2] => 4 [6,1,1] => 3 [5,3] => 5 [5,2,1] => 5 [5,1,1,1] => 4 [4,4] => 4 [4,3,1] => 8 [4,2,2] => 7 [4,2,1,1] => 6 [4,1,1,1,1] => 3 [3,3,2] => 6 [3,3,1,1] => 5 [3,2,2,1] => 4 [3,2,1,1,1] => 4 [3,1,1,1,1,1] => 3 [2,2,2,2] => 3 [2,2,2,1,1] => 2 [2,2,1,1,1,1] => 2 [2,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1] => 1 [9] => 2 [8,1] => 3 [7,2] => 4 [7,1,1] => 3 [6,3] => 4 [6,2,1] => 5 [6,1,1,1] => 3 [5,4] => 5 [5,3,1] => 7 [5,2,2] => 6 [5,2,1,1] => 5 [5,1,1,1,1] => 4 [4,4,1] => 6 [4,3,2] => 9 [4,3,1,1] => 8 [4,2,2,1] => 7 [4,2,1,1,1] => 5 [4,1,1,1,1,1] => 3 [3,3,3] => 5 [3,3,2,1] => 6 [3,3,1,1,1] => 4 [3,2,2,2] => 4 [3,2,2,1,1] => 3 [3,2,1,1,1,1] => 4 [3,1,1,1,1,1,1] => 3 [2,2,2,2,1] => 3 [2,2,2,1,1,1] => 2 [2,2,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1] => 1 [10] => 2 [9,1] => 3 [8,2] => 4 [8,1,1] => 3 [7,3] => 4 [7,2,1] => 5 [7,1,1,1] => 3 [6,4] => 5 [6,3,1] => 6 [6,2,2] => 5 [6,2,1,1] => 4 [6,1,1,1,1] => 4 [5,5] => 4 [5,4,1] => 6 [5,3,2] => 9 [5,3,1,1] => 8 [5,2,2,1] => 7 [5,2,1,1,1] => 5 [5,1,1,1,1,1] => 3 [4,4,2] => 8 [4,4,1,1] => 7 [4,3,3] => 7 [4,3,2,1] => 10 [4,3,1,1,1] => 6 [4,2,2,2] => 6 [4,2,2,1,1] => 5 [4,2,1,1,1,1] => 5 [4,1,1,1,1,1,1] => 3 [3,3,3,1] => 6 [3,3,2,2] => 5 [3,3,2,1,1] => 4 [3,3,1,1,1,1] => 4 [3,2,2,2,1] => 4 [3,2,2,1,1,1] => 3 [3,2,1,1,1,1,1] => 4 [3,1,1,1,1,1,1,1] => 3 [2,2,2,2,2] => 3 [2,2,2,2,1,1] => 2 [2,2,2,1,1,1,1] => 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 [11] => 2 [10,1] => 3 [9,2] => 4 [9,1,1] => 3 [8,3] => 4 [8,2,1] => 5 [8,1,1,1] => 3 [7,4] => 4 [7,3,1] => 6 [7,2,2] => 5 [7,2,1,1] => 4 [7,1,1,1,1] => 3 [6,5] => 5 [6,4,1] => 6 [6,3,2] => 7 [6,3,1,1] => 6 [6,2,2,1] => 5 [6,2,1,1,1] => 5 [6,1,1,1,1,1] => 4 [5,5,1] => 5 [5,4,2] => 9 [5,4,1,1] => 8 [5,3,3] => 8 [5,3,2,1] => 11 [5,3,1,1,1] => 7 [5,2,2,2] => 7 [5,2,2,1,1] => 6 [5,2,1,1,1,1] => 4 [5,1,1,1,1,1,1] => 3 [4,4,3] => 7 [4,4,2,1] => 10 [4,4,1,1,1] => 6 [4,3,3,1] => 9 [4,3,2,2] => 8 [4,3,2,1,1] => 7 [4,3,1,1,1,1] => 6 [4,2,2,2,1] => 5 [4,2,2,1,1,1] => 5 [4,2,1,1,1,1,1] => 5 [4,1,1,1,1,1,1,1] => 3 [3,3,3,2] => 6 [3,3,3,1,1] => 5 [3,3,2,2,1] => 4 [3,3,2,1,1,1] => 4 [3,3,1,1,1,1,1] => 4 [3,2,2,2,2] => 4 [3,2,2,2,1,1] => 3 [3,2,2,1,1,1,1] => 3 [3,2,1,1,1,1,1,1] => 4 [3,1,1,1,1,1,1,1,1] => 3 [2,2,2,2,2,1] => 3 [2,2,2,2,1,1,1] => 2 [2,2,2,1,1,1,1,1] => 2 [2,2,1,1,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 2 [11,1] => 3 [10,2] => 4 [10,1,1] => 3 [9,3] => 4 [9,2,1] => 5 [9,1,1,1] => 3 [8,4] => 4 [8,3,1] => 6 [8,2,2] => 5 [8,2,1,1] => 4 [8,1,1,1,1] => 3 [7,5] => 5 [7,4,1] => 5 [7,3,2] => 7 [7,3,1,1] => 6 [7,2,2,1] => 5 [7,2,1,1,1] => 4 [7,1,1,1,1,1] => 4 [6,6] => 4 [6,5,1] => 6 [6,4,2] => 8 [6,4,1,1] => 7 [6,3,3] => 7 [6,3,2,1] => 8 [6,3,1,1,1] => 6 [6,2,2,2] => 6 [6,2,2,1,1] => 5 [6,2,1,1,1,1] => 5 [6,1,1,1,1,1,1] => 3 [5,5,2] => 7 [5,5,1,1] => 6 [5,4,3] => 9 [5,4,2,1] => 12 [5,4,1,1,1] => 8 [5,3,3,1] => 11 [5,3,2,2] => 10 [5,3,2,1,1] => 9 [5,3,1,1,1,1] => 6 [5,2,2,2,1] => 7 [5,2,2,1,1,1] => 5 [5,2,1,1,1,1,1] => 4 [5,1,1,1,1,1,1,1] => 3 [4,4,4] => 6 [4,4,3,1] => 10 [4,4,2,2] => 9 [4,4,2,1,1] => 8 [4,4,1,1,1,1] => 5 [4,3,3,2] => 8 [4,3,3,1,1] => 7 [4,3,2,2,1] => 6 [4,3,2,1,1,1] => 7 [4,3,1,1,1,1,1] => 6 [4,2,2,2,2] => 5 [4,2,2,2,1,1] => 4 [4,2,2,1,1,1,1] => 5 [4,2,1,1,1,1,1,1] => 5 [4,1,1,1,1,1,1,1,1] => 3 [3,3,3,3] => 5 [3,3,3,2,1] => 6 [3,3,3,1,1,1] => 4 [3,3,2,2,2] => 4 [3,3,2,2,1,1] => 3 [3,3,2,1,1,1,1] => 4 [3,3,1,1,1,1,1,1] => 4 [3,2,2,2,2,1] => 4 [3,2,2,2,1,1,1] => 3 [3,2,2,1,1,1,1,1] => 3 [3,2,1,1,1,1,1,1,1] => 4 [3,1,1,1,1,1,1,1,1,1] => 3 [2,2,2,2,2,2] => 3 [2,2,2,2,2,1,1] => 2 [2,2,2,2,1,1,1,1] => 2 [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] => 2 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 [13] => 2 [12,1] => 3 [10,3] => 4 [9,2,2] => 5 [8,5] => 4 [8,4,1] => 5 [8,3,2] => 7 [8,3,1,1] => 6 [7,6] => 5 [7,5,1] => 6 [7,4,2] => 7 [7,3,3] => 6 [6,6,1] => 5 [6,5,2] => 7 [6,5,1,1] => 6 [6,4,3] => 9 [6,4,2,1] => 10 [6,3,2,2] => 8 [6,3,1,1,1,1] => 6 [6,2,2,1,1,1] => 5 [6,1,1,1,1,1,1,1] => 3 [5,5,3] => 8 [5,4,4] => 7 [5,4,3,1] => 13 [5,4,2,2] => 12 [5,4,2,1,1] => 11 [5,4,1,1,1,1] => 6 [5,3,3,2] => 11 [5,3,3,1,1] => 10 [5,3,2,2,1] => 9 [5,3,2,1,1,1] => 8 [5,3,1,1,1,1,1] => 6 [5,2,2,2,1,1] => 5 [5,2,2,1,1,1,1] => 5 [5,2,1,1,1,1,1,1] => 4 [4,4,4,1] => 8 [4,4,3,2] => 10 [4,4,3,1,1] => 9 [4,4,2,2,1] => 8 [4,3,3,3] => 7 [4,3,3,2,1] => 7 [3,3,3,3,1] => 6 [3,3,3,2,2] => 5 [3,3,2,2,2,1] => 4 [3,3,2,1,1,1,1,1] => 4 [3,2,2,2,2,2] => 4 [3,2,2,2,2,1,1] => 3 [3,1,1,1,1,1,1,1,1,1,1] => 3 [2,2,2,2,2,2,1] => 3 [2,2,2,2,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [14] => 2 [13,1] => 3 [12,2] => 4 [12,1,1] => 3 [9,5] => 4 [8,6] => 5 [8,5,1] => 5 [8,4,2] => 7 [8,3,1,1,1] => 5 [7,7] => 4 [7,5,2] => 7 [7,4,3] => 7 [6,6,2] => 6 [6,6,1,1] => 5 [6,5,3] => 9 [6,5,1,1,1] => 8 [6,4,4] => 8 [6,4,2,2] => 11 [6,2,2,2,2] => 7 [6,1,1,1,1,1,1,1,1] => 3 [5,5,4] => 7 [5,5,1,1,1,1] => 6 [5,4,3,2] => 14 [5,4,3,1,1] => 13 [5,4,2,2,1] => 12 [5,4,2,1,1,1] => 9 [5,4,1,1,1,1,1] => 6 [5,3,3,3] => 9 [5,3,3,2,1] => 11 [5,3,2,2,2] => 8 [5,3,2,2,1,1] => 7 [5,3,2,1,1,1,1] => 8 [5,3,1,1,1,1,1,1] => 6 [5,2,2,2,2,1] => 5 [5,2,2,1,1,1,1,1] => 5 [4,4,4,2] => 9 [4,4,4,1,1] => 8 [4,4,3,3] => 8 [4,4,3,2,1] => 10 [4,3,2,2,2,1] => 6 [3,3,3,3,2] => 6 [3,3,3,3,1,1] => 5 [3,3,3,2,2,1] => 4 [3,3,2,2,2,2] => 4 [3,3,1,1,1,1,1,1,1,1] => 4 [3,2,2,2,2,1,1,1] => 3 [2,2,2,2,2,2,2] => 3 [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [15] => 2 [14,1] => 3 [12,3] => 4 [11,2,2] => 5 [9,6] => 4 [9,5,1] => 5 [8,7] => 5 [8,5,2] => 6 [7,5,3] => 8 [6,6,3] => 7 [6,5,4] => 9 [6,5,1,1,1,1] => 8 [6,4,3,1,1] => 13 [6,3,3,3] => 9 [6,3,1,1,1,1,1,1] => 5 [6,2,2,2,2,1] => 7 [5,5,5] => 6 [5,4,3,3] => 11 [5,4,3,2,1] => 15 [5,4,3,1,1,1] => 10 [5,3,2,2,2,1] => 7 [5,3,2,2,1,1,1] => 7 [5,3,1,1,1,1,1,1,1] => 6 [4,4,4,3] => 8 [4,4,4,1,1,1] => 7 [4,3,3,3,2] => 7 [3,3,3,3,3] => 5 [3,3,3,3,2,1] => 6 [3,3,3,2,2,2] => 4 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [16] => 2 [15,1] => 3 [12,4] => 4 [12,1,1,1,1] => 3 [10,6] => 4 [8,8] => 4 [8,6,2] => 7 [8,5,3] => 7 [7,6,3] => 7 [7,5,3,1] => 10 [6,6,4] => 8 [6,6,2,2] => 8 [6,5,5] => 7 [6,4,3,3] => 12 [5,5,3,3] => 11 [5,5,2,2,2] => 10 [5,4,4,3] => 10 [5,4,3,2,1,1] => 11 [5,4,2,2,2,1] => 8 [4,4,4,4] => 7 [4,4,4,2,2] => 9 [4,4,3,3,2] => 8 [4,3,3,3,3] => 6 [4,3,3,3,2,1] => 7 [3,3,3,3,2,2] => 5 [3,3,3,3,1,1,1,1] => 4 [3,3,2,2,2,2,2] => 4 [2,2,2,2,2,2,2,2] => 3 [2,2,2,2,2,2,1,1,1,1] => 2 [2,2,2,2,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [17] => 2 [9,8] => 5 [8,8,1] => 5 [8,6,3] => 7 [7,5,3,2] => 12 [6,5,3,3] => 13 [6,5,2,2,2] => 12 [6,4,4,3] => 12 [6,4,4,1,1,1] => 11 [6,3,3,3,2] => 11 [6,3,3,3,1,1] => 10 [5,5,4,3] => 11 [5,5,4,1,1,1] => 10 [5,5,2,2,2,1] => 9 [5,4,4,4] => 9 [5,4,3,2,2,1] => 9 [5,3,3,3,2,1] => 8 [4,4,4,3,2] => 10 [4,4,4,3,1,1] => 9 [4,4,4,2,2,1] => 8 [4,4,3,3,3] => 7 [3,3,3,3,3,2] => 6 [3,2,2,2,2,2,2,2] => 4 [3,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 3 [4,4,4,3,2,1] => 10 [5,4,3,3,2,1] => 9 [6,3,3,3,2,1] => 11 [6,5,2,2,2,1] => 12 [5,5,3,3,1,1] => 12 [6,5,4,1,1,1] => 13 [5,5,3,3,2] => 13 [5,5,4,2,2] => 14 [6,4,4,2,2] => 15 [6,5,4,3] => 14 [4,4,4,3,3] => 8 [4,4,4,4,2] => 9 [5,5,4,4] => 9 [6,4,4,4] => 10 [2,2,2,2,2,2,2,2,2] => 3 [3,3,3,3,3,3] => 5 [18] => 2 [6,2,2,2,2,2,2] => 5 [6,6,6] => 6 [4,2,2,2,2,2,2,2] => 5 [10,4,4] => 6 [9,6,3] => 6 [8,6,4] => 8 [10,8] => 5 [3,3,3,3,1,1,1,1,1,1] => 4 [9,9] => 4 [7,7,1,1,1,1] => 6 [5,4,4,3,2,1] => 11 [5,5,3,3,2,1] => 12 [5,5,4,2,2,1] => 13 [6,4,4,2,2,1] => 14 [5,5,4,3,1,1] => 14 [6,4,4,3,1,1] => 15 [6,5,3,3,1,1] => 16 [5,5,4,3,2] => 15 [6,4,4,3,2] => 16 [6,5,3,3,2] => 17 [6,5,4,2,2] => 18 [6,5,4,3,1] => 19 [6,5,4,1,1,1,1] => 10 [4,4,4,4,3] => 8 [4,3,3,3,3,3] => 6 [19] => 2 [7,2,2,2,2,2,2] => 6 [7,6,6] => 7 [5,2,2,2,2,2,2,2] => 5 [13,6] => 4 [11,4,4] => 6 [9,6,4] => 7 [8,5,4,2] => 11 [8,5,5,1] => 8 [5,5,4,3,2,1] => 15 [6,4,4,3,2,1] => 16 [6,5,3,3,2,1] => 17 [6,5,4,2,2,1] => 18 [6,5,4,3,1,1] => 19 [6,5,4,3,2] => 20 [6,5,2,2,2,2,1] => 8 [6,5,4,2,1,1,1] => 14 [6,6,2,2,2,2] => 10 [7,5,4,3,1] => 18 [2,2,2,2,2,2,2,2,2,2] => 3 [3,3,3,3,3,3,2] => 6 [4,4,3,3,3,3] => 6 [4,4,4,4,4] => 7 [5,5,5,5] => 8 [20] => 2 [10,10] => 4 [8,2,2,2,2,2,2] => 7 [8,6,4,2] => 11 [8,6,6] => 8 [10,6,4] => 7 [9,8,3] => 7 [6,6,6,2] => 9 [10,7,3] => 6 [9,7,4] => 7 [9,5,5,1] => 7 [6,5,4,3,2,1] => 21 [6,3,3,3,3,2,1] => 8 [6,5,3,2,2,2,1] => 11 [6,5,4,3,1,1,1] => 15 [7,6,2,2,2,2] => 12 [6,6,5,4] => 11 [3,3,3,3,3,3,3] => 5 [4,4,4,3,3,3] => 7 [21] => 2 [11,7,3] => 6 [5,4,2,2,2,2,2,2] => 8 [11,10] => 5 [13,4,4] => 6 [7,6,6,2] => 9 [4,4,4,4,3,2,1] => 10 [6,4,3,3,3,2,1] => 10 [6,5,4,2,2,2,1] => 12 [6,5,4,3,2,1,1] => 16 [4,4,4,4,3,3] => 8 [8,6,2,2,2,2] => 11 [22] => 2 [9,6,4,3] => 11 [8,7,7] => 7 [5,4,4,4,3,2,1] => 11 [6,5,3,3,3,2,1] => 11 [6,5,4,3,2,2,1] => 13 [15,2,2,2,2] => 5 [9,6,5,3] => 11 [8,6,5,3,1] => 15 [6,4,4,4,3,2,1] => 12 [6,5,4,3,3,2,1] => 12 [3,3,3,3,3,3,3,3] => 5 [4,4,4,4,4,4] => 7 [6,6,6,6] => 8 [24] => 2 [11,7,5,1] => 8 [9,7,5,3] => 11 [8,8,8] => 6 [17,7] => 4 [12,9,3] => 6 [5,5,5,4,3,2,1] => 15 [6,5,4,4,3,2,1] => 13 [7,6,6,6] => 9 [5,4,4,4,2,2,2,2] => 7 [13,12] => 5 [15,5,5] => 6 [9,7,5,3,1] => 13 [10,7,5,3] => 10 [3,3,3,3,3,1,1,1,1,1,1,1,1,1,1] => 4 [5,5,5,5,5] => 9 [12,10,3] => 7 [6,5,5,4,3,2,1] => 16 [6,6,4,3,3,2,2] => 13 [8,6,6,6] => 10 [9,7,5,4,1] => 13 [16,8,2] => 6 [6,6,5,4,3,2,1] => 21 [7,6,5,4,3,2] => 27 [3,3,3,3,3,3,3,3,3] => 5 [15,6,6] => 6 [7,6,5,4,3,2,1] => 28 [6,6,6,4,3,3] => 17 [4,3,3,3,3,3,3,3,3] => 6 [7,6,5,4,3,1,1,1] => 21 [10,7,6,4,1] => 12 [9,7,6,4,2] => 16 [10,8,5,4,1] => 11 [15,14] => 5 [7,6,5,4,3,2,1,1] => 22 [7,6,5,4,2,2,2,1] => 17 [10,8,6,4,1] => 12 [6,6,6,6,6] => 10 [15,15] => 4 [5,5,5,5,5,5] => 9 [6,6,6,6,3,3] => 15 [10,10,10] => 6 [9,7,5,5,3,1] => 20 [7,6,5,4,3,2,2,1] => 18 [7,6,5,3,3,3,2,1] => 15 [11,8,6,4,1] => 11 [10,8,6,4,2] => 14 [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => 3 [7,6,6,6,6] => 11 [7,6,5,4,3,3,2,1] => 16 [7,6,4,4,4,3,2,1] => 15 [11,8,6,5,1] => 12 [10,10,9,2] => 9 [14,9,8] => 7 [4,4,4,4,4,4,4,4] => 7 [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => 3 [7,6,5,4,4,3,2,1] => 16 [7,5,5,5,4,3,2,1] => 17 [16,16] => 4 [10,10,10,2] => 8 [16,8,4,2,2] => 11 [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => 4 [5,4,4,4,4,4,4,4] => 8 [7,6,5,5,4,3,2,1] => 18 [6,6,6,5,4,3,2,1] => 21 [22,7,4] => 6 [12,12,4,3,2] => 13 [4,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => 5 [7,6,6,5,4,3,2,1] => 22 [12,9,7,5,1] => 11 [7,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => 5 [7,7,6,5,4,3,2,1] => 28 [13,9,7,5,1] => 11 [18,18] => 4 [24,12] => 4 [11,9,7,5,3,1] => 16 [11,8,7,5,4,1] => 17 [8,7,6,5,4,3,2,1] => 36 [8,7,7,7,7] => 11 [7,6,2,2,2,2,2,2,2,2,2,2,2,2] => 8 [7,6,6,6,6,6] => 13 [8,7,6,5,4,3,2,1,1] => 29 [8,7,6,5,4,3,2,2,1] => 24 [8,7,6,5,4,3,3,2,1] => 21 [10,10,10,10] => 8 [12,12,12,4] => 8 [8,7,6,5,4,4,3,2,1] => 20 [11,9,7,5,5,3] => 20 [4,4,4,4,4,4,4,4,4,4] => 7 [14,14,8,4] => 8 [7,6,6,6,2,2,2,2,2,2,2,2] => 12 [8,7,6,5,5,4,3,2,1] => 21 [9,9,9,9,3,3] => 13 [8,7,6,6,5,4,3,2,1] => 24 [10,10,10,5,5,2] => 14 [16,10,8,8] => 10 [26,9,7] => 7 [8,7,7,6,5,4,3,2,1] => 29 [8,8,7,6,5,4,3,2,1] => 36 [18,18,9] => 6 [9,8,7,6,5,4,3,2,1] => 45 [11,9,7,7,5,3,3] => 27 [48] => 2 [7,6,6,6,6,6,6,6] => 12 [22,9,8,5,5] => 13 [6,6,6,6,6,6,6,3,3,2] => 17 [27,14,9] => 6 [17,14,13,8] => 9 [5,5,5,5,5,5,5,5,5,5,5] => 9 [27,20,8] => 6 [26,13,7,7,2] => 10 [31,12,10,4] => 9 [6,6,6,6,6,6,6,6,6,6] => 11 [5,5,5,5,5,5,5,5,5,5,5,5] => 9 [12,12,12,12,12] => 10 [12,12,12,12,11,4] => 13 [12,12,12,12,12,4] => 12 [3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2] => 4 [11,11,11,11,11,10] => 13 [11,11,11,11,11,11] => 12 [16,14,9,9,9,9] => 14 [8,8,8,8,8,8,8,8,4,2] => 20 [24,24,24] => 6 [16,16,16,16,8,4,2] => 15 [12,12,12,12,12,12,4,4] => 17 [22,20,20,12,8] => 12 [20,20,8,4,4,4,4,4,4,4,4,4] => 13 [33,32,9,9,5] => 11 [18,18,18,18,18] => 10 [18,18,18,18,9,9] => 12 [6,6,6,6,6,6,6,6,6,6,6,6,6,6,3,3] => 15 [56,17,11,10,5] => 11 [33,26,23,12,9] => 10 [22,22,22,10,10,9,4,4] => 17 [22,22,22,10,10,10,4,4] => 16 [12,12,12,12,12,12,12,12,4,3,2] => 31 [24,24,24,24,14] => 10 [76,22,9,6] => 8 [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5] => 9 [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5] => 9 [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6] => 11 [12,12,12,12,12,12,12,12,12,12] => 20 [24,24,24,24,24] => 10 [12,12,12,12,12,12,12,12,12,6,6,6,4,2] => 38 [54,48,18,10,8] => 11 [24,24,24,24,24,24] => 12 [37,33,20,20,13,10,10,10,7] => 22 [42,34,30,22,22,14,2] => 14 [54,52,34,18,11,7] => 13 [65,27,27,25,13,9,6,6] => 19 [18,18,18,18,18,18,18,18,18,18] => 20 [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6] => 11 [16,16,16,16,16,16,16,8,8,8,8,8,8,8,8,4,2] => 49 [58,38,38,38,30] => 10 [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12] => 23 [10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,2] => 21 [74,58,34,34,22,22,13,11,2] => 19 [61,48,48,48,45,22,12,2] => 18 [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3] => 40 [98,62,36,26,14,14,14,14,14,14,12,2] => 26 [14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,13,2] => 30 [14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,2] => 29 [74,74,26,26,26,26,12,12,12,12,12,12,6,6,4,4,3,3,3,3] => 45 [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6] => 11 [64,57,40,39,28,23,19,19,16,9,8,8,8,8,8,8,6,6] => 42 [70,70,46,28,22,22,10,9,9,9,9,8,8,8,8,8,7,7,7,3,3,3] => 38 [14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,7,7,7,7,7,2] => 41 [170,92,36,28,27,24,20,14,13,10,8,8] => 28 [12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,6,6,6,6,4,4,4,3,3] => 34 [145,83,62,44,44,42,19,16,16,10,8] => 26 [98,98,37,37,37,37,24,24,24,24,24,24,24,24,10,10,8,8] => 39 [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6] => 11 [11,9,7,6,5,3,1] => 24 [13,11,9,7,5,3,1] => 19 [13,11,9,7,7,5,3,1] => 26 [17,13,11,9,7,5,1] => 17 [15,13,11,9,7,5,3,1] => 22 [29,23,19,17,13,11,7,1] => 17 ----------------------------------------------------------------------------- Created: Feb 06, 2016 at 17:13 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Jan 15, 2024 at 12:24 by Martin Rubey