***************************************************************************** * 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: St001389 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$ ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(la): k = len(la) return det(matrix([[binomial(la[k-i], j-i+1) for j in range(1, k+1)] for i in range(1, k+1)])) ----------------------------------------------------------------------------- Statistic values: [1] => 1 [2] => 2 [1,1] => 1 [3] => 3 [2,1] => 2 [1,1,1] => 1 [4] => 4 [3,1] => 3 [2,2] => 3 [2,1,1] => 2 [1,1,1,1] => 1 [5] => 5 [4,1] => 4 [3,2] => 5 [3,1,1] => 3 [2,2,1] => 3 [2,1,1,1] => 2 [1,1,1,1,1] => 1 [6] => 6 [5,1] => 5 [4,2] => 7 [4,1,1] => 4 [3,3] => 6 [3,2,1] => 5 [3,1,1,1] => 3 [2,2,2] => 4 [2,2,1,1] => 3 [2,1,1,1,1] => 2 [1,1,1,1,1,1] => 1 [7] => 7 [6,1] => 6 [5,2] => 9 [5,1,1] => 5 [4,3] => 9 [4,2,1] => 7 [4,1,1,1] => 4 [3,3,1] => 6 [3,2,2] => 7 [3,2,1,1] => 5 [3,1,1,1,1] => 3 [2,2,2,1] => 4 [2,2,1,1,1] => 3 [2,1,1,1,1,1] => 2 [1,1,1,1,1,1,1] => 1 [8] => 8 [7,1] => 7 [6,2] => 11 [6,1,1] => 6 [5,3] => 12 [5,2,1] => 9 [5,1,1,1] => 5 [4,4] => 10 [4,3,1] => 9 [4,2,2] => 10 [4,2,1,1] => 7 [4,1,1,1,1] => 4 [3,3,2] => 9 [3,3,1,1] => 6 [3,2,2,1] => 7 [3,2,1,1,1] => 5 [3,1,1,1,1,1] => 3 [2,2,2,2] => 5 [2,2,2,1,1] => 4 [2,2,1,1,1,1] => 3 [2,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1] => 1 [9] => 9 [8,1] => 8 [7,2] => 13 [7,1,1] => 7 [6,3] => 15 [6,2,1] => 11 [6,1,1,1] => 6 [5,4] => 14 [5,3,1] => 12 [5,2,2] => 13 [5,2,1,1] => 9 [5,1,1,1,1] => 5 [4,4,1] => 10 [4,3,2] => 14 [4,3,1,1] => 9 [4,2,2,1] => 10 [4,2,1,1,1] => 7 [4,1,1,1,1,1] => 4 [3,3,3] => 10 [3,3,2,1] => 9 [3,3,1,1,1] => 6 [3,2,2,2] => 9 [3,2,2,1,1] => 7 [3,2,1,1,1,1] => 5 [3,1,1,1,1,1,1] => 3 [2,2,2,2,1] => 5 [2,2,2,1,1,1] => 4 [2,2,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1] => 1 [10] => 10 [9,1] => 9 [8,2] => 15 [8,1,1] => 8 [7,3] => 18 [7,2,1] => 13 [7,1,1,1] => 7 [6,4] => 18 [6,3,1] => 15 [6,2,2] => 16 [6,2,1,1] => 11 [6,1,1,1,1] => 6 [5,5] => 15 [5,4,1] => 14 [5,3,2] => 19 [5,3,1,1] => 12 [5,2,2,1] => 13 [5,2,1,1,1] => 9 [5,1,1,1,1,1] => 5 [4,4,2] => 16 [4,4,1,1] => 10 [4,3,3] => 16 [4,3,2,1] => 14 [4,3,1,1,1] => 9 [4,2,2,2] => 13 [4,2,2,1,1] => 10 [4,2,1,1,1,1] => 7 [4,1,1,1,1,1,1] => 4 [3,3,3,1] => 10 [3,3,2,2] => 12 [3,3,2,1,1] => 9 [3,3,1,1,1,1] => 6 [3,2,2,2,1] => 9 [3,2,2,1,1,1] => 7 [3,2,1,1,1,1,1] => 5 [3,1,1,1,1,1,1,1] => 3 [2,2,2,2,2] => 6 [2,2,2,2,1,1] => 5 [2,2,2,1,1,1,1] => 4 [2,2,1,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1] => 1 [11] => 11 [10,1] => 10 [9,2] => 17 [9,1,1] => 9 [8,3] => 21 [8,2,1] => 15 [8,1,1,1] => 8 [7,4] => 22 [7,3,1] => 18 [7,2,2] => 19 [7,2,1,1] => 13 [7,1,1,1,1] => 7 [6,5] => 20 [6,4,1] => 18 [6,3,2] => 24 [6,3,1,1] => 15 [6,2,2,1] => 16 [6,2,1,1,1] => 11 [6,1,1,1,1,1] => 6 [5,5,1] => 15 [5,4,2] => 23 [5,4,1,1] => 14 [5,3,3] => 22 [5,3,2,1] => 19 [5,3,1,1,1] => 12 [5,2,2,2] => 17 [5,2,2,1,1] => 13 [5,2,1,1,1,1] => 9 [5,1,1,1,1,1,1] => 5 [4,4,3] => 19 [4,4,2,1] => 16 [4,4,1,1,1] => 10 [4,3,3,1] => 16 [4,3,2,2] => 19 [4,3,2,1,1] => 14 [4,3,1,1,1,1] => 9 [4,2,2,2,1] => 13 [4,2,2,1,1,1] => 10 [4,2,1,1,1,1,1] => 7 [4,1,1,1,1,1,1,1] => 4 [3,3,3,2] => 14 [3,3,3,1,1] => 10 [3,3,2,2,1] => 12 [3,3,2,1,1,1] => 9 [3,3,1,1,1,1,1] => 6 [3,2,2,2,2] => 11 [3,2,2,2,1,1] => 9 [3,2,2,1,1,1,1] => 7 [3,2,1,1,1,1,1,1] => 5 [3,1,1,1,1,1,1,1,1] => 3 [2,2,2,2,2,1] => 6 [2,2,2,2,1,1,1] => 5 [2,2,2,1,1,1,1,1] => 4 [2,2,1,1,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 12 [11,1] => 11 [10,2] => 19 [10,1,1] => 10 [9,3] => 24 [9,2,1] => 17 [9,1,1,1] => 9 [8,4] => 26 [8,3,1] => 21 [8,2,2] => 22 [8,2,1,1] => 15 [8,1,1,1,1] => 8 [7,5] => 25 [7,4,1] => 22 [7,3,2] => 29 [7,3,1,1] => 18 [7,2,2,1] => 19 [7,2,1,1,1] => 13 [7,1,1,1,1,1] => 7 [6,6] => 21 [6,5,1] => 20 [6,4,2] => 30 [6,4,1,1] => 18 [6,3,3] => 28 [6,3,2,1] => 24 [6,3,1,1,1] => 15 [6,2,2,2] => 21 [6,2,2,1,1] => 16 [6,2,1,1,1,1] => 11 [6,1,1,1,1,1,1] => 6 [5,5,2] => 25 [5,5,1,1] => 15 [5,4,3] => 28 [5,4,2,1] => 23 [5,4,1,1,1] => 14 [5,3,3,1] => 22 [5,3,2,2] => 26 [5,3,2,1,1] => 19 [5,3,1,1,1,1] => 12 [5,2,2,2,1] => 17 [5,2,2,1,1,1] => 13 [5,2,1,1,1,1,1] => 9 [5,1,1,1,1,1,1,1] => 5 [4,4,4] => 20 [4,4,3,1] => 19 [4,4,2,2] => 22 [4,4,2,1,1] => 16 [4,4,1,1,1,1] => 10 [4,3,3,2] => 23 [4,3,3,1,1] => 16 [4,3,2,2,1] => 19 [4,3,2,1,1,1] => 14 [4,3,1,1,1,1,1] => 9 [4,2,2,2,2] => 16 [4,2,2,2,1,1] => 13 [4,2,2,1,1,1,1] => 10 [4,2,1,1,1,1,1,1] => 7 [4,1,1,1,1,1,1,1,1] => 4 [3,3,3,3] => 15 [3,3,3,2,1] => 14 [3,3,3,1,1,1] => 10 [3,3,2,2,2] => 15 [3,3,2,2,1,1] => 12 [3,3,2,1,1,1,1] => 9 [3,3,1,1,1,1,1,1] => 6 [3,2,2,2,2,1] => 11 [3,2,2,2,1,1,1] => 9 [3,2,2,1,1,1,1,1] => 7 [3,2,1,1,1,1,1,1,1] => 5 [3,1,1,1,1,1,1,1,1,1] => 3 [2,2,2,2,2,2] => 7 [2,2,2,2,2,1,1] => 6 [2,2,2,2,1,1,1,1] => 5 [2,2,2,1,1,1,1,1,1] => 4 [2,2,1,1,1,1,1,1,1,1] => 3 [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] => 13 [12,1] => 12 [11,2] => 21 [11,1,1] => 11 [10,3] => 27 [10,2,1] => 19 [10,1,1,1] => 10 [9,4] => 30 [9,3,1] => 24 [9,2,2] => 25 [9,2,1,1] => 17 [9,1,1,1,1] => 9 [8,5] => 30 [8,4,1] => 26 [8,3,2] => 34 [8,3,1,1] => 21 [8,2,2,1] => 22 [8,2,1,1,1] => 15 [8,1,1,1,1,1] => 8 [7,6] => 27 [7,5,1] => 25 [7,4,2] => 37 [7,4,1,1] => 22 [7,3,3] => 34 [7,3,2,1] => 29 [7,3,1,1,1] => 18 [7,2,2,2] => 25 [7,2,2,1,1] => 19 [7,2,1,1,1,1] => 13 [7,1,1,1,1,1,1] => 7 [6,6,1] => 21 [6,5,2] => 34 [6,5,1,1] => 20 [6,4,3] => 37 [6,4,2,1] => 30 [6,4,1,1,1] => 18 [6,3,3,1] => 28 [6,3,2,2] => 33 [6,3,2,1,1] => 24 [6,3,1,1,1,1] => 15 [6,2,2,2,1] => 21 [6,2,2,1,1,1] => 16 [6,2,1,1,1,1,1] => 11 [6,1,1,1,1,1,1,1] => 6 [5,5,3] => 31 [5,5,2,1] => 25 [5,5,1,1,1] => 15 [5,4,4] => 30 [5,4,3,1] => 28 [5,4,2,2] => 32 [5,4,2,1,1] => 23 [5,4,1,1,1,1] => 14 [5,3,3,2] => 32 [5,3,3,1,1] => 22 [5,3,2,2,1] => 26 [5,3,2,1,1,1] => 19 [5,3,1,1,1,1,1] => 12 [5,2,2,2,2] => 21 [5,2,2,2,1,1] => 17 [5,2,2,1,1,1,1] => 13 [5,2,1,1,1,1,1,1] => 9 [5,1,1,1,1,1,1,1,1] => 5 [4,4,4,1] => 20 [4,4,3,2] => 28 [4,4,3,1,1] => 19 [4,4,2,2,1] => 22 [4,4,2,1,1,1] => 16 [4,4,1,1,1,1,1] => 10 [4,3,3,3] => 25 [4,3,3,2,1] => 23 [4,3,3,1,1,1] => 16 [4,3,2,2,2] => 24 [4,3,2,2,1,1] => 19 [4,3,2,1,1,1,1] => 14 [4,3,1,1,1,1,1,1] => 9 [4,2,2,2,2,1] => 16 [4,2,2,2,1,1,1] => 13 [4,2,2,1,1,1,1,1] => 10 [4,2,1,1,1,1,1,1,1] => 7 [4,1,1,1,1,1,1,1,1,1] => 4 [3,3,3,3,1] => 15 [3,3,3,2,2] => 18 [3,3,3,2,1,1] => 14 [3,3,3,1,1,1,1] => 10 [3,3,2,2,2,1] => 15 [3,3,2,2,1,1,1] => 12 [3,3,2,1,1,1,1,1] => 9 [3,3,1,1,1,1,1,1,1] => 6 [3,2,2,2,2,2] => 13 [3,2,2,2,2,1,1] => 11 [3,2,2,2,1,1,1,1] => 9 [3,2,2,1,1,1,1,1,1] => 7 [3,2,1,1,1,1,1,1,1,1] => 5 [3,1,1,1,1,1,1,1,1,1,1] => 3 [2,2,2,2,2,2,1] => 7 [2,2,2,2,2,1,1,1] => 6 [2,2,2,2,1,1,1,1,1] => 5 [2,2,2,1,1,1,1,1,1,1] => 4 [2,2,1,1,1,1,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [14] => 14 [13,1] => 13 [12,2] => 23 [12,1,1] => 12 [11,3] => 30 [11,2,1] => 21 [11,1,1,1] => 11 [10,4] => 34 [10,3,1] => 27 [10,2,2] => 28 [10,2,1,1] => 19 [10,1,1,1,1] => 10 [9,5] => 35 [9,4,1] => 30 [9,3,2] => 39 [9,3,1,1] => 24 [9,2,2,1] => 25 [9,2,1,1,1] => 17 [9,1,1,1,1,1] => 9 [8,6] => 33 [8,5,1] => 30 [8,4,2] => 44 [8,4,1,1] => 26 [8,3,3] => 40 [8,3,2,1] => 34 [8,3,1,1,1] => 21 [8,2,2,2] => 29 [8,2,2,1,1] => 22 [8,2,1,1,1,1] => 15 [8,1,1,1,1,1,1] => 8 [7,7] => 28 [7,6,1] => 27 [7,5,2] => 43 [7,5,1,1] => 25 [7,4,3] => 46 [7,4,2,1] => 37 [7,4,1,1,1] => 22 [7,3,3,1] => 34 [7,3,2,2] => 40 [7,3,2,1,1] => 29 [7,3,1,1,1,1] => 18 [7,2,2,2,1] => 25 [7,2,2,1,1,1] => 19 [7,2,1,1,1,1,1] => 13 [7,1,1,1,1,1,1,1] => 7 [6,6,2] => 36 [6,6,1,1] => 21 [6,5,3] => 43 [6,5,2,1] => 34 [6,5,1,1,1] => 20 [6,4,4] => 40 [6,4,3,1] => 37 [6,4,2,2] => 42 [6,4,2,1,1] => 30 [6,4,1,1,1,1] => 18 [6,3,3,2] => 41 [6,3,3,1,1] => 28 [6,3,2,2,1] => 33 [6,3,2,1,1,1] => 24 [6,3,1,1,1,1,1] => 15 [6,2,2,2,2] => 26 [6,2,2,2,1,1] => 21 [6,2,2,1,1,1,1] => 16 [6,2,1,1,1,1,1,1] => 11 [6,1,1,1,1,1,1,1,1] => 6 [5,5,4] => 34 [5,5,3,1] => 31 [5,5,2,2] => 35 [5,5,2,1,1] => 25 [5,5,1,1,1,1] => 15 [5,4,4,1] => 30 [5,4,3,2] => 42 [5,4,3,1,1] => 28 [5,4,2,2,1] => 32 [5,4,2,1,1,1] => 23 [5,4,1,1,1,1,1] => 14 [5,3,3,3] => 35 [5,3,3,2,1] => 32 [5,3,3,1,1,1] => 22 [5,3,2,2,2] => 33 [5,3,2,2,1,1] => 26 [5,3,2,1,1,1,1] => 19 [5,3,1,1,1,1,1,1] => 12 [5,2,2,2,2,1] => 21 [5,2,2,2,1,1,1] => 17 [5,2,2,1,1,1,1,1] => 13 [5,2,1,1,1,1,1,1,1] => 9 [5,1,1,1,1,1,1,1,1,1] => 5 [4,4,4,2] => 30 [4,4,4,1,1] => 20 [4,4,3,3] => 31 [4,4,3,2,1] => 28 [4,4,3,1,1,1] => 19 [4,4,2,2,2] => 28 [4,4,2,2,1,1] => 22 [4,4,2,1,1,1,1] => 16 [4,4,1,1,1,1,1,1] => 10 [4,3,3,3,1] => 25 [4,3,3,2,2] => 30 [4,3,3,2,1,1] => 23 [4,3,3,1,1,1,1] => 16 [4,3,2,2,2,1] => 24 [4,3,2,2,1,1,1] => 19 [4,3,2,1,1,1,1,1] => 14 [4,3,1,1,1,1,1,1,1] => 9 [4,2,2,2,2,2] => 19 [4,2,2,2,2,1,1] => 16 [4,2,2,2,1,1,1,1] => 13 [4,2,2,1,1,1,1,1,1] => 10 [4,2,1,1,1,1,1,1,1,1] => 7 [4,1,1,1,1,1,1,1,1,1,1] => 4 [3,3,3,3,2] => 20 [3,3,3,3,1,1] => 15 [3,3,3,2,2,1] => 18 [3,3,3,2,1,1,1] => 14 [3,3,3,1,1,1,1,1] => 10 [3,3,2,2,2,2] => 18 [3,3,2,2,2,1,1] => 15 [3,3,2,2,1,1,1,1] => 12 [3,3,2,1,1,1,1,1,1] => 9 [3,3,1,1,1,1,1,1,1,1] => 6 [3,2,2,2,2,2,1] => 13 [3,2,2,2,2,1,1,1] => 11 [3,2,2,2,1,1,1,1,1] => 9 [3,2,2,1,1,1,1,1,1,1] => 7 [3,2,1,1,1,1,1,1,1,1,1] => 5 [3,1,1,1,1,1,1,1,1,1,1,1] => 3 [2,2,2,2,2,2,2] => 8 [2,2,2,2,2,2,1,1] => 7 [2,2,2,2,2,1,1,1,1] => 6 [2,2,2,2,1,1,1,1,1,1] => 5 [2,2,2,1,1,1,1,1,1,1,1] => 4 [2,2,1,1,1,1,1,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [15] => 15 [14,1] => 14 [13,2] => 25 [13,1,1] => 13 [12,3] => 33 [12,2,1] => 23 [12,1,1,1] => 12 [11,4] => 38 [11,3,1] => 30 [11,2,2] => 31 [11,2,1,1] => 21 [11,1,1,1,1] => 11 [10,5] => 40 [10,4,1] => 34 [10,3,2] => 44 [10,3,1,1] => 27 [10,2,2,1] => 28 [10,2,1,1,1] => 19 [10,1,1,1,1,1] => 10 [9,6] => 39 [9,5,1] => 35 [9,4,2] => 51 [9,4,1,1] => 30 [9,3,3] => 46 [9,3,2,1] => 39 [9,3,1,1,1] => 24 [9,2,2,2] => 33 [9,2,2,1,1] => 25 [9,2,1,1,1,1] => 17 [9,1,1,1,1,1,1] => 9 [8,7] => 35 [8,6,1] => 33 [8,5,2] => 52 [8,5,1,1] => 30 [8,4,3] => 55 [8,4,2,1] => 44 [8,4,1,1,1] => 26 [8,3,3,1] => 40 [8,3,2,2] => 47 [8,3,2,1,1] => 34 [8,3,1,1,1,1] => 21 [8,2,2,2,1] => 29 [8,2,2,1,1,1] => 22 [8,2,1,1,1,1,1] => 15 [8,1,1,1,1,1,1,1] => 8 [7,7,1] => 28 [7,6,2] => 47 [7,6,1,1] => 27 [7,5,3] => 55 [7,5,2,1] => 43 [7,5,1,1,1] => 25 [7,4,4] => 50 [7,4,3,1] => 46 [7,4,2,2] => 52 [7,4,2,1,1] => 37 [7,4,1,1,1,1] => 22 [7,3,3,2] => 50 [7,3,3,1,1] => 34 [7,3,2,2,1] => 40 [7,3,2,1,1,1] => 29 [7,3,1,1,1,1,1] => 18 [7,2,2,2,2] => 31 [7,2,2,2,1,1] => 25 [7,2,2,1,1,1,1] => 19 [7,2,1,1,1,1,1,1] => 13 [7,1,1,1,1,1,1,1,1] => 7 [6,6,3] => 46 [6,6,2,1] => 36 [6,6,1,1,1] => 21 [6,5,4] => 48 [6,5,3,1] => 43 [6,5,2,2] => 48 [6,5,2,1,1] => 34 [6,5,1,1,1,1] => 20 [6,4,4,1] => 40 [6,4,3,2] => 56 [6,4,3,1,1] => 37 [6,4,2,2,1] => 42 [6,4,2,1,1,1] => 30 [6,4,1,1,1,1,1] => 18 [6,3,3,3] => 45 [6,3,3,2,1] => 41 [6,3,3,1,1,1] => 28 [6,3,2,2,2] => 42 [6,3,2,2,1,1] => 33 [6,3,2,1,1,1,1] => 24 [6,3,1,1,1,1,1,1] => 15 [6,2,2,2,2,1] => 26 [6,2,2,2,1,1,1] => 21 [6,2,2,1,1,1,1,1] => 16 [6,2,1,1,1,1,1,1,1] => 11 [6,1,1,1,1,1,1,1,1,1] => 6 [5,5,5] => 35 [5,5,4,1] => 34 [5,5,3,2] => 47 [5,5,3,1,1] => 31 [5,5,2,2,1] => 35 [5,5,2,1,1,1] => 25 [5,5,1,1,1,1,1] => 15 [5,4,4,2] => 46 [5,4,4,1,1] => 30 [5,4,3,3] => 47 [5,4,3,2,1] => 42 [5,4,3,1,1,1] => 28 [5,4,2,2,2] => 41 [5,4,2,2,1,1] => 32 [5,4,2,1,1,1,1] => 23 [5,4,1,1,1,1,1,1] => 14 [5,3,3,3,1] => 35 [5,3,3,2,2] => 42 [5,3,3,2,1,1] => 32 [5,3,3,1,1,1,1] => 22 [5,3,2,2,2,1] => 33 [5,3,2,2,1,1,1] => 26 [5,3,2,1,1,1,1,1] => 19 [5,3,1,1,1,1,1,1,1] => 12 [5,2,2,2,2,2] => 25 [5,2,2,2,2,1,1] => 21 [5,2,2,2,1,1,1,1] => 17 [5,2,2,1,1,1,1,1,1] => 13 [5,2,1,1,1,1,1,1,1,1] => 9 [5,1,1,1,1,1,1,1,1,1,1] => 5 [4,4,4,3] => 34 [4,4,4,2,1] => 30 [4,4,4,1,1,1] => 20 [4,4,3,3,1] => 31 [4,4,3,2,2] => 37 [4,4,3,2,1,1] => 28 [4,4,3,1,1,1,1] => 19 [4,4,2,2,2,1] => 28 [4,4,2,2,1,1,1] => 22 [4,4,2,1,1,1,1,1] => 16 [4,4,1,1,1,1,1,1,1] => 10 [4,3,3,3,2] => 34 [4,3,3,3,1,1] => 25 [4,3,3,2,2,1] => 30 [4,3,3,2,1,1,1] => 23 [4,3,3,1,1,1,1,1] => 16 [4,3,2,2,2,2] => 29 [4,3,2,2,2,1,1] => 24 [4,3,2,2,1,1,1,1] => 19 [4,3,2,1,1,1,1,1,1] => 14 [4,3,1,1,1,1,1,1,1,1] => 9 [4,2,2,2,2,2,1] => 19 [4,2,2,2,2,1,1,1] => 16 [4,2,2,2,1,1,1,1,1] => 13 [4,2,2,1,1,1,1,1,1,1] => 10 [4,2,1,1,1,1,1,1,1,1,1] => 7 [4,1,1,1,1,1,1,1,1,1,1,1] => 4 [3,3,3,3,3] => 21 [3,3,3,3,2,1] => 20 [3,3,3,3,1,1,1] => 15 [3,3,3,2,2,2] => 22 [3,3,3,2,2,1,1] => 18 [3,3,3,2,1,1,1,1] => 14 [3,3,3,1,1,1,1,1,1] => 10 [3,3,2,2,2,2,1] => 18 [3,3,2,2,2,1,1,1] => 15 [3,3,2,2,1,1,1,1,1] => 12 [3,3,2,1,1,1,1,1,1,1] => 9 [3,3,1,1,1,1,1,1,1,1,1] => 6 [3,2,2,2,2,2,2] => 15 [3,2,2,2,2,2,1,1] => 13 [3,2,2,2,2,1,1,1,1] => 11 [3,2,2,2,1,1,1,1,1,1] => 9 [3,2,2,1,1,1,1,1,1,1,1] => 7 [3,2,1,1,1,1,1,1,1,1,1,1] => 5 [3,1,1,1,1,1,1,1,1,1,1,1,1] => 3 [2,2,2,2,2,2,2,1] => 8 [2,2,2,2,2,2,1,1,1] => 7 [2,2,2,2,2,1,1,1,1,1] => 6 [2,2,2,2,1,1,1,1,1,1,1] => 5 [2,2,2,1,1,1,1,1,1,1,1,1] => 4 [2,2,1,1,1,1,1,1,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [16] => 16 [15,1] => 15 [14,2] => 27 [14,1,1] => 14 [13,3] => 36 [13,2,1] => 25 [13,1,1,1] => 13 [12,4] => 42 [12,3,1] => 33 [12,2,2] => 34 [12,2,1,1] => 23 [12,1,1,1,1] => 12 [11,5] => 45 [11,4,1] => 38 [11,3,2] => 49 [11,3,1,1] => 30 [11,2,2,1] => 31 [11,2,1,1,1] => 21 [11,1,1,1,1,1] => 11 [10,6] => 45 [10,5,1] => 40 [10,4,2] => 58 [10,4,1,1] => 34 [10,3,3] => 52 [10,3,2,1] => 44 [10,3,1,1,1] => 27 [10,2,2,2] => 37 [10,2,2,1,1] => 28 [10,2,1,1,1,1] => 19 [10,1,1,1,1,1,1] => 10 [9,7] => 42 [9,6,1] => 39 [9,5,2] => 61 [9,5,1,1] => 35 [9,4,3] => 64 [9,4,2,1] => 51 [9,4,1,1,1] => 30 [9,3,3,1] => 46 [9,3,2,2] => 54 [9,3,2,1,1] => 39 [9,3,1,1,1,1] => 24 [9,2,2,2,1] => 33 [9,2,2,1,1,1] => 25 [9,2,1,1,1,1,1] => 17 [9,1,1,1,1,1,1,1] => 9 [8,8] => 36 [8,7,1] => 35 [8,6,2] => 58 [8,6,1,1] => 33 [8,5,3] => 67 [8,5,2,1] => 52 [8,5,1,1,1] => 30 [8,4,4] => 60 [8,4,3,1] => 55 [8,4,2,2] => 62 [8,4,2,1,1] => 44 [8,4,1,1,1,1] => 26 [8,3,3,2] => 59 [8,3,3,1,1] => 40 [8,3,2,2,1] => 47 [8,3,2,1,1,1] => 34 [8,3,1,1,1,1,1] => 21 [8,2,2,2,2] => 36 [8,2,2,2,1,1] => 29 [8,2,2,1,1,1,1] => 22 [8,2,1,1,1,1,1,1] => 15 [8,1,1,1,1,1,1,1,1] => 8 [7,7,2] => 49 [7,7,1,1] => 28 [7,6,3] => 61 [7,6,2,1] => 47 [7,6,1,1,1] => 27 [7,5,4] => 62 [7,5,3,1] => 55 [7,5,2,2] => 61 [7,5,2,1,1] => 43 [7,5,1,1,1,1] => 25 [7,4,4,1] => 50 [7,4,3,2] => 70 [7,4,3,1,1] => 46 [7,4,2,2,1] => 52 [7,4,2,1,1,1] => 37 [7,4,1,1,1,1,1] => 22 [7,3,3,3] => 55 [7,3,3,2,1] => 50 [7,3,3,1,1,1] => 34 [7,3,2,2,2] => 51 [7,3,2,2,1,1] => 40 [7,3,2,1,1,1,1] => 29 [7,3,1,1,1,1,1,1] => 18 [7,2,2,2,2,1] => 31 [7,2,2,2,1,1,1] => 25 [7,2,2,1,1,1,1,1] => 19 [7,2,1,1,1,1,1,1,1] => 13 [7,1,1,1,1,1,1,1,1,1] => 7 [6,6,4] => 52 [6,6,3,1] => 46 [6,6,2,2] => 51 [6,6,2,1,1] => 36 [6,6,1,1,1,1] => 21 [6,5,5] => 50 [6,5,4,1] => 48 [6,5,3,2] => 66 [6,5,3,1,1] => 43 [6,5,2,2,1] => 48 [6,5,2,1,1,1] => 34 [6,5,1,1,1,1,1] => 20 [6,4,4,2] => 62 [6,4,4,1,1] => 40 [6,4,3,3] => 63 [6,4,3,2,1] => 56 [6,4,3,1,1,1] => 37 [6,4,2,2,2] => 54 [6,4,2,2,1,1] => 42 [6,4,2,1,1,1,1] => 30 [6,4,1,1,1,1,1,1] => 18 [6,3,3,3,1] => 45 [6,3,3,2,2] => 54 [6,3,3,2,1,1] => 41 [6,3,3,1,1,1,1] => 28 [6,3,2,2,2,1] => 42 [6,3,2,2,1,1,1] => 33 [6,3,2,1,1,1,1,1] => 24 [6,3,1,1,1,1,1,1,1] => 15 [6,2,2,2,2,2] => 31 [6,2,2,2,2,1,1] => 26 [6,2,2,2,1,1,1,1] => 21 [6,2,2,1,1,1,1,1,1] => 16 [6,2,1,1,1,1,1,1,1,1] => 11 [6,1,1,1,1,1,1,1,1,1,1] => 6 [5,5,5,1] => 35 [5,5,4,2] => 53 [5,5,4,1,1] => 34 [5,5,3,3] => 53 [5,5,3,2,1] => 47 [5,5,3,1,1,1] => 31 [5,5,2,2,2] => 45 [5,5,2,2,1,1] => 35 [5,5,2,1,1,1,1] => 25 [5,5,1,1,1,1,1,1] => 15 [5,4,4,3] => 53 [5,4,4,2,1] => 46 [5,4,4,1,1,1] => 30 [5,4,3,3,1] => 47 [5,4,3,2,2] => 56 [5,4,3,2,1,1] => 42 [5,4,3,1,1,1,1] => 28 [5,4,2,2,2,1] => 41 [5,4,2,2,1,1,1] => 32 [5,4,2,1,1,1,1,1] => 23 [5,4,1,1,1,1,1,1,1] => 14 [5,3,3,3,2] => 48 [5,3,3,3,1,1] => 35 [5,3,3,2,2,1] => 42 [5,3,3,2,1,1,1] => 32 [5,3,3,1,1,1,1,1] => 22 [5,3,2,2,2,2] => 40 [5,3,2,2,2,1,1] => 33 [5,3,2,2,1,1,1,1] => 26 [5,3,2,1,1,1,1,1,1] => 19 [5,3,1,1,1,1,1,1,1,1] => 12 [5,2,2,2,2,2,1] => 25 [5,2,2,2,2,1,1,1] => 21 [5,2,2,2,1,1,1,1,1] => 17 [5,2,2,1,1,1,1,1,1,1] => 13 [5,2,1,1,1,1,1,1,1,1,1] => 9 [5,1,1,1,1,1,1,1,1,1,1,1] => 5 [4,4,4,4] => 35 [4,4,4,3,1] => 34 [4,4,4,2,2] => 40 [4,4,4,2,1,1] => 30 [4,4,4,1,1,1,1] => 20 [4,4,3,3,2] => 43 [4,4,3,3,1,1] => 31 [4,4,3,2,2,1] => 37 [4,4,3,2,1,1,1] => 28 [4,4,3,1,1,1,1,1] => 19 [4,4,2,2,2,2] => 34 [4,4,2,2,2,1,1] => 28 [4,4,2,2,1,1,1,1] => 22 [4,4,2,1,1,1,1,1,1] => 16 [4,4,1,1,1,1,1,1,1,1] => 10 [4,3,3,3,3] => 36 [4,3,3,3,2,1] => 34 [4,3,3,3,1,1,1] => 25 [4,3,3,2,2,2] => 37 [4,3,3,2,2,1,1] => 30 [4,3,3,2,1,1,1,1] => 23 [4,3,3,1,1,1,1,1,1] => 16 [4,3,2,2,2,2,1] => 29 [4,3,2,2,2,1,1,1] => 24 [4,3,2,2,1,1,1,1,1] => 19 [4,3,2,1,1,1,1,1,1,1] => 14 [4,3,1,1,1,1,1,1,1,1,1] => 9 [4,2,2,2,2,2,2] => 22 [4,2,2,2,2,2,1,1] => 19 [4,2,2,2,2,1,1,1,1] => 16 [4,2,2,2,1,1,1,1,1,1] => 13 [4,2,2,1,1,1,1,1,1,1,1] => 10 [4,2,1,1,1,1,1,1,1,1,1,1] => 7 [4,1,1,1,1,1,1,1,1,1,1,1,1] => 4 [3,3,3,3,3,1] => 21 [3,3,3,3,2,2] => 25 [3,3,3,3,2,1,1] => 20 [3,3,3,3,1,1,1,1] => 15 [3,3,3,2,2,2,1] => 22 [3,3,3,2,2,1,1,1] => 18 [3,3,3,2,1,1,1,1,1] => 14 [3,3,3,1,1,1,1,1,1,1] => 10 [3,3,2,2,2,2,2] => 21 [3,3,2,2,2,2,1,1] => 18 [3,3,2,2,2,1,1,1,1] => 15 [3,3,2,2,1,1,1,1,1,1] => 12 [3,3,2,1,1,1,1,1,1,1,1] => 9 [3,3,1,1,1,1,1,1,1,1,1,1] => 6 [3,2,2,2,2,2,2,1] => 15 [3,2,2,2,2,2,1,1,1] => 13 [3,2,2,2,2,1,1,1,1,1] => 11 [3,2,2,2,1,1,1,1,1,1,1] => 9 [3,2,2,1,1,1,1,1,1,1,1,1] => 7 [3,2,1,1,1,1,1,1,1,1,1,1,1] => 5 [3,1,1,1,1,1,1,1,1,1,1,1,1,1] => 3 [2,2,2,2,2,2,2,2] => 9 [2,2,2,2,2,2,2,1,1] => 8 [2,2,2,2,2,2,1,1,1,1] => 7 [2,2,2,2,2,1,1,1,1,1,1] => 6 [2,2,2,2,1,1,1,1,1,1,1,1] => 5 [2,2,2,1,1,1,1,1,1,1,1,1,1] => 4 [2,2,1,1,1,1,1,1,1,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [17] => 17 [16,1] => 16 [15,2] => 29 [15,1,1] => 15 [14,3] => 39 [14,2,1] => 27 [14,1,1,1] => 14 [13,4] => 46 [13,3,1] => 36 [13,2,2] => 37 [13,2,1,1] => 25 [13,1,1,1,1] => 13 [12,5] => 50 [12,4,1] => 42 [12,3,2] => 54 [12,3,1,1] => 33 [12,2,2,1] => 34 [12,2,1,1,1] => 23 [12,1,1,1,1,1] => 12 [11,6] => 51 [11,5,1] => 45 [11,4,2] => 65 [11,4,1,1] => 38 [11,3,3] => 58 [11,3,2,1] => 49 [11,3,1,1,1] => 30 [11,2,2,2] => 41 [11,2,2,1,1] => 31 [11,2,1,1,1,1] => 21 [11,1,1,1,1,1,1] => 11 [10,7] => 49 [10,6,1] => 45 [10,5,2] => 70 [10,5,1,1] => 40 [10,4,3] => 73 [10,4,2,1] => 58 [10,4,1,1,1] => 34 [10,3,3,1] => 52 [10,3,2,2] => 61 [10,3,2,1,1] => 44 [10,3,1,1,1,1] => 27 [10,2,2,2,1] => 37 [10,2,2,1,1,1] => 28 [10,2,1,1,1,1,1] => 19 [10,1,1,1,1,1,1,1] => 10 [9,8] => 44 [9,7,1] => 42 [9,6,2] => 69 [9,6,1,1] => 39 [9,5,3] => 79 [9,5,2,1] => 61 [9,5,1,1,1] => 35 [9,4,4] => 70 [9,4,3,1] => 64 [9,4,2,2] => 72 [9,4,2,1,1] => 51 [9,4,1,1,1,1] => 30 [9,3,3,2] => 68 [9,3,3,1,1] => 46 [9,3,2,2,1] => 54 [9,3,2,1,1,1] => 39 [9,3,1,1,1,1,1] => 24 [9,2,2,2,2] => 41 [9,2,2,2,1,1] => 33 [9,2,2,1,1,1,1] => 25 [9,2,1,1,1,1,1,1] => 17 [9,1,1,1,1,1,1,1,1] => 9 [8,8,1] => 36 [8,7,2] => 62 [8,7,1,1] => 35 [8,6,3] => 76 [8,6,2,1] => 58 [8,6,1,1,1] => 33 [8,5,4] => 76 [8,5,3,1] => 67 [8,5,2,2] => 74 [8,5,2,1,1] => 52 [8,5,1,1,1,1] => 30 [8,4,4,1] => 60 [8,4,3,2] => 84 [8,4,3,1,1] => 55 [8,4,2,2,1] => 62 [8,4,2,1,1,1] => 44 [8,4,1,1,1,1,1] => 26 [8,3,3,3] => 65 [8,3,3,2,1] => 59 [8,3,3,1,1,1] => 40 [8,3,2,2,2] => 60 [8,3,2,2,1,1] => 47 [8,3,2,1,1,1,1] => 34 [8,3,1,1,1,1,1,1] => 21 [8,2,2,2,2,1] => 36 [8,2,2,2,1,1,1] => 29 [8,2,2,1,1,1,1,1] => 22 [8,2,1,1,1,1,1,1,1] => 15 [8,1,1,1,1,1,1,1,1,1] => 8 [7,7,3] => 64 [7,7,2,1] => 49 [7,7,1,1,1] => 28 [7,6,4] => 70 [7,6,3,1] => 61 [7,6,2,2] => 67 [7,6,2,1,1] => 47 [7,6,1,1,1,1] => 27 [7,5,5] => 65 [7,5,4,1] => 62 [7,5,3,2] => 85 [7,5,3,1,1] => 55 [7,5,2,2,1] => 61 [7,5,2,1,1,1] => 43 [7,5,1,1,1,1,1] => 25 [7,4,4,2] => 78 [7,4,4,1,1] => 50 [7,4,3,3] => 79 [7,4,3,2,1] => 70 [7,4,3,1,1,1] => 46 [7,4,2,2,2] => 67 [7,4,2,2,1,1] => 52 [7,4,2,1,1,1,1] => 37 [7,4,1,1,1,1,1,1] => 22 [7,3,3,3,1] => 55 [7,3,3,2,2] => 66 [7,3,3,2,1,1] => 50 [7,3,3,1,1,1,1] => 34 [7,3,2,2,2,1] => 51 [7,3,2,2,1,1,1] => 40 [7,3,2,1,1,1,1,1] => 29 [7,3,1,1,1,1,1,1,1] => 18 [7,2,2,2,2,2] => 37 [7,2,2,2,2,1,1] => 31 [7,2,2,2,1,1,1,1] => 25 [7,2,2,1,1,1,1,1,1] => 19 [7,2,1,1,1,1,1,1,1,1] => 13 [7,1,1,1,1,1,1,1,1,1,1] => 7 [6,6,5] => 55 [6,6,4,1] => 52 [6,6,3,2] => 71 [6,6,3,1,1] => 46 [6,6,2,2,1] => 51 [6,6,2,1,1,1] => 36 [6,6,1,1,1,1,1] => 21 [6,5,5,1] => 50 [6,5,4,2] => 76 [6,5,4,1,1] => 48 [6,5,3,3] => 75 [6,5,3,2,1] => 66 [6,5,3,1,1,1] => 43 [6,5,2,2,2] => 62 [6,5,2,2,1,1] => 48 [6,5,2,1,1,1,1] => 34 [6,5,1,1,1,1,1,1] => 20 [6,4,4,3] => 72 [6,4,4,2,1] => 62 [6,4,4,1,1,1] => 40 [6,4,3,3,1] => 63 [6,4,3,2,2] => 75 [6,4,3,2,1,1] => 56 [6,4,3,1,1,1,1] => 37 [6,4,2,2,2,1] => 54 [6,4,2,2,1,1,1] => 42 [6,4,2,1,1,1,1,1] => 30 [6,4,1,1,1,1,1,1,1] => 18 [6,3,3,3,2] => 62 [6,3,3,3,1,1] => 45 [6,3,3,2,2,1] => 54 [6,3,3,2,1,1,1] => 41 [6,3,3,1,1,1,1,1] => 28 [6,3,2,2,2,2] => 51 [6,3,2,2,2,1,1] => 42 [6,3,2,2,1,1,1,1] => 33 [6,3,2,1,1,1,1,1,1] => 24 [6,3,1,1,1,1,1,1,1,1] => 15 [6,2,2,2,2,2,1] => 31 [6,2,2,2,2,1,1,1] => 26 [6,2,2,2,1,1,1,1,1] => 21 [6,2,2,1,1,1,1,1,1,1] => 16 [6,2,1,1,1,1,1,1,1,1,1] => 11 [6,1,1,1,1,1,1,1,1,1,1,1] => 6 [5,5,5,2] => 55 [5,5,5,1,1] => 35 [5,5,4,3] => 62 [5,5,4,2,1] => 53 [5,5,4,1,1,1] => 34 [5,5,3,3,1] => 53 [5,5,3,2,2] => 63 [5,5,3,2,1,1] => 47 [5,5,3,1,1,1,1] => 31 [5,5,2,2,2,1] => 45 [5,5,2,2,1,1,1] => 35 [5,5,2,1,1,1,1,1] => 25 [5,5,1,1,1,1,1,1,1] => 15 [5,4,4,4] => 55 [5,4,4,3,1] => 53 [5,4,4,2,2] => 62 [5,4,4,2,1,1] => 46 [5,4,4,1,1,1,1] => 30 [5,4,3,3,2] => 66 [5,4,3,3,1,1] => 47 [5,4,3,2,2,1] => 56 [5,4,3,2,1,1,1] => 42 [5,4,3,1,1,1,1,1] => 28 [5,4,2,2,2,2] => 50 [5,4,2,2,2,1,1] => 41 [5,4,2,2,1,1,1,1] => 32 [5,4,2,1,1,1,1,1,1] => 23 [5,4,1,1,1,1,1,1,1,1] => 14 [5,3,3,3,3] => 51 [5,3,3,3,2,1] => 48 [5,3,3,3,1,1,1] => 35 [5,3,3,2,2,2] => 52 [5,3,3,2,2,1,1] => 42 [5,3,3,2,1,1,1,1] => 32 [5,3,3,1,1,1,1,1,1] => 22 [5,3,2,2,2,2,1] => 40 [5,3,2,2,2,1,1,1] => 33 [5,3,2,2,1,1,1,1,1] => 26 [5,3,2,1,1,1,1,1,1,1] => 19 [5,3,1,1,1,1,1,1,1,1,1] => 12 [5,2,2,2,2,2,2] => 29 [5,2,2,2,2,2,1,1] => 25 [5,2,2,2,2,1,1,1,1] => 21 [5,2,2,2,1,1,1,1,1,1] => 17 [5,2,2,1,1,1,1,1,1,1,1] => 13 [5,2,1,1,1,1,1,1,1,1,1,1] => 9 [5,1,1,1,1,1,1,1,1,1,1,1,1] => 5 [4,4,4,4,1] => 35 [4,4,4,3,2] => 48 [4,4,4,3,1,1] => 34 [4,4,4,2,2,1] => 40 [4,4,4,2,1,1,1] => 30 [4,4,4,1,1,1,1,1] => 20 [4,4,3,3,3] => 46 [4,4,3,3,2,1] => 43 [4,4,3,3,1,1,1] => 31 [4,4,3,2,2,2] => 46 [4,4,3,2,2,1,1] => 37 [4,4,3,2,1,1,1,1] => 28 [4,4,3,1,1,1,1,1,1] => 19 [4,4,2,2,2,2,1] => 34 [4,4,2,2,2,1,1,1] => 28 [4,4,2,2,1,1,1,1,1] => 22 [4,4,2,1,1,1,1,1,1,1] => 16 [4,4,1,1,1,1,1,1,1,1,1] => 10 [4,3,3,3,3,1] => 36 [4,3,3,3,2,2] => 43 [4,3,3,3,2,1,1] => 34 [4,3,3,3,1,1,1,1] => 25 [4,3,3,2,2,2,1] => 37 [4,3,3,2,2,1,1,1] => 30 [4,3,3,2,1,1,1,1,1] => 23 [4,3,3,1,1,1,1,1,1,1] => 16 [4,3,2,2,2,2,2] => 34 [4,3,2,2,2,2,1,1] => 29 [4,3,2,2,2,1,1,1,1] => 24 [4,3,2,2,1,1,1,1,1,1] => 19 [4,3,2,1,1,1,1,1,1,1,1] => 14 [4,3,1,1,1,1,1,1,1,1,1,1] => 9 [4,2,2,2,2,2,2,1] => 22 [4,2,2,2,2,2,1,1,1] => 19 [4,2,2,2,2,1,1,1,1,1] => 16 [4,2,2,2,1,1,1,1,1,1,1] => 13 [4,2,2,1,1,1,1,1,1,1,1,1] => 10 [4,2,1,1,1,1,1,1,1,1,1,1,1] => 7 [4,1,1,1,1,1,1,1,1,1,1,1,1,1] => 4 [3,3,3,3,3,2] => 27 [3,3,3,3,3,1,1] => 21 [3,3,3,3,2,2,1] => 25 [3,3,3,3,2,1,1,1] => 20 [3,3,3,3,1,1,1,1,1] => 15 [3,3,3,2,2,2,2] => 26 [3,3,3,2,2,2,1,1] => 22 [3,3,3,2,2,1,1,1,1] => 18 [3,3,3,2,1,1,1,1,1,1] => 14 [3,3,3,1,1,1,1,1,1,1,1] => 10 [3,3,2,2,2,2,2,1] => 21 [3,3,2,2,2,2,1,1,1] => 18 [3,3,2,2,2,1,1,1,1,1] => 15 [3,3,2,2,1,1,1,1,1,1,1] => 12 [3,3,2,1,1,1,1,1,1,1,1,1] => 9 [3,3,1,1,1,1,1,1,1,1,1,1,1] => 6 [3,2,2,2,2,2,2,2] => 17 [3,2,2,2,2,2,2,1,1] => 15 [3,2,2,2,2,2,1,1,1,1] => 13 [3,2,2,2,2,1,1,1,1,1,1] => 11 [3,2,2,2,1,1,1,1,1,1,1,1] => 9 [3,2,2,1,1,1,1,1,1,1,1,1,1] => 7 ----------------------------------------------------------------------------- Created: May 11, 2019 at 18:25 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: May 11, 2019 at 18:25 by Martin Rubey