***************************************************************************** * 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: St000170 ----------------------------------------------------------------------------- Collection: Semistandard tableaux ----------------------------------------------------------------------------- Description: The trace of a semistandard tableau. This is the sum of the entries on the diagonal. ----------------------------------------------------------------------------- References: [1] Stanley, R. P. The conjugate trace and trace of a plane partition [[MathSciNet:0309748]] [2] Gansner, E. R. The Hillman-Grassl correspondence and the enumeration of reverse plane partitions [[MathSciNet:0607040]] [3] Gansner, E. R. The enumeration of plane partitions via the Burge correspondence [[MathSciNet:0630832]] [4] Krattenthaler, C. Generating functions for plane partitions of a given shape [[MathSciNet:1072987]] ----------------------------------------------------------------------------- Code: def statistic(T): m = 0 for i, row in enumerate(T): try: m += row[i] except IndexError: break return m ----------------------------------------------------------------------------- Statistic values: [[1]] => 1 [[2]] => 2 [[3]] => 3 [[4]] => 4 [[5]] => 5 [[6]] => 6 [[1,1]] => 1 [[1,2]] => 1 [[2,2]] => 2 [[1],[2]] => 1 [[1,3]] => 1 [[2,3]] => 2 [[3,3]] => 3 [[1],[3]] => 1 [[2],[3]] => 2 [[1,4]] => 1 [[2,4]] => 2 [[3,4]] => 3 [[4,4]] => 4 [[1],[4]] => 1 [[2],[4]] => 2 [[3],[4]] => 3 [[1,5]] => 1 [[2,5]] => 2 [[3,5]] => 3 [[4,5]] => 4 [[5,5]] => 5 [[1],[5]] => 1 [[2],[5]] => 2 [[3],[5]] => 3 [[4],[5]] => 4 [[1,6]] => 1 [[2,6]] => 2 [[3,6]] => 3 [[4,6]] => 4 [[5,6]] => 5 [[6,6]] => 6 [[1],[6]] => 1 [[2],[6]] => 2 [[3],[6]] => 3 [[4],[6]] => 4 [[5],[6]] => 5 [[1,1,1]] => 1 [[1,1,2]] => 1 [[1,2,2]] => 1 [[2,2,2]] => 2 [[1,1],[2]] => 1 [[1,2],[2]] => 1 [[1,1,3]] => 1 [[1,2,3]] => 1 [[1,3,3]] => 1 [[2,2,3]] => 2 [[2,3,3]] => 2 [[3,3,3]] => 3 [[1,1],[3]] => 1 [[1,2],[3]] => 1 [[1,3],[2]] => 1 [[1,3],[3]] => 1 [[2,2],[3]] => 2 [[2,3],[3]] => 2 [[1],[2],[3]] => 1 [[1,1,4]] => 1 [[1,2,4]] => 1 [[1,3,4]] => 1 [[1,4,4]] => 1 [[2,2,4]] => 2 [[2,3,4]] => 2 [[2,4,4]] => 2 [[3,3,4]] => 3 [[3,4,4]] => 3 [[4,4,4]] => 4 [[1,1],[4]] => 1 [[1,2],[4]] => 1 [[1,4],[2]] => 1 [[1,3],[4]] => 1 [[1,4],[3]] => 1 [[1,4],[4]] => 1 [[2,2],[4]] => 2 [[2,3],[4]] => 2 [[2,4],[3]] => 2 [[2,4],[4]] => 2 [[3,3],[4]] => 3 [[3,4],[4]] => 3 [[1],[2],[4]] => 1 [[1],[3],[4]] => 1 [[2],[3],[4]] => 2 [[1,1,5]] => 1 [[1,2,5]] => 1 [[1,3,5]] => 1 [[1,4,5]] => 1 [[1,5,5]] => 1 [[2,2,5]] => 2 [[2,3,5]] => 2 [[2,4,5]] => 2 [[2,5,5]] => 2 [[3,3,5]] => 3 [[3,4,5]] => 3 [[3,5,5]] => 3 [[4,4,5]] => 4 [[4,5,5]] => 4 [[5,5,5]] => 5 [[1,1],[5]] => 1 [[1,2],[5]] => 1 [[1,5],[2]] => 1 [[1,3],[5]] => 1 [[1,5],[3]] => 1 [[1,4],[5]] => 1 [[1,5],[4]] => 1 [[1,5],[5]] => 1 [[2,2],[5]] => 2 [[2,3],[5]] => 2 [[2,5],[3]] => 2 [[2,4],[5]] => 2 [[2,5],[4]] => 2 [[2,5],[5]] => 2 [[3,3],[5]] => 3 [[3,4],[5]] => 3 [[3,5],[4]] => 3 [[3,5],[5]] => 3 [[4,4],[5]] => 4 [[4,5],[5]] => 4 [[1],[2],[5]] => 1 [[1],[3],[5]] => 1 [[1],[4],[5]] => 1 [[2],[3],[5]] => 2 [[2],[4],[5]] => 2 [[3],[4],[5]] => 3 [[1,1,1,1]] => 1 [[1,1,1,2]] => 1 [[1,1,2,2]] => 1 [[1,2,2,2]] => 1 [[2,2,2,2]] => 2 [[1,1,1],[2]] => 1 [[1,1,2],[2]] => 1 [[1,2,2],[2]] => 1 [[1,1],[2,2]] => 3 [[1,1,1,3]] => 1 [[1,1,2,3]] => 1 [[1,1,3,3]] => 1 [[1,2,2,3]] => 1 [[1,2,3,3]] => 1 [[1,3,3,3]] => 1 [[2,2,2,3]] => 2 [[2,2,3,3]] => 2 [[2,3,3,3]] => 2 [[3,3,3,3]] => 3 [[1,1,1],[3]] => 1 [[1,1,2],[3]] => 1 [[1,1,3],[2]] => 1 [[1,1,3],[3]] => 1 [[1,2,2],[3]] => 1 [[1,2,3],[2]] => 1 [[1,2,3],[3]] => 1 [[1,3,3],[2]] => 1 [[1,3,3],[3]] => 1 [[2,2,2],[3]] => 2 [[2,2,3],[3]] => 2 [[2,3,3],[3]] => 2 [[1,1],[2,3]] => 4 [[1,1],[3,3]] => 4 [[1,2],[2,3]] => 4 [[1,2],[3,3]] => 4 [[2,2],[3,3]] => 5 [[1,1],[2],[3]] => 1 [[1,2],[2],[3]] => 1 [[1,3],[2],[3]] => 1 [[1,1,1,4]] => 1 [[1,1,2,4]] => 1 [[1,1,3,4]] => 1 [[1,1,4,4]] => 1 [[1,2,2,4]] => 1 [[1,2,3,4]] => 1 [[1,2,4,4]] => 1 [[1,3,3,4]] => 1 [[1,3,4,4]] => 1 [[1,4,4,4]] => 1 [[2,2,2,4]] => 2 [[2,2,3,4]] => 2 [[2,2,4,4]] => 2 [[2,3,3,4]] => 2 [[2,3,4,4]] => 2 [[2,4,4,4]] => 2 [[3,3,3,4]] => 3 [[3,3,4,4]] => 3 [[3,4,4,4]] => 3 [[4,4,4,4]] => 4 [[1,1,1],[4]] => 1 [[1,1,2],[4]] => 1 [[1,1,4],[2]] => 1 [[1,1,3],[4]] => 1 [[1,1,4],[3]] => 1 [[1,1,4],[4]] => 1 [[1,2,2],[4]] => 1 [[1,2,4],[2]] => 1 [[1,2,3],[4]] => 1 [[1,2,4],[3]] => 1 [[1,3,4],[2]] => 1 [[1,2,4],[4]] => 1 [[1,4,4],[2]] => 1 [[1,3,3],[4]] => 1 [[1,3,4],[3]] => 1 [[1,3,4],[4]] => 1 [[1,4,4],[3]] => 1 [[1,4,4],[4]] => 1 [[2,2,2],[4]] => 2 [[2,2,3],[4]] => 2 [[2,2,4],[3]] => 2 [[2,2,4],[4]] => 2 [[2,3,3],[4]] => 2 [[2,3,4],[3]] => 2 [[2,3,4],[4]] => 2 [[2,4,4],[3]] => 2 [[2,4,4],[4]] => 2 [[3,3,3],[4]] => 3 [[3,3,4],[4]] => 3 [[3,4,4],[4]] => 3 [[1,1],[2,4]] => 5 [[1,1],[3,4]] => 5 [[1,1],[4,4]] => 5 [[1,2],[2,4]] => 5 [[1,2],[3,4]] => 5 [[1,3],[2,4]] => 5 [[1,2],[4,4]] => 5 [[1,3],[3,4]] => 5 [[1,3],[4,4]] => 5 [[2,2],[3,4]] => 6 [[2,2],[4,4]] => 6 [[2,3],[3,4]] => 6 [[2,3],[4,4]] => 6 [[3,3],[4,4]] => 7 [[1,1],[2],[4]] => 1 [[1,1],[3],[4]] => 1 [[1,2],[2],[4]] => 1 [[1,2],[3],[4]] => 1 [[1,3],[2],[4]] => 1 [[1,4],[2],[3]] => 1 [[1,4],[2],[4]] => 1 [[1,3],[3],[4]] => 1 [[1,4],[3],[4]] => 1 [[2,2],[3],[4]] => 2 [[2,3],[3],[4]] => 2 [[2,4],[3],[4]] => 2 [[1],[2],[3],[4]] => 1 [[1,1,1,1,1]] => 1 [[1,1,1,1,2]] => 1 [[1,1,1,2,2]] => 1 [[1,1,2,2,2]] => 1 [[1,2,2,2,2]] => 1 [[2,2,2,2,2]] => 2 [[1,1,1,1],[2]] => 1 [[1,1,1,2],[2]] => 1 [[1,1,2,2],[2]] => 1 [[1,2,2,2],[2]] => 1 [[1,1,1],[2,2]] => 3 [[1,1,2],[2,2]] => 3 [[1,1,1,1,3]] => 1 [[1,1,1,2,3]] => 1 [[1,1,1,3,3]] => 1 [[1,1,2,2,3]] => 1 [[1,1,2,3,3]] => 1 [[1,1,3,3,3]] => 1 [[1,2,2,2,3]] => 1 [[1,2,2,3,3]] => 1 [[1,2,3,3,3]] => 1 [[1,3,3,3,3]] => 1 [[2,2,2,2,3]] => 2 [[2,2,2,3,3]] => 2 [[2,2,3,3,3]] => 2 [[2,3,3,3,3]] => 2 [[3,3,3,3,3]] => 3 [[1,1,1,1],[3]] => 1 [[1,1,1,2],[3]] => 1 [[1,1,1,3],[2]] => 1 [[1,1,1,3],[3]] => 1 [[1,1,2,2],[3]] => 1 [[1,1,2,3],[2]] => 1 [[1,1,2,3],[3]] => 1 [[1,1,3,3],[2]] => 1 [[1,1,3,3],[3]] => 1 [[1,2,2,2],[3]] => 1 [[1,2,2,3],[2]] => 1 [[1,2,2,3],[3]] => 1 [[1,2,3,3],[2]] => 1 [[1,2,3,3],[3]] => 1 [[1,3,3,3],[2]] => 1 [[1,3,3,3],[3]] => 1 [[2,2,2,2],[3]] => 2 [[2,2,2,3],[3]] => 2 [[2,2,3,3],[3]] => 2 [[2,3,3,3],[3]] => 2 [[1,1,1],[2,3]] => 4 [[1,1,1],[3,3]] => 4 [[1,1,2],[2,3]] => 4 [[1,1,3],[2,2]] => 3 [[1,1,2],[3,3]] => 4 [[1,1,3],[2,3]] => 4 [[1,1,3],[3,3]] => 4 [[1,2,2],[2,3]] => 4 [[1,2,2],[3,3]] => 4 [[1,2,3],[2,3]] => 4 [[1,2,3],[3,3]] => 4 [[2,2,2],[3,3]] => 5 [[2,2,3],[3,3]] => 5 [[1,1,1],[2],[3]] => 1 [[1,1,2],[2],[3]] => 1 [[1,1,3],[2],[3]] => 1 [[1,2,2],[2],[3]] => 1 [[1,2,3],[2],[3]] => 1 [[1,3,3],[2],[3]] => 1 [[1,1],[2,2],[3]] => 3 [[1,1],[2,3],[3]] => 4 [[1,2],[2,3],[3]] => 4 [[1,2,3,4,5]] => 1 [[1,2,3,4],[5]] => 1 [[1,2,3,5],[4]] => 1 [[1,2,4,5],[3]] => 1 [[1,3,4,5],[2]] => 1 [[1,2,3],[4,5]] => 6 [[1,2,4],[3,5]] => 6 [[1,2,5],[3,4]] => 5 [[1,3,4],[2,5]] => 6 [[1,3,5],[2,4]] => 5 [[1,2,3],[4],[5]] => 1 [[1,2,4],[3],[5]] => 1 [[1,2,5],[3],[4]] => 1 [[1,3,4],[2],[5]] => 1 [[1,3,5],[2],[4]] => 1 [[1,4,5],[2],[3]] => 1 [[1,2],[3,4],[5]] => 5 [[1,2],[3,5],[4]] => 6 [[1,3],[2,4],[5]] => 5 [[1,3],[2,5],[4]] => 6 [[1,4],[2,5],[3]] => 6 [[1,2],[3],[4],[5]] => 1 [[1,3],[2],[4],[5]] => 1 [[1,4],[2],[3],[5]] => 1 [[1,5],[2],[3],[4]] => 1 [[1],[2],[3],[4],[5]] => 1 [[1,1,1,1,1,2]] => 1 [[1,1,1,1,2,2]] => 1 [[1,1,1,2,2,2]] => 1 [[1,1,2,2,2,2]] => 1 [[1,2,2,2,2,2]] => 1 [[2,2,2,2,2,2]] => 2 [[1,1,1,1,1],[2]] => 1 [[1,1,1,1,2],[2]] => 1 [[1,1,1,2,2],[2]] => 1 [[1,1,2,2,2],[2]] => 1 [[1,2,2,2,2],[2]] => 1 [[1,1,1,1],[2,2]] => 3 [[1,1,1,2],[2,2]] => 3 [[1,1,2,2],[2,2]] => 3 [[1,1,1],[2,2,2]] => 3 [[1,1,1,2,2],[3]] => 1 [[1,1,1,2,3],[3]] => 1 [[1,1,2,2,2],[3]] => 1 [[1,1,2,2,3],[3]] => 1 [[1,1,2,3,3],[3]] => 1 [[1,2,2,2,3],[3]] => 1 [[1,2,2,3,3],[3]] => 1 [[1,1,1],[2,2],[3]] => 3 [[1,1,1],[2,3],[3]] => 4 [[1,1,2],[2,2],[3]] => 3 [[1,1,2],[2,3],[3]] => 4 [[1,1,3],[2,3],[3]] => 4 [[1,2,2],[2,3],[3]] => 4 [[1,2,3],[2,3],[3]] => 4 [[1,2,3,4,5,6]] => 1 [[1,2,3,4,5],[6]] => 1 [[1,2,3,4,6],[5]] => 1 [[1,2,3,4],[5],[6]] => 1 [[1,2,3,5,6],[4]] => 1 [[1,2,3,5],[4,6]] => 7 [[1,2,3,5],[4],[6]] => 1 [[1,2,3,4],[5,6]] => 7 [[1,2,3,6],[4],[5]] => 1 [[1,2,3],[4],[5],[6]] => 1 [[1,2,4,5,6],[3]] => 1 [[1,2,4,5],[3,6]] => 7 [[1,2,4,6],[3,5]] => 6 [[1,2,4],[3,5],[6]] => 6 [[1,2,4,5],[3],[6]] => 1 [[1,2,4,6],[3],[5]] => 1 [[1,2,4],[3],[5],[6]] => 1 [[1,2,3,6],[4,5]] => 6 [[1,2,3],[4,5],[6]] => 6 [[1,2,5,6],[3],[4]] => 1 [[1,2,5],[3,6],[4]] => 7 [[1,2,5],[3],[4],[6]] => 1 [[1,2,3],[4,6],[5]] => 7 [[1,2,4],[3,6],[5]] => 7 [[1,2,6],[3],[4],[5]] => 1 [[1,2],[3],[4],[5],[6]] => 1 [[1,3,4,5,6],[2]] => 1 [[1,3,4,5],[2,6]] => 7 [[1,3,4,6],[2,5]] => 6 [[1,3,4],[2,5],[6]] => 6 [[1,3,5,6],[2,4]] => 5 [[1,3,5],[2,4,6]] => 5 [[1,3,5],[2,4],[6]] => 5 [[1,3,4],[2,5,6]] => 6 [[1,3,6],[2,4],[5]] => 5 [[1,3],[2,4],[5],[6]] => 5 [[1,3,4,5],[2],[6]] => 1 [[1,3,4,6],[2],[5]] => 1 [[1,3,4],[2],[5],[6]] => 1 [[1,3,5,6],[2],[4]] => 1 [[1,3,5],[2,6],[4]] => 7 [[1,3,5],[2],[4],[6]] => 1 [[1,3,4],[2,6],[5]] => 7 [[1,3,6],[2],[4],[5]] => 1 [[1,3],[2],[4],[5],[6]] => 1 [[1,2,5,6],[3,4]] => 5 [[1,2,5],[3,4,6]] => 5 [[1,2,5],[3,4],[6]] => 5 [[1,2,4],[3,5,6]] => 6 [[1,2,6],[3,4],[5]] => 5 [[1,2],[3,4],[5],[6]] => 5 [[1,4,5,6],[2],[3]] => 1 [[1,4,5],[2,6],[3]] => 7 [[1,4,6],[2,5],[3]] => 6 [[1,4],[2,5],[3,6]] => 6 [[1,4],[2,5],[3],[6]] => 6 [[1,4,5],[2],[3],[6]] => 1 [[1,4,6],[2],[3],[5]] => 1 [[1,4],[2],[3],[5],[6]] => 1 [[1,2,3],[4,5,6]] => 6 [[1,2,6],[3,5],[4]] => 6 [[1,2],[3,5],[4],[6]] => 6 [[1,3,6],[2,5],[4]] => 6 [[1,3],[2,5],[4,6]] => 6 [[1,3],[2,5],[4],[6]] => 6 [[1,2],[3,5],[4,6]] => 6 [[1,5,6],[2],[3],[4]] => 1 [[1,5],[2,6],[3],[4]] => 7 [[1,5],[2],[3],[4],[6]] => 1 [[1,2],[3,6],[4],[5]] => 7 [[1,3],[2,4],[5,6]] => 5 [[1,3],[2,6],[4],[5]] => 7 [[1,2],[3,4],[5,6]] => 5 [[1,4],[2,6],[3],[5]] => 7 [[1,6],[2],[3],[4],[5]] => 1 [[1],[2],[3],[4],[5],[6]] => 1 [[1,1,1,1],[2,2,2],[3,3],[4]] => 3 [[1,1,1,2],[2,2,2],[3,3],[4]] => 3 [[1,1,1,1],[2,2,3],[3,3],[4]] => 3 [[1,1,1,2],[2,2,3],[3,3],[4]] => 3 [[1,1,1,3],[2,2,3],[3,3],[4]] => 3 [[1,1,2,2],[2,2,3],[3,3],[4]] => 3 [[1,1,2,3],[2,2,3],[3,3],[4]] => 3 [[1,1,1,1],[2,2,2],[3,4],[4]] => 3 [[1,1,1,2],[2,2,2],[3,4],[4]] => 3 [[1,1,1,1],[2,2,3],[3,4],[4]] => 3 [[1,1,1,2],[2,2,3],[3,4],[4]] => 3 [[1,1,1,3],[2,2,3],[3,4],[4]] => 3 [[1,1,2,2],[2,2,3],[3,4],[4]] => 3 [[1,1,2,3],[2,2,3],[3,4],[4]] => 3 [[1,1,1,1],[2,2,4],[3,4],[4]] => 3 [[1,1,1,2],[2,2,4],[3,4],[4]] => 3 [[1,1,1,3],[2,2,4],[3,4],[4]] => 3 [[1,1,1,4],[2,2,4],[3,4],[4]] => 3 [[1,1,2,2],[2,2,4],[3,4],[4]] => 3 [[1,1,2,3],[2,2,4],[3,4],[4]] => 3 [[1,1,2,4],[2,2,4],[3,4],[4]] => 3 [[1,1,1,1],[2,3,3],[3,4],[4]] => 4 [[1,1,1,2],[2,3,3],[3,4],[4]] => 4 [[1,1,1,3],[2,3,3],[3,4],[4]] => 4 [[1,1,2,2],[2,3,3],[3,4],[4]] => 4 [[1,1,2,3],[2,3,3],[3,4],[4]] => 4 [[1,1,1,1],[2,3,4],[3,4],[4]] => 4 [[1,1,1,2],[2,3,4],[3,4],[4]] => 4 [[1,1,1,3],[2,3,4],[3,4],[4]] => 4 [[1,1,1,4],[2,3,4],[3,4],[4]] => 4 [[1,1,2,2],[2,3,4],[3,4],[4]] => 4 [[1,1,2,3],[2,3,4],[3,4],[4]] => 4 [[1,1,2,4],[2,3,4],[3,4],[4]] => 4 [[1,1,3,3],[2,3,4],[3,4],[4]] => 4 [[1,1,3,4],[2,3,4],[3,4],[4]] => 4 [[1,2,2,2],[2,3,3],[3,4],[4]] => 4 [[1,2,2,3],[2,3,3],[3,4],[4]] => 4 [[1,2,2,2],[2,3,4],[3,4],[4]] => 4 [[1,2,2,3],[2,3,4],[3,4],[4]] => 4 [[1,2,2,4],[2,3,4],[3,4],[4]] => 4 [[1,2,3,3],[2,3,4],[3,4],[4]] => 4 [[1,2,3,4],[2,3,4],[3,4],[4]] => 4 [[1,1,1,1,2,2,2],[3,3],[4]] => 4 [[1,1,1,2,2,2,2],[3,3],[4]] => 4 [[1,1,1,1,2,2,3],[3,3],[4]] => 4 [[1,1,1,2,2,2,3],[3,3],[4]] => 4 [[1,1,1,2,2,3,3],[3,3],[4]] => 4 [[1,1,2,2,2,2,3],[3,3],[4]] => 4 [[1,1,2,2,2,3,3],[3,3],[4]] => 4 [[1,1,1,1,2,2,2],[3,4],[4]] => 5 [[1,1,1,2,2,2,2],[3,4],[4]] => 5 [[1,1,1,1,2,2,3],[3,4],[4]] => 5 [[1,1,1,2,2,2,3],[3,4],[4]] => 5 [[1,1,1,2,2,3],[3,3,4],[4]] => 4 [[1,1,2,2,2,2,3],[3,4],[4]] => 5 [[1,1,2,2,2,3],[3,3,4],[4]] => 4 [[1,1,1,1,2,2,4],[3,4],[4]] => 5 [[1,1,1,2,2,2,4],[3,4],[4]] => 5 [[1,1,1,2,2,4,4],[3,3],[4]] => 4 [[1,1,1,2,2,4,4],[3,4],[4]] => 5 [[1,1,2,2,2,2,4],[3,4],[4]] => 5 [[1,1,2,2,2,4,4],[3,3],[4]] => 4 [[1,1,2,2,2,4,4],[3,4],[4]] => 5 [[1,1,1,1,2,3,3],[3,4],[4]] => 5 [[1,1,1,2,2,3,3],[3,4],[4]] => 5 [[1,1,1,2,3,3,3],[3,4],[4]] => 5 [[1,1,2,2,2,3,3],[3,4],[4]] => 5 [[1,1,2,2,3,3,3],[3,4],[4]] => 5 [[1,1,1,1,2,3,4],[3,4],[4]] => 5 [[1,1,1,2,2,3,4],[3,4],[4]] => 5 [[1,1,1,2,3,3,4],[3,4],[4]] => 5 [[1,1,1,2,3,4,4],[3,4],[4]] => 5 [[1,1,2,2,2,3,4],[3,4],[4]] => 5 [[1,1,2,2,3,3,4],[3,4],[4]] => 5 [[1,1,2,2,3,4,4],[3,4],[4]] => 5 [[1,1,2,3,3,3,4],[3,4],[4]] => 5 [[1,1,2,3,3,4,4],[3,4],[4]] => 5 [[1,2,2,2,2,3,3],[3,4],[4]] => 5 [[1,2,2,2,3,3,3],[3,4],[4]] => 5 [[1,2,2,2,2,3,4],[3,4],[4]] => 5 [[1,2,2,2,3,3,4],[3,4],[4]] => 5 [[1,2,2,2,3,4,4],[3,4],[4]] => 5 [[1,2,2,3,3,3,4],[3,4],[4]] => 5 [[1,2,2,3,3,4,4],[3,4],[4]] => 5 [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]] => 6 [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]] => 6 [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]] => 6 [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]] => 6 [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]] => 6 [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]] => 6 [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]] => 6 [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]] => 7 [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]] => 7 [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]] => 7 [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,4],[5]] => 7 [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,4],[5]] => 7 [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,4],[5]] => 7 [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]] => 7 [[1,1,1,1,2],[2,2,2,4],[3,3,4],[4,4],[5]] => 7 [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,4],[5]] => 7 [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]] => 7 [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]] => 7 [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]] => 7 [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]] => 7 [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]] => 7 [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]] => 7 [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,4],[5]] => 7 [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]] => 7 [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]] => 7 [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]] => 7 [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]] => 7 [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]] => 7 [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,4],[5]] => 7 [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,4],[5]] => 7 [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]] => 7 [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,4],[5]] => 7 [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]] => 7 [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]] => 6 [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]] => 6 [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]] => 6 [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,5],[5]] => 6 [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]] => 6 [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]] => 6 [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]] => 6 [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]] => 7 [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]] => 7 [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,5],[5]] => 7 [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,5],[5]] => 7 [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,5],[5]] => 7 [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,5],[5]] => 7 [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,5],[5]] => 7 [[1,1,1,1,2],[2,2,2,4],[3,3,4],[4,5],[5]] => 7 [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,5],[5]] => 7 [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,5],[5]] => 7 [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,5],[5]] => 7 [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,5],[5]] => 7 [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,5],[5]] => 7 [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,5],[5]] => 7 [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,1],[2,2,2,3],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,2],[2,2,2,3],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,3],[2,2,2,3],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,1],[2,2,2,4],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,2],[2,2,2,4],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,3],[2,2,2,4],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,4],[2,2,2,4],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,2],[2,2,2,5],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,3],[2,2,2,5],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,4],[2,2,2,5],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]] => 8 [[1,1,1,2,2],[2,2,2,5],[3,3,5],[4,5],[5]] => 8 [[1,1,1,2,5],[2,2,2,5],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,1],[2,2,3,5],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,5],[2,2,3,5],[3,3,5],[4,5],[5]] => 8 [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]] => 8 [[1,1,1,3,5],[2,2,3,5],[3,3,5],[4,5],[5]] => 8 [[1,1,2,2,2],[2,2,3,3],[3,3,5],[4,5],[5]] => 8 [[1,1,2,2,3],[2,2,3,3],[3,3,5],[4,5],[5]] => 8 [[1,1,2,2,2],[2,2,3,5],[3,3,5],[4,5],[5]] => 8 [[1,1,2,2,5],[2,2,3,5],[3,3,5],[4,5],[5]] => 8 [[1,1,2,3,3],[2,2,3,5],[3,3,5],[4,5],[5]] => 8 [[1,1,2,3,5],[2,2,3,5],[3,3,5],[4,5],[5]] => 8 [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]] => 7 [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]] => 7 [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]] => 7 [[1,1,1,1,4],[2,2,2,4],[3,4,4],[4,5],[5]] => 7 [[1,1,1,2,2],[2,2,2,4],[3,4,4],[4,5],[5]] => 7 [[1,1,1,2,4],[2,2,2,4],[3,4,4],[4,5],[5]] => 7 [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]] => 8 [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]] => 8 [[1,1,1,1,1],[2,2,2,3],[3,4,5],[4,5],[5]] => 8 [[1,1,1,1,2],[2,2,2,3],[3,4,5],[4,5],[5]] => 8 [[1,1,1,2,3],[2,2,2,3],[3,4,5],[4,5],[5]] => 8 [[1,1,1,1,1],[2,2,2,5],[3,4,5],[4,5],[5]] => 8 [[1,1,1,1,5],[2,2,2,5],[3,4,5],[4,5],[5]] => 8 [[1,1,1,2,2],[2,2,2,5],[3,4,5],[4,5],[5]] => 8 [[1,1,1,2,5],[2,2,2,5],[3,4,5],[4,5],[5]] => 8 [[1,1,1,1,1],[2,2,3,4],[3,4,5],[4,5],[5]] => 8 [[1,1,1,1,2],[2,2,3,4],[3,4,5],[4,5],[5]] => 8 [[1,1,1,2,3],[2,2,3,4],[3,4,5],[4,5],[5]] => 8 [[1,1,2,3,4],[2,2,3,4],[3,4,5],[4,5],[5]] => 8 [[1,1,1,1,1],[2,2,4,4],[3,4,5],[4,5],[5]] => 8 [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]] => 8 [[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]] => 8 [[1,1,1,1,5],[2,2,4,5],[3,4,5],[4,5],[5]] => 8 [[1,1,1,4,4],[2,2,4,5],[3,4,5],[4,5],[5]] => 8 [[1,1,1,4,5],[2,2,4,5],[3,4,5],[4,5],[5]] => 8 [[1,1,2,2,2],[2,2,4,4],[3,4,5],[4,5],[5]] => 8 [[1,1,2,2,4],[2,2,4,4],[3,4,5],[4,5],[5]] => 8 [[1,1,2,2,2],[2,2,4,5],[3,4,5],[4,5],[5]] => 8 [[1,1,2,2,5],[2,2,4,5],[3,4,5],[4,5],[5]] => 8 [[1,1,2,4,4],[2,2,4,5],[3,4,5],[4,5],[5]] => 8 [[1,1,2,4,5],[2,2,4,5],[3,4,5],[4,5],[5]] => 8 [[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]] => 8 [[1,1,1,1,3],[2,3,3,3],[3,4,4],[4,5],[5]] => 8 [[1,1,1,1,1],[2,3,3,4],[3,4,4],[4,5],[5]] => 8 [[1,1,1,1,4],[2,3,3,4],[3,4,4],[4,5],[5]] => 8 [[1,1,1,3,3],[2,3,3,4],[3,4,4],[4,5],[5]] => 8 [[1,1,1,3,4],[2,3,3,4],[3,4,4],[4,5],[5]] => 8 [[1,1,1,1,1],[2,3,3,3],[3,4,5],[4,5],[5]] => 9 [[1,1,1,1,3],[2,3,3,3],[3,4,5],[4,5],[5]] => 9 [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]] => 9 [[1,1,1,1,5],[2,3,3,5],[3,4,5],[4,5],[5]] => 9 [[1,1,1,3,3],[2,3,3,5],[3,4,5],[4,5],[5]] => 9 [[1,1,1,3,5],[2,3,3,5],[3,4,5],[4,5],[5]] => 9 [[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]] => 9 [[1,1,1,1,4],[2,3,4,4],[3,4,5],[4,5],[5]] => 9 [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,1,1,1,2],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,1,1,1,5],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,1,1,2,3],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,1,1,4,4],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,1,1,4,5],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,1,2,3,4],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,1,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]] => 9 [[1,1,3,3,4],[2,3,4,4],[3,4,5],[4,5],[5]] => 9 [[1,1,3,3,3],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,1,3,3,5],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,1,3,4,4],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,1,3,4,5],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,2,2,2,2],[2,3,3,3],[3,4,4],[4,5],[5]] => 8 [[1,2,2,2,3],[2,3,3,3],[3,4,4],[4,5],[5]] => 8 [[1,2,2,2,2],[2,3,3,4],[3,4,4],[4,5],[5]] => 8 [[1,2,2,2,4],[2,3,3,4],[3,4,4],[4,5],[5]] => 8 [[1,2,2,3,3],[2,3,3,4],[3,4,4],[4,5],[5]] => 8 [[1,2,2,3,4],[2,3,3,4],[3,4,4],[4,5],[5]] => 8 [[1,2,2,2,2],[2,3,3,3],[3,4,5],[4,5],[5]] => 9 [[1,2,2,2,3],[2,3,3,3],[3,4,5],[4,5],[5]] => 9 [[1,2,2,2,2],[2,3,3,5],[3,4,5],[4,5],[5]] => 9 [[1,2,2,2,5],[2,3,3,5],[3,4,5],[4,5],[5]] => 9 [[1,2,2,3,3],[2,3,3,5],[3,4,5],[4,5],[5]] => 9 [[1,2,2,3,5],[2,3,3,5],[3,4,5],[4,5],[5]] => 9 [[1,2,2,2,2],[2,3,4,4],[3,4,5],[4,5],[5]] => 9 [[1,2,2,2,4],[2,3,4,4],[3,4,5],[4,5],[5]] => 9 [[1,2,2,2,2],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,2,2,2,5],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,2,2,4,4],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,2,2,4,5],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,2,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]] => 9 [[1,2,3,3,4],[2,3,4,4],[3,4,5],[4,5],[5]] => 9 [[1,2,3,3,3],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,2,3,3,5],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,2,3,4,4],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,2,3,4,5],[2,3,4,5],[3,4,5],[4,5],[5]] => 9 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]] => 6 [[1,1,1,2,3,4],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,4],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,4],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,4],[3,3,3,4],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,4],[3,3,3,4],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,5],[6]] => 6 [[1,1,1,2,3,4],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,5],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,5],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,5],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,5],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,4],[3,3,3,5],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,4],[3,3,3,5],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,5],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,4],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,5],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]] => 7 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]] => 7 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,4,5],[5,5],[6]] => 7 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,4,5],[4,4,5],[5,5],[6]] => 7 [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,4,5],[4,4,5],[5,5],[6]] => 7 [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,5],[6]] => 7 [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,5],[6]] => 7 [[1,1,1,1,2,3],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,5],[6]] => 7 [[1,1,1,2,3,4],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,5],[6]] => 7 [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,5],[6]] => 7 [[1,1,1,1,1,2],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,5],[6]] => 7 [[1,1,1,1,2,3],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,5],[6]] => 7 [[1,1,1,2,3,4],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,5],[6]] => 7 [[1,1,2,3,4,5],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,5],[6]] => 7 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]] => 6 [[1,1,1,2,3,4],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,4],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,4],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,4],[3,3,3,4],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,4],[3,3,3,4],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,6],[6]] => 6 [[1,1,1,2,3,4],[2,2,2,3,4],[3,3,3,4],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,5],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,5],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,5],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,5],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,4],[3,3,3,5],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,4],[3,3,3,5],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,5],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,4],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,5],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,6],[6]] => 7 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,6],[6]] => 7 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,4,5],[5,6],[6]] => 7 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,4,5],[4,4,5],[5,6],[6]] => 7 [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,4,5],[4,4,5],[5,6],[6]] => 7 [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]] => 7 [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]] => 7 [[1,1,1,1,2,3],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]] => 7 [[1,1,1,2,3,4],[2,2,2,3,4],[3,3,4,5],[4,4,5],[5,6],[6]] => 7 [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,6],[6]] => 7 [[1,1,1,1,1,2],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,6],[6]] => 7 [[1,1,1,1,2,3],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,6],[6]] => 7 [[1,1,1,2,3,4],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,6],[6]] => 7 [[1,1,2,3,4,5],[2,2,3,4,5],[3,3,4,5],[4,4,5],[5,6],[6]] => 7 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,4],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,4],[3,3,3,4],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,4],[3,3,3,4],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,5],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,5],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,5],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,5],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,4],[3,3,3,5],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,4],[3,3,3,5],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,5],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,5],[3,3,3,5],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,5],[3,3,3,5],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,4],[2,2,2,2,5],[3,3,3,5],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,5],[2,2,2,2,5],[3,3,3,5],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,4],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,4],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,5],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,5],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,4],[2,2,2,2,5],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,5],[2,2,2,2,5],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,6],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,6],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,3],[2,2,2,2,6],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,4],[2,2,2,2,6],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,5],[2,2,2,2,6],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,6],[2,2,2,2,6],[3,3,3,6],[4,4,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,5,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,5,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,4],[4,5,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,5,6],[5,6],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,4],[4,5,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,5,6],[5,6],[6]] => 6 [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,3,4],[4,5,6],[5,6],[6]] => 6 [[1,1,1,1,2,3],[2,2,2,3,4],[3,3,3,4],[4,5,6],[5,6],[6]] => 6 [[1,1,1,2,3,4],[2,2,2,3,4],[3,3,3,4],[4,5,6],[5,6],[6]] => 6 [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,5,6],[5,6],[6]] => 7 [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,5],[4,5,6],[5,6],[6]] => 7 [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,4,5],[4,5,6],[5,6],[6]] => 7 [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,4,5],[4,5,6],[5,6],[6]] => 7 [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,4,5],[4,5,6],[5,6],[6]] => 7 [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,4,5],[4,5,6],[5,6],[6]] => 7 [[1,1,1,1,1,2],[2,2,2,3,4],[3,3,4,5],[4,5,6],[5,6],[6]] => 7 [[1,1,1,1,2,3],[2,2,2,3,4],[3,3,4,5],[4,5,6],[5,6],[6]] => 7 [[1,1,1,2,3,4],[2,2,2,3,4],[3,3,4,5],[4,5,6],[5,6],[6]] => 7 [[1,1,1,1,1,1],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]] => 7 [[1,1,1,1,1,2],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]] => 7 [[1,1,1,1,2,3],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]] => 7 [[1,1,1,2,3,4],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]] => 7 [[1,1,2,3,4,5],[2,2,3,4,5],[3,3,4,5],[4,5,6],[5,6],[6]] => 7 [[1,1,1,1,1,1],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]] => 8 [[1,1,1,1,1,2],[2,2,2,2,2],[3,4,5,6],[4,5,6],[5,6],[6]] => 8 [[1,1,1,1,1,1],[2,2,2,2,3],[3,4,5,6],[4,5,6],[5,6],[6]] => 8 [[1,1,1,1,1,2],[2,2,2,2,3],[3,4,5,6],[4,5,6],[5,6],[6]] => 8 [[1,1,1,1,2,3],[2,2,2,2,3],[3,4,5,6],[4,5,6],[5,6],[6]] => 8 [[1,1,1,1,1,1],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]] => 8 [[1,1,1,1,1,2],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]] => 8 [[1,1,1,1,2,3],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]] => 8 [[1,1,1,2,3,4],[2,2,2,3,4],[3,4,5,6],[4,5,6],[5,6],[6]] => 8 [[1,1,1,1,1,1],[2,2,3,4,5],[3,4,5,6],[4,5,6],[5,6],[6]] => 8 [[1,1,1,1,1,2],[2,2,3,4,5],[3,4,5,6],[4,5,6],[5,6],[6]] => 8 [[1,1,1,1,2,3],[2,2,3,4,5],[3,4,5,6],[4,5,6],[5,6],[6]] => 8 [[1,1,1,2,3,4],[2,2,3,4,5],[3,4,5,6],[4,5,6],[5,6],[6]] => 8 [[1,1,2,3,4,5],[2,2,3,4,5],[3,4,5,6],[4,5,6],[5,6],[6]] => 8 [[1,1,1,1,1,1],[2,3,4,5,6],[3,4,5,6],[4,5,6],[5,6],[6]] => 9 [[1,1,1,1,1,2],[2,3,4,5,6],[3,4,5,6],[4,5,6],[5,6],[6]] => 9 [[1,1,1,1,2,3],[2,3,4,5,6],[3,4,5,6],[4,5,6],[5,6],[6]] => 9 [[1,1,1,2,3,4],[2,3,4,5,6],[3,4,5,6],[4,5,6],[5,6],[6]] => 9 [[1,1,2,3,4,5],[2,3,4,5,6],[3,4,5,6],[4,5,6],[5,6],[6]] => 9 [[1,2,3,4,5,6],[2,3,4,5,6],[3,4,5,6],[4,5,6],[5,6],[6]] => 9 ----------------------------------------------------------------------------- Created: Nov 29, 2013 at 23:28 by Christian Krattenthaler ----------------------------------------------------------------------------- Last Updated: Aug 09, 2019 at 12:47 by Martin Rubey