***************************************************************************** * 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: St000112 ----------------------------------------------------------------------------- Collection: Semistandard tableaux ----------------------------------------------------------------------------- Description: The sum of the entries reduced by the index of their row in a semistandard tableau. This is also the depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is the shape of $T$, independent of the Cartan rank. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(T): return sum(e-i for i, row in enumerate(T, 1) for e in row) ----------------------------------------------------------------------------- Statistic values: [[1]] => 0 [[2]] => 1 [[1,1]] => 0 [[1,2]] => 1 [[2,2]] => 2 [[1],[2]] => 0 [[1,3]] => 2 [[2,3]] => 3 [[3,3]] => 4 [[1],[3]] => 1 [[2],[3]] => 2 [[1,4]] => 3 [[2,4]] => 4 [[3,4]] => 5 [[4,4]] => 6 [[1],[4]] => 2 [[2],[4]] => 3 [[3],[4]] => 4 [[1,5]] => 4 [[2,5]] => 5 [[3,5]] => 6 [[4,5]] => 7 [[5,5]] => 8 [[1],[5]] => 3 [[2],[5]] => 4 [[3],[5]] => 5 [[4],[5]] => 6 [[1,6]] => 5 [[2,6]] => 6 [[3,6]] => 7 [[4,6]] => 8 [[5,6]] => 9 [[6,6]] => 10 [[1],[6]] => 4 [[2],[6]] => 5 [[3],[6]] => 6 [[4],[6]] => 7 [[5],[6]] => 8 [[1,1,1]] => 0 [[1,1,2]] => 1 [[1,2,2]] => 2 [[2,2,2]] => 3 [[1,1],[2]] => 0 [[1,2],[2]] => 1 [[1,1,3]] => 2 [[1,2,3]] => 3 [[1,3,3]] => 4 [[2,2,3]] => 4 [[2,3,3]] => 5 [[3,3,3]] => 6 [[1,1],[3]] => 1 [[1,2],[3]] => 2 [[1,3],[2]] => 2 [[1,3],[3]] => 3 [[2,2],[3]] => 3 [[2,3],[3]] => 4 [[1],[2],[3]] => 0 [[1,1,4]] => 3 [[1,2,4]] => 4 [[1,3,4]] => 5 [[1,4,4]] => 6 [[2,2,4]] => 5 [[2,3,4]] => 6 [[2,4,4]] => 7 [[3,3,4]] => 7 [[3,4,4]] => 8 [[4,4,4]] => 9 [[1,1],[4]] => 2 [[1,2],[4]] => 3 [[1,4],[2]] => 3 [[1,3],[4]] => 4 [[1,4],[3]] => 4 [[1,4],[4]] => 5 [[2,2],[4]] => 4 [[2,3],[4]] => 5 [[2,4],[3]] => 5 [[2,4],[4]] => 6 [[3,3],[4]] => 6 [[3,4],[4]] => 7 [[1],[2],[4]] => 1 [[1],[3],[4]] => 2 [[2],[3],[4]] => 3 [[1,1,5]] => 4 [[1,2,5]] => 5 [[1,3,5]] => 6 [[1,4,5]] => 7 [[1,5,5]] => 8 [[2,2,5]] => 6 [[2,3,5]] => 7 [[2,4,5]] => 8 [[2,5,5]] => 9 [[3,3,5]] => 8 [[3,4,5]] => 9 [[3,5,5]] => 10 [[4,4,5]] => 10 [[4,5,5]] => 11 [[5,5,5]] => 12 [[1,1],[5]] => 3 [[1,2],[5]] => 4 [[1,5],[2]] => 4 [[1,3],[5]] => 5 [[1,5],[3]] => 5 [[1,4],[5]] => 6 [[1,5],[4]] => 6 [[1,5],[5]] => 7 [[2,2],[5]] => 5 [[2,3],[5]] => 6 [[2,5],[3]] => 6 [[2,4],[5]] => 7 [[2,5],[4]] => 7 [[2,5],[5]] => 8 [[3,3],[5]] => 7 [[3,4],[5]] => 8 [[3,5],[4]] => 8 [[3,5],[5]] => 9 [[4,4],[5]] => 9 [[4,5],[5]] => 10 [[1],[2],[5]] => 2 [[1],[3],[5]] => 3 [[1],[4],[5]] => 4 [[2],[3],[5]] => 4 [[2],[4],[5]] => 5 [[3],[4],[5]] => 6 [[1,1,1,1]] => 0 [[1,1,1,2]] => 1 [[1,1,2,2]] => 2 [[1,2,2,2]] => 3 [[2,2,2,2]] => 4 [[1,1,1],[2]] => 0 [[1,1,2],[2]] => 1 [[1,2,2],[2]] => 2 [[1,1],[2,2]] => 0 [[1,1,1,3]] => 2 [[1,1,2,3]] => 3 [[1,1,3,3]] => 4 [[1,2,2,3]] => 4 [[1,2,3,3]] => 5 [[1,3,3,3]] => 6 [[2,2,2,3]] => 5 [[2,2,3,3]] => 6 [[2,3,3,3]] => 7 [[3,3,3,3]] => 8 [[1,1,1],[3]] => 1 [[1,1,2],[3]] => 2 [[1,1,3],[2]] => 2 [[1,1,3],[3]] => 3 [[1,2,2],[3]] => 3 [[1,2,3],[2]] => 3 [[1,2,3],[3]] => 4 [[1,3,3],[2]] => 4 [[1,3,3],[3]] => 5 [[2,2,2],[3]] => 4 [[2,2,3],[3]] => 5 [[2,3,3],[3]] => 6 [[1,1],[2,3]] => 1 [[1,1],[3,3]] => 2 [[1,2],[2,3]] => 2 [[1,2],[3,3]] => 3 [[2,2],[3,3]] => 4 [[1,1],[2],[3]] => 0 [[1,2],[2],[3]] => 1 [[1,3],[2],[3]] => 2 [[1,1,1,4]] => 3 [[1,1,2,4]] => 4 [[1,1,3,4]] => 5 [[1,1,4,4]] => 6 [[1,2,2,4]] => 5 [[1,2,3,4]] => 6 [[1,2,4,4]] => 7 [[1,3,3,4]] => 7 [[1,3,4,4]] => 8 [[1,4,4,4]] => 9 [[2,2,2,4]] => 6 [[2,2,3,4]] => 7 [[2,2,4,4]] => 8 [[2,3,3,4]] => 8 [[2,3,4,4]] => 9 [[2,4,4,4]] => 10 [[3,3,3,4]] => 9 [[3,3,4,4]] => 10 [[3,4,4,4]] => 11 [[4,4,4,4]] => 12 [[1,1,1],[4]] => 2 [[1,1,2],[4]] => 3 [[1,1,4],[2]] => 3 [[1,1,3],[4]] => 4 [[1,1,4],[3]] => 4 [[1,1,4],[4]] => 5 [[1,2,2],[4]] => 4 [[1,2,4],[2]] => 4 [[1,2,3],[4]] => 5 [[1,2,4],[3]] => 5 [[1,3,4],[2]] => 5 [[1,2,4],[4]] => 6 [[1,4,4],[2]] => 6 [[1,3,3],[4]] => 6 [[1,3,4],[3]] => 6 [[1,3,4],[4]] => 7 [[1,4,4],[3]] => 7 [[1,4,4],[4]] => 8 [[2,2,2],[4]] => 5 [[2,2,3],[4]] => 6 [[2,2,4],[3]] => 6 [[2,2,4],[4]] => 7 [[2,3,3],[4]] => 7 [[2,3,4],[3]] => 7 [[2,3,4],[4]] => 8 [[2,4,4],[3]] => 8 [[2,4,4],[4]] => 9 [[3,3,3],[4]] => 8 [[3,3,4],[4]] => 9 [[3,4,4],[4]] => 10 [[1,1],[2,4]] => 2 [[1,1],[3,4]] => 3 [[1,1],[4,4]] => 4 [[1,2],[2,4]] => 3 [[1,2],[3,4]] => 4 [[1,3],[2,4]] => 4 [[1,2],[4,4]] => 5 [[1,3],[3,4]] => 5 [[1,3],[4,4]] => 6 [[2,2],[3,4]] => 5 [[2,2],[4,4]] => 6 [[2,3],[3,4]] => 6 [[2,3],[4,4]] => 7 [[3,3],[4,4]] => 8 [[1,1],[2],[4]] => 1 [[1,1],[3],[4]] => 2 [[1,2],[2],[4]] => 2 [[1,2],[3],[4]] => 3 [[1,3],[2],[4]] => 3 [[1,4],[2],[3]] => 3 [[1,4],[2],[4]] => 4 [[1,3],[3],[4]] => 4 [[1,4],[3],[4]] => 5 [[2,2],[3],[4]] => 4 [[2,3],[3],[4]] => 5 [[2,4],[3],[4]] => 6 [[1],[2],[3],[4]] => 0 [[1,1,1,1,2]] => 1 [[1,1,1,2,2]] => 2 [[1,1,2,2,2]] => 3 [[1,2,2,2,2]] => 4 [[2,2,2,2,2]] => 5 [[1,1,1,1],[2]] => 0 [[1,1,1,2],[2]] => 1 [[1,1,2,2],[2]] => 2 [[1,2,2,2],[2]] => 3 [[1,1,1],[2,2]] => 0 [[1,1,2],[2,2]] => 1 [[1,1,1,1,3]] => 2 [[1,1,1,2,3]] => 3 [[1,1,1,3,3]] => 4 [[1,1,2,2,3]] => 4 [[1,1,2,3,3]] => 5 [[1,1,3,3,3]] => 6 [[1,2,2,2,3]] => 5 [[1,2,2,3,3]] => 6 [[1,2,3,3,3]] => 7 [[1,3,3,3,3]] => 8 [[2,2,2,2,3]] => 6 [[2,2,2,3,3]] => 7 [[2,2,3,3,3]] => 8 [[2,3,3,3,3]] => 9 [[3,3,3,3,3]] => 10 [[1,1,1,1],[3]] => 1 [[1,1,1,2],[3]] => 2 [[1,1,1,3],[2]] => 2 [[1,1,1,3],[3]] => 3 [[1,1,2,2],[3]] => 3 [[1,1,2,3],[2]] => 3 [[1,1,2,3],[3]] => 4 [[1,1,3,3],[2]] => 4 [[1,1,3,3],[3]] => 5 [[1,2,2,2],[3]] => 4 [[1,2,2,3],[2]] => 4 [[1,2,2,3],[3]] => 5 [[1,2,3,3],[2]] => 5 [[1,2,3,3],[3]] => 6 [[1,3,3,3],[2]] => 6 [[1,3,3,3],[3]] => 7 [[2,2,2,2],[3]] => 5 [[2,2,2,3],[3]] => 6 [[2,2,3,3],[3]] => 7 [[2,3,3,3],[3]] => 8 [[1,1,1],[2,3]] => 1 [[1,1,1],[3,3]] => 2 [[1,1,2],[2,3]] => 2 [[1,1,3],[2,2]] => 2 [[1,1,2],[3,3]] => 3 [[1,1,3],[2,3]] => 3 [[1,1,3],[3,3]] => 4 [[1,2,2],[2,3]] => 3 [[1,2,2],[3,3]] => 4 [[1,2,3],[2,3]] => 4 [[1,2,3],[3,3]] => 5 [[2,2,2],[3,3]] => 5 [[2,2,3],[3,3]] => 6 [[1,1,1],[2],[3]] => 0 [[1,1,2],[2],[3]] => 1 [[1,1,3],[2],[3]] => 2 [[1,2,2],[2],[3]] => 2 [[1,2,3],[2],[3]] => 3 [[1,3,3],[2],[3]] => 4 [[1,1],[2,2],[3]] => 0 [[1,1],[2,3],[3]] => 1 [[1,2],[2,3],[3]] => 2 [[1,1,1,1,1,2]] => 1 [[1,1,1,1,2,2]] => 2 [[1,1,1,2,2,2]] => 3 [[1,1,2,2,2,2]] => 4 [[1,2,2,2,2,2]] => 5 [[2,2,2,2,2,2]] => 6 [[1,1,1,1,1],[2]] => 0 [[1,1,1,1,2],[2]] => 1 [[1,1,1,2,2],[2]] => 2 [[1,1,2,2,2],[2]] => 3 [[1,2,2,2,2],[2]] => 4 [[1,1,1,1],[2,2]] => 0 [[1,1,1,2],[2,2]] => 1 [[1,1,2,2],[2,2]] => 2 [[1,1,1],[2,2,2]] => 0 ----------------------------------------------------------------------------- Created: Jun 15, 2013 at 15:48 by Travis Scrimshaw ----------------------------------------------------------------------------- Last Updated: Feb 21, 2021 at 14:57 by Martin Rubey