***************************************************************************** * 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: St000229 ----------------------------------------------------------------------------- Collection: Set partitions ----------------------------------------------------------------------------- Description: Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. This is, for a set partition $P = \{B_1,\ldots,B_k\}$ of $\{1,\ldots,n\}$, the statistic is $$d(P) = \sum_i \big(\operatorname{max}(B_i)-\operatorname{min}(B_i)+1\big).$$ This statistic is called ''dimension index'' in [2] ----------------------------------------------------------------------------- References: [1] Chern, B., Diaconis, P., Kane, D. M., Rhoades, R. C. Central Limit Theorems for some Set Partition Statistics [[arXiv:1502.00938]] [2] Grubb, T., Rajasekaran, F. Set Partition Patterns and the Dimension Index [[arXiv:2009.00650]] ----------------------------------------------------------------------------- Code: def statistic(S): return sum(max(B)-min(B)+1 for B in S) ----------------------------------------------------------------------------- Statistic values: {{1}} => 1 {{1,2}} => 2 {{1},{2}} => 2 {{1,2,3}} => 3 {{1,2},{3}} => 3 {{1,3},{2}} => 4 {{1},{2,3}} => 3 {{1},{2},{3}} => 3 {{1,2,3,4}} => 4 {{1,2,3},{4}} => 4 {{1,2,4},{3}} => 5 {{1,2},{3,4}} => 4 {{1,2},{3},{4}} => 4 {{1,3,4},{2}} => 5 {{1,3},{2,4}} => 6 {{1,3},{2},{4}} => 5 {{1,4},{2,3}} => 6 {{1},{2,3,4}} => 4 {{1},{2,3},{4}} => 4 {{1,4},{2},{3}} => 6 {{1},{2,4},{3}} => 5 {{1},{2},{3,4}} => 4 {{1},{2},{3},{4}} => 4 {{1,2,3,4,5}} => 5 {{1,2,3,4},{5}} => 5 {{1,2,3,5},{4}} => 6 {{1,2,3},{4,5}} => 5 {{1,2,3},{4},{5}} => 5 {{1,2,4,5},{3}} => 6 {{1,2,4},{3,5}} => 7 {{1,2,4},{3},{5}} => 6 {{1,2,5},{3,4}} => 7 {{1,2},{3,4,5}} => 5 {{1,2},{3,4},{5}} => 5 {{1,2,5},{3},{4}} => 7 {{1,2},{3,5},{4}} => 6 {{1,2},{3},{4,5}} => 5 {{1,2},{3},{4},{5}} => 5 {{1,3,4,5},{2}} => 6 {{1,3,4},{2,5}} => 8 {{1,3,4},{2},{5}} => 6 {{1,3,5},{2,4}} => 8 {{1,3},{2,4,5}} => 7 {{1,3},{2,4},{5}} => 7 {{1,3,5},{2},{4}} => 7 {{1,3},{2,5},{4}} => 8 {{1,3},{2},{4,5}} => 6 {{1,3},{2},{4},{5}} => 6 {{1,4,5},{2,3}} => 7 {{1,4},{2,3,5}} => 8 {{1,4},{2,3},{5}} => 7 {{1,5},{2,3,4}} => 8 {{1},{2,3,4,5}} => 5 {{1},{2,3,4},{5}} => 5 {{1,5},{2,3},{4}} => 8 {{1},{2,3,5},{4}} => 6 {{1},{2,3},{4,5}} => 5 {{1},{2,3},{4},{5}} => 5 {{1,4,5},{2},{3}} => 7 {{1,4},{2,5},{3}} => 9 {{1,4},{2},{3,5}} => 8 {{1,4},{2},{3},{5}} => 7 {{1,5},{2,4},{3}} => 9 {{1},{2,4,5},{3}} => 6 {{1},{2,4},{3,5}} => 7 {{1},{2,4},{3},{5}} => 6 {{1,5},{2},{3,4}} => 8 {{1},{2,5},{3,4}} => 7 {{1},{2},{3,4,5}} => 5 {{1},{2},{3,4},{5}} => 5 {{1,5},{2},{3},{4}} => 8 {{1},{2,5},{3},{4}} => 7 {{1},{2},{3,5},{4}} => 6 {{1},{2},{3},{4,5}} => 5 {{1},{2},{3},{4},{5}} => 5 {{1,2,3,4,5,6}} => 6 {{1,2,3,4,5},{6}} => 6 {{1,2,3,4,6},{5}} => 7 {{1,2,3,4},{5,6}} => 6 {{1,2,3,4},{5},{6}} => 6 {{1,2,3,5,6},{4}} => 7 {{1,2,3,5},{4,6}} => 8 {{1,2,3,5},{4},{6}} => 7 {{1,2,3,6},{4,5}} => 8 {{1,2,3},{4,5,6}} => 6 {{1,2,3},{4,5},{6}} => 6 {{1,2,3,6},{4},{5}} => 8 {{1,2,3},{4,6},{5}} => 7 {{1,2,3},{4},{5,6}} => 6 {{1,2,3},{4},{5},{6}} => 6 {{1,2,4,5,6},{3}} => 7 {{1,2,4,5},{3,6}} => 9 {{1,2,4,5},{3},{6}} => 7 {{1,2,4,6},{3,5}} => 9 {{1,2,4},{3,5,6}} => 8 {{1,2,4},{3,5},{6}} => 8 {{1,2,4,6},{3},{5}} => 8 {{1,2,4},{3,6},{5}} => 9 {{1,2,4},{3},{5,6}} => 7 {{1,2,4},{3},{5},{6}} => 7 {{1,2,5,6},{3,4}} => 8 {{1,2,5},{3,4,6}} => 9 {{1,2,5},{3,4},{6}} => 8 {{1,2,6},{3,4,5}} => 9 {{1,2},{3,4,5,6}} => 6 {{1,2},{3,4,5},{6}} => 6 {{1,2,6},{3,4},{5}} => 9 {{1,2},{3,4,6},{5}} => 7 {{1,2},{3,4},{5,6}} => 6 {{1,2},{3,4},{5},{6}} => 6 {{1,2,5,6},{3},{4}} => 8 {{1,2,5},{3,6},{4}} => 10 {{1,2,5},{3},{4,6}} => 9 {{1,2,5},{3},{4},{6}} => 8 {{1,2,6},{3,5},{4}} => 10 {{1,2},{3,5,6},{4}} => 7 {{1,2},{3,5},{4,6}} => 8 {{1,2},{3,5},{4},{6}} => 7 {{1,2,6},{3},{4,5}} => 9 {{1,2},{3,6},{4,5}} => 8 {{1,2},{3},{4,5,6}} => 6 {{1,2},{3},{4,5},{6}} => 6 {{1,2,6},{3},{4},{5}} => 9 {{1,2},{3,6},{4},{5}} => 8 {{1,2},{3},{4,6},{5}} => 7 {{1,2},{3},{4},{5,6}} => 6 {{1,2},{3},{4},{5},{6}} => 6 {{1,3,4,5,6},{2}} => 7 {{1,3,4,5},{2,6}} => 10 {{1,3,4,5},{2},{6}} => 7 {{1,3,4,6},{2,5}} => 10 {{1,3,4},{2,5,6}} => 9 {{1,3,4},{2,5},{6}} => 9 {{1,3,4,6},{2},{5}} => 8 {{1,3,4},{2,6},{5}} => 10 {{1,3,4},{2},{5,6}} => 7 {{1,3,4},{2},{5},{6}} => 7 {{1,3,5,6},{2,4}} => 9 {{1,3,5},{2,4,6}} => 10 {{1,3,5},{2,4},{6}} => 9 {{1,3,6},{2,4,5}} => 10 {{1,3},{2,4,5,6}} => 8 {{1,3},{2,4,5},{6}} => 8 {{1,3,6},{2,4},{5}} => 10 {{1,3},{2,4,6},{5}} => 9 {{1,3},{2,4},{5,6}} => 8 {{1,3},{2,4},{5},{6}} => 8 {{1,3,5,6},{2},{4}} => 8 {{1,3,5},{2,6},{4}} => 11 {{1,3,5},{2},{4,6}} => 9 {{1,3,5},{2},{4},{6}} => 8 {{1,3,6},{2,5},{4}} => 11 {{1,3},{2,5,6},{4}} => 9 {{1,3},{2,5},{4,6}} => 10 {{1,3},{2,5},{4},{6}} => 9 {{1,3,6},{2},{4,5}} => 9 {{1,3},{2,6},{4,5}} => 10 {{1,3},{2},{4,5,6}} => 7 {{1,3},{2},{4,5},{6}} => 7 {{1,3,6},{2},{4},{5}} => 9 {{1,3},{2,6},{4},{5}} => 10 {{1,3},{2},{4,6},{5}} => 8 {{1,3},{2},{4},{5,6}} => 7 {{1,3},{2},{4},{5},{6}} => 7 {{1,4,5,6},{2,3}} => 8 {{1,4,5},{2,3,6}} => 10 {{1,4,5},{2,3},{6}} => 8 {{1,4,6},{2,3,5}} => 10 {{1,4},{2,3,5,6}} => 9 {{1,4},{2,3,5},{6}} => 9 {{1,4,6},{2,3},{5}} => 9 {{1,4},{2,3,6},{5}} => 10 {{1,4},{2,3},{5,6}} => 8 {{1,4},{2,3},{5},{6}} => 8 {{1,5,6},{2,3,4}} => 9 {{1,5},{2,3,4,6}} => 10 {{1,5},{2,3,4},{6}} => 9 {{1,6},{2,3,4,5}} => 10 {{1},{2,3,4,5,6}} => 6 {{1},{2,3,4,5},{6}} => 6 {{1,6},{2,3,4},{5}} => 10 {{1},{2,3,4,6},{5}} => 7 {{1},{2,3,4},{5,6}} => 6 {{1},{2,3,4},{5},{6}} => 6 {{1,5,6},{2,3},{4}} => 9 {{1,5},{2,3,6},{4}} => 11 {{1,5},{2,3},{4,6}} => 10 {{1,5},{2,3},{4},{6}} => 9 {{1,6},{2,3,5},{4}} => 11 {{1},{2,3,5,6},{4}} => 7 {{1},{2,3,5},{4,6}} => 8 {{1},{2,3,5},{4},{6}} => 7 {{1,6},{2,3},{4,5}} => 10 {{1},{2,3,6},{4,5}} => 8 {{1},{2,3},{4,5,6}} => 6 {{1},{2,3},{4,5},{6}} => 6 {{1,6},{2,3},{4},{5}} => 10 {{1},{2,3,6},{4},{5}} => 8 {{1},{2,3},{4,6},{5}} => 7 {{1},{2,3},{4},{5,6}} => 6 {{1},{2,3},{4},{5},{6}} => 6 {{1,4,5,6},{2},{3}} => 8 {{1,4,5},{2,6},{3}} => 11 {{1,4,5},{2},{3,6}} => 10 {{1,4,5},{2},{3},{6}} => 8 {{1,4,6},{2,5},{3}} => 11 {{1,4},{2,5,6},{3}} => 10 {{1,4},{2,5},{3,6}} => 12 {{1,4},{2,5},{3},{6}} => 10 {{1,4,6},{2},{3,5}} => 10 {{1,4},{2,6},{3,5}} => 12 {{1,4},{2},{3,5,6}} => 9 {{1,4},{2},{3,5},{6}} => 9 {{1,4,6},{2},{3},{5}} => 9 {{1,4},{2,6},{3},{5}} => 11 {{1,4},{2},{3,6},{5}} => 10 {{1,4},{2},{3},{5,6}} => 8 {{1,4},{2},{3},{5},{6}} => 8 {{1,5,6},{2,4},{3}} => 10 {{1,5},{2,4,6},{3}} => 11 {{1,5},{2,4},{3,6}} => 12 {{1,5},{2,4},{3},{6}} => 10 {{1,6},{2,4,5},{3}} => 11 {{1},{2,4,5,6},{3}} => 7 {{1},{2,4,5},{3,6}} => 9 {{1},{2,4,5},{3},{6}} => 7 {{1,6},{2,4},{3,5}} => 12 {{1},{2,4,6},{3,5}} => 9 {{1},{2,4},{3,5,6}} => 8 {{1},{2,4},{3,5},{6}} => 8 {{1,6},{2,4},{3},{5}} => 11 {{1},{2,4,6},{3},{5}} => 8 {{1},{2,4},{3,6},{5}} => 9 {{1},{2,4},{3},{5,6}} => 7 {{1},{2,4},{3},{5},{6}} => 7 {{1,5,6},{2},{3,4}} => 9 {{1,5},{2,6},{3,4}} => 12 {{1,5},{2},{3,4,6}} => 10 {{1,5},{2},{3,4},{6}} => 9 {{1,6},{2,5},{3,4}} => 12 {{1},{2,5,6},{3,4}} => 8 {{1},{2,5},{3,4,6}} => 9 {{1},{2,5},{3,4},{6}} => 8 {{1,6},{2},{3,4,5}} => 10 {{1},{2,6},{3,4,5}} => 9 {{1},{2},{3,4,5,6}} => 6 {{1},{2},{3,4,5},{6}} => 6 {{1,6},{2},{3,4},{5}} => 10 {{1},{2,6},{3,4},{5}} => 9 {{1},{2},{3,4,6},{5}} => 7 {{1},{2},{3,4},{5,6}} => 6 {{1},{2},{3,4},{5},{6}} => 6 {{1,5,6},{2},{3},{4}} => 9 {{1,5},{2,6},{3},{4}} => 12 {{1,5},{2},{3,6},{4}} => 11 {{1,5},{2},{3},{4,6}} => 10 {{1,5},{2},{3},{4},{6}} => 9 {{1,6},{2,5},{3},{4}} => 12 {{1},{2,5,6},{3},{4}} => 8 {{1},{2,5},{3,6},{4}} => 10 {{1},{2,5},{3},{4,6}} => 9 {{1},{2,5},{3},{4},{6}} => 8 {{1,6},{2},{3,5},{4}} => 11 {{1},{2,6},{3,5},{4}} => 10 {{1},{2},{3,5,6},{4}} => 7 {{1},{2},{3,5},{4,6}} => 8 {{1},{2},{3,5},{4},{6}} => 7 {{1,6},{2},{3},{4,5}} => 10 {{1},{2,6},{3},{4,5}} => 9 {{1},{2},{3,6},{4,5}} => 8 {{1},{2},{3},{4,5,6}} => 6 {{1},{2},{3},{4,5},{6}} => 6 {{1,6},{2},{3},{4},{5}} => 10 {{1},{2,6},{3},{4},{5}} => 9 {{1},{2},{3,6},{4},{5}} => 8 {{1},{2},{3},{4,6},{5}} => 7 {{1},{2},{3},{4},{5,6}} => 6 {{1},{2},{3},{4},{5},{6}} => 6 ----------------------------------------------------------------------------- Created: Feb 04, 2015 at 11:18 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Sep 29, 2020 at 10:54 by Christian Stump