***************************************************************************** * 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: St000284 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The Plancherel distribution on integer partitions. This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions. Equivalently, this is given by the square of the number of standard Young tableaux of the given shape. ----------------------------------------------------------------------------- References: [1] [[wikipedia:Plancherel_measure]] ----------------------------------------------------------------------------- Code: def statistic(L): return L.standard_tableaux().cardinality()^2 ----------------------------------------------------------------------------- Statistic values: [2] => 1 [1,1] => 1 [3] => 1 [2,1] => 4 [1,1,1] => 1 [4] => 1 [3,1] => 9 [2,2] => 4 [2,1,1] => 9 [1,1,1,1] => 1 [5] => 1 [4,1] => 16 [3,2] => 25 [3,1,1] => 36 [2,2,1] => 25 [2,1,1,1] => 16 [1,1,1,1,1] => 1 [6] => 1 [5,1] => 25 [4,2] => 81 [4,1,1] => 100 [3,3] => 25 [3,2,1] => 256 [3,1,1,1] => 100 [2,2,2] => 25 [2,2,1,1] => 81 [2,1,1,1,1] => 25 [1,1,1,1,1,1] => 1 [7] => 1 [6,1] => 36 [5,2] => 196 [5,1,1] => 225 [4,3] => 196 [4,2,1] => 1225 [4,1,1,1] => 400 [3,3,1] => 441 [3,2,2] => 441 [3,2,1,1] => 1225 [3,1,1,1,1] => 225 [2,2,2,1] => 196 [2,2,1,1,1] => 196 [2,1,1,1,1,1] => 36 [1,1,1,1,1,1,1] => 1 [8] => 1 [7,1] => 49 [6,2] => 400 [6,1,1] => 441 [5,3] => 784 [5,2,1] => 4096 [5,1,1,1] => 1225 [4,4] => 196 [4,3,1] => 4900 [4,2,2] => 3136 [4,2,1,1] => 8100 [4,1,1,1,1] => 1225 [3,3,2] => 1764 [3,3,1,1] => 3136 [3,2,2,1] => 4900 [3,2,1,1,1] => 4096 [3,1,1,1,1,1] => 441 [2,2,2,2] => 196 [2,2,2,1,1] => 784 [2,2,1,1,1,1] => 400 [2,1,1,1,1,1,1] => 49 [1,1,1,1,1,1,1,1] => 1 [9] => 1 [8,1] => 64 [7,2] => 729 [7,1,1] => 784 [6,3] => 2304 [6,2,1] => 11025 [6,1,1,1] => 3136 [5,4] => 1764 [5,3,1] => 26244 [5,2,2] => 14400 [5,2,1,1] => 35721 [5,1,1,1,1] => 4900 [4,4,1] => 7056 [4,3,2] => 28224 [4,3,1,1] => 46656 [4,2,2,1] => 46656 [4,2,1,1,1] => 35721 [4,1,1,1,1,1] => 3136 [3,3,3] => 1764 [3,3,2,1] => 28224 [3,3,1,1,1] => 14400 [3,2,2,2] => 7056 [3,2,2,1,1] => 26244 [3,2,1,1,1,1] => 11025 [3,1,1,1,1,1,1] => 784 [2,2,2,2,1] => 1764 [2,2,2,1,1,1] => 2304 [2,2,1,1,1,1,1] => 729 [2,1,1,1,1,1,1,1] => 64 [1,1,1,1,1,1,1,1,1] => 1 [10] => 1 [9,1] => 81 [8,2] => 1225 [8,1,1] => 1296 [7,3] => 5625 [7,2,1] => 25600 [7,1,1,1] => 7056 [6,4] => 8100 [6,3,1] => 99225 [6,2,2] => 50625 [6,2,1,1] => 122500 [6,1,1,1,1] => 15876 [5,5] => 1764 [5,4,1] => 82944 [5,3,2] => 202500 [5,3,1,1] => 321489 [5,2,2,1] => 275625 [5,2,1,1,1] => 200704 [5,1,1,1,1,1] => 15876 [4,4,2] => 63504 [4,4,1,1] => 90000 [4,3,3] => 44100 [4,3,2,1] => 589824 [4,3,1,1,1] => 275625 [4,2,2,2] => 90000 [4,2,2,1,1] => 321489 [4,2,1,1,1,1] => 122500 [4,1,1,1,1,1,1] => 7056 [3,3,3,1] => 44100 [3,3,2,2] => 63504 [3,3,2,1,1] => 202500 [3,3,1,1,1,1] => 50625 [3,2,2,2,1] => 82944 [3,2,2,1,1,1] => 99225 [3,2,1,1,1,1,1] => 25600 [3,1,1,1,1,1,1,1] => 1296 [2,2,2,2,2] => 1764 [2,2,2,2,1,1] => 8100 [2,2,2,1,1,1,1] => 5625 [2,2,1,1,1,1,1,1] => 1225 [2,1,1,1,1,1,1,1,1] => 81 [1,1,1,1,1,1,1,1,1,1] => 1 [11] => 1 [10,1] => 100 [9,2] => 1936 [9,1,1] => 2025 [8,3] => 12100 [8,2,1] => 53361 [8,1,1,1] => 14400 [7,4] => 27225 [7,3,1] => 302500 [7,2,2] => 148225 [7,2,1,1] => 352836 [7,1,1,1,1] => 44100 [6,5] => 17424 [6,4,1] => 480249 [6,3,2] => 980100 [6,3,1,1] => 1517824 [6,2,2,1] => 1210000 [6,2,1,1,1] => 853776 [6,1,1,1,1,1] => 63504 [5,5,1] => 108900 [5,4,2] => 980100 [5,4,1,1] => 1334025 [5,3,3] => 435600 [5,3,2,1] => 5336100 [5,3,1,1,1] => 2371600 [5,2,2,2] => 680625 [5,2,2,1,1] => 2371600 [5,2,1,1,1,1] => 853776 [5,1,1,1,1,1,1] => 44100 [4,4,3] => 213444 [4,4,2,1] => 1742400 [4,4,1,1,1] => 680625 [4,3,3,1] => 1411344 [4,3,2,2] => 1742400 [4,3,2,1,1] => 5336100 [4,3,1,1,1,1] => 1210000 [4,2,2,2,1] => 1334025 [4,2,2,1,1,1] => 1517824 [4,2,1,1,1,1,1] => 352836 [4,1,1,1,1,1,1,1] => 14400 [3,3,3,2] => 213444 [3,3,3,1,1] => 435600 [3,3,2,2,1] => 980100 [3,3,2,1,1,1] => 980100 [3,3,1,1,1,1,1] => 148225 [3,2,2,2,2] => 108900 [3,2,2,2,1,1] => 480249 [3,2,2,1,1,1,1] => 302500 [3,2,1,1,1,1,1,1] => 53361 [3,1,1,1,1,1,1,1,1] => 2025 [2,2,2,2,2,1] => 17424 [2,2,2,2,1,1,1] => 27225 [2,2,2,1,1,1,1,1] => 12100 [2,2,1,1,1,1,1,1,1] => 1936 [2,1,1,1,1,1,1,1,1,1] => 100 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 1 [11,1] => 121 [10,2] => 2916 [10,1,1] => 3025 [9,3] => 23716 [9,2,1] => 102400 [9,1,1,1] => 27225 [8,4] => 75625 [8,3,1] => 793881 [8,2,2] => 379456 [8,2,1,1] => 893025 [8,1,1,1,1] => 108900 [7,5] => 88209 [7,4,1] => 1982464 [7,3,2] => 3705625 [7,3,1,1] => 5645376 [7,2,2,1] => 4322241 [7,2,1,1,1] => 2985984 [7,1,1,1,1,1] => 213444 [6,6] => 17424 [6,5,1] => 1334025 [6,4,2] => 7144929 [6,4,1,1] => 9486400 [6,3,3] => 2722500 [6,3,2,1] => 31719424 [6,3,1,1,1] => 13660416 [6,2,2,2] => 3705625 [6,2,2,1,1] => 12702096 [6,2,1,1,1,1] => 4410000 [6,1,1,1,1,1,1] => 213444 [5,5,2] => 1742400 [5,5,1,1] => 2205225 [5,4,3] => 4460544 [5,4,2,1] => 33350625 [5,4,1,1,1] => 12390400 [5,3,3,1] => 17288964 [5,3,2,2] => 19847025 [5,3,2,1,1] => 59290000 [5,3,1,1,1,1] => 12702096 [5,2,2,2,1] => 12390400 [5,2,2,1,1,1] => 13660416 [5,2,1,1,1,1,1] => 2985984 [5,1,1,1,1,1,1,1] => 108900 [4,4,4] => 213444 [4,4,3,1] => 8820900 [4,4,2,2] => 6969600 [4,4,2,1,1] => 19847025 [4,4,1,1,1,1] => 3705625 [4,3,3,2] => 8820900 [4,3,3,1,1] => 17288964 [4,3,2,2,1] => 33350625 [4,3,2,1,1,1] => 31719424 [4,3,1,1,1,1,1] => 4322241 [4,2,2,2,2] => 2205225 [4,2,2,2,1,1] => 9486400 [4,2,2,1,1,1,1] => 5645376 [4,2,1,1,1,1,1,1] => 893025 [4,1,1,1,1,1,1,1,1] => 27225 [3,3,3,3] => 213444 [3,3,3,2,1] => 4460544 [3,3,3,1,1,1] => 2722500 [3,3,2,2,2] => 1742400 [3,3,2,2,1,1] => 7144929 [3,3,2,1,1,1,1] => 3705625 [3,3,1,1,1,1,1,1] => 379456 [3,2,2,2,2,1] => 1334025 [3,2,2,2,1,1,1] => 1982464 [3,2,2,1,1,1,1,1] => 793881 [3,2,1,1,1,1,1,1,1] => 102400 [3,1,1,1,1,1,1,1,1,1] => 3025 [2,2,2,2,2,2] => 17424 [2,2,2,2,2,1,1] => 88209 [2,2,2,2,1,1,1,1] => 75625 [2,2,2,1,1,1,1,1,1] => 23716 [2,2,1,1,1,1,1,1,1,1] => 2916 [2,1,1,1,1,1,1,1,1,1,1] => 121 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 ----------------------------------------------------------------------------- Created: Sep 15, 2015 at 08:40 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jul 12, 2017 at 10:03 by Christian Stump