***************************************************************************** * 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: St000003 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of [[/StandardTableaux|standard Young tableaux]] of the partition. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(x): return StandardTableaux(x).cardinality() ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 1 [1,1] => 1 [3] => 1 [2,1] => 2 [1,1,1] => 1 [4] => 1 [3,1] => 3 [2,2] => 2 [2,1,1] => 3 [1,1,1,1] => 1 [5] => 1 [4,1] => 4 [3,2] => 5 [3,1,1] => 6 [2,2,1] => 5 [2,1,1,1] => 4 [1,1,1,1,1] => 1 [6] => 1 [5,1] => 5 [4,2] => 9 [4,1,1] => 10 [3,3] => 5 [3,2,1] => 16 [3,1,1,1] => 10 [2,2,2] => 5 [2,2,1,1] => 9 [2,1,1,1,1] => 5 [1,1,1,1,1,1] => 1 [7] => 1 [6,1] => 6 [5,2] => 14 [5,1,1] => 15 [4,3] => 14 [4,2,1] => 35 [4,1,1,1] => 20 [3,3,1] => 21 [3,2,2] => 21 [3,2,1,1] => 35 [3,1,1,1,1] => 15 [2,2,2,1] => 14 [2,2,1,1,1] => 14 [2,1,1,1,1,1] => 6 [1,1,1,1,1,1,1] => 1 [8] => 1 [7,1] => 7 [6,2] => 20 [6,1,1] => 21 [5,3] => 28 [5,2,1] => 64 [5,1,1,1] => 35 [4,4] => 14 [4,3,1] => 70 [4,2,2] => 56 [4,2,1,1] => 90 [4,1,1,1,1] => 35 [3,3,2] => 42 [3,3,1,1] => 56 [3,2,2,1] => 70 [3,2,1,1,1] => 64 [3,1,1,1,1,1] => 21 [2,2,2,2] => 14 [2,2,2,1,1] => 28 [2,2,1,1,1,1] => 20 [2,1,1,1,1,1,1] => 7 [1,1,1,1,1,1,1,1] => 1 [9] => 1 [8,1] => 8 [7,2] => 27 [7,1,1] => 28 [6,3] => 48 [6,2,1] => 105 [6,1,1,1] => 56 [5,4] => 42 [5,3,1] => 162 [5,2,2] => 120 [5,2,1,1] => 189 [5,1,1,1,1] => 70 [4,4,1] => 84 [4,3,2] => 168 [4,3,1,1] => 216 [4,2,2,1] => 216 [4,2,1,1,1] => 189 [4,1,1,1,1,1] => 56 [3,3,3] => 42 [3,3,2,1] => 168 [3,3,1,1,1] => 120 [3,2,2,2] => 84 [3,2,2,1,1] => 162 [3,2,1,1,1,1] => 105 [3,1,1,1,1,1,1] => 28 [2,2,2,2,1] => 42 [2,2,2,1,1,1] => 48 [2,2,1,1,1,1,1] => 27 [2,1,1,1,1,1,1,1] => 8 [1,1,1,1,1,1,1,1,1] => 1 [10] => 1 [9,1] => 9 [8,2] => 35 [8,1,1] => 36 [7,3] => 75 [7,2,1] => 160 [7,1,1,1] => 84 [6,4] => 90 [6,3,1] => 315 [6,2,2] => 225 [6,2,1,1] => 350 [6,1,1,1,1] => 126 [5,5] => 42 [5,4,1] => 288 [5,3,2] => 450 [5,3,1,1] => 567 [5,2,2,1] => 525 [5,2,1,1,1] => 448 [5,1,1,1,1,1] => 126 [4,4,2] => 252 [4,4,1,1] => 300 [4,3,3] => 210 [4,3,2,1] => 768 [4,3,1,1,1] => 525 [4,2,2,2] => 300 [4,2,2,1,1] => 567 [4,2,1,1,1,1] => 350 [4,1,1,1,1,1,1] => 84 [3,3,3,1] => 210 [3,3,2,2] => 252 [3,3,2,1,1] => 450 [3,3,1,1,1,1] => 225 [3,2,2,2,1] => 288 [3,2,2,1,1,1] => 315 [3,2,1,1,1,1,1] => 160 [3,1,1,1,1,1,1,1] => 36 [2,2,2,2,2] => 42 [2,2,2,2,1,1] => 90 [2,2,2,1,1,1,1] => 75 [2,2,1,1,1,1,1,1] => 35 [2,1,1,1,1,1,1,1,1] => 9 [1,1,1,1,1,1,1,1,1,1] => 1 [11] => 1 [10,1] => 10 [9,2] => 44 [9,1,1] => 45 [8,3] => 110 [8,2,1] => 231 [8,1,1,1] => 120 [7,4] => 165 [7,3,1] => 550 [7,2,2] => 385 [7,2,1,1] => 594 [7,1,1,1,1] => 210 [6,5] => 132 [6,4,1] => 693 [6,3,2] => 990 [6,3,1,1] => 1232 [6,2,2,1] => 1100 [6,2,1,1,1] => 924 [6,1,1,1,1,1] => 252 [5,5,1] => 330 [5,4,2] => 990 [5,4,1,1] => 1155 [5,3,3] => 660 [5,3,2,1] => 2310 [5,3,1,1,1] => 1540 [5,2,2,2] => 825 [5,2,2,1,1] => 1540 [5,2,1,1,1,1] => 924 [5,1,1,1,1,1,1] => 210 [4,4,3] => 462 [4,4,2,1] => 1320 [4,4,1,1,1] => 825 [4,3,3,1] => 1188 [4,3,2,2] => 1320 [4,3,2,1,1] => 2310 [4,3,1,1,1,1] => 1100 [4,2,2,2,1] => 1155 [4,2,2,1,1,1] => 1232 [4,2,1,1,1,1,1] => 594 [4,1,1,1,1,1,1,1] => 120 [3,3,3,2] => 462 [3,3,3,1,1] => 660 [3,3,2,2,1] => 990 [3,3,2,1,1,1] => 990 [3,3,1,1,1,1,1] => 385 [3,2,2,2,2] => 330 [3,2,2,2,1,1] => 693 [3,2,2,1,1,1,1] => 550 [3,2,1,1,1,1,1,1] => 231 [3,1,1,1,1,1,1,1,1] => 45 [2,2,2,2,2,1] => 132 [2,2,2,2,1,1,1] => 165 [2,2,2,1,1,1,1,1] => 110 [2,2,1,1,1,1,1,1,1] => 44 [2,1,1,1,1,1,1,1,1,1] => 10 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 1 [11,1] => 11 [10,2] => 54 [10,1,1] => 55 [9,3] => 154 [9,2,1] => 320 [9,1,1,1] => 165 [8,4] => 275 [8,3,1] => 891 [8,2,2] => 616 [8,2,1,1] => 945 [8,1,1,1,1] => 330 [7,5] => 297 [7,4,1] => 1408 [7,3,2] => 1925 [7,3,1,1] => 2376 [7,2,2,1] => 2079 [7,2,1,1,1] => 1728 [7,1,1,1,1,1] => 462 [6,6] => 132 [6,5,1] => 1155 [6,4,2] => 2673 [6,4,1,1] => 3080 [6,3,3] => 1650 [6,3,2,1] => 5632 [6,3,1,1,1] => 3696 [6,2,2,2] => 1925 [6,2,2,1,1] => 3564 [6,2,1,1,1,1] => 2100 [6,1,1,1,1,1,1] => 462 [5,5,2] => 1320 [5,5,1,1] => 1485 [5,4,3] => 2112 [5,4,2,1] => 5775 [5,4,1,1,1] => 3520 [5,3,3,1] => 4158 [5,3,2,2] => 4455 [5,3,2,1,1] => 7700 [5,3,1,1,1,1] => 3564 [5,2,2,2,1] => 3520 [5,2,2,1,1,1] => 3696 [5,2,1,1,1,1,1] => 1728 [5,1,1,1,1,1,1,1] => 330 [4,4,4] => 462 [4,4,3,1] => 2970 [4,4,2,2] => 2640 [4,4,2,1,1] => 4455 [4,4,1,1,1,1] => 1925 [4,3,3,2] => 2970 [4,3,3,1,1] => 4158 [4,3,2,2,1] => 5775 [4,3,2,1,1,1] => 5632 [4,3,1,1,1,1,1] => 2079 [4,2,2,2,2] => 1485 [4,2,2,2,1,1] => 3080 [4,2,2,1,1,1,1] => 2376 [4,2,1,1,1,1,1,1] => 945 [4,1,1,1,1,1,1,1,1] => 165 [3,3,3,3] => 462 [3,3,3,2,1] => 2112 [3,3,3,1,1,1] => 1650 [3,3,2,2,2] => 1320 [3,3,2,2,1,1] => 2673 [3,3,2,1,1,1,1] => 1925 [3,3,1,1,1,1,1,1] => 616 [3,2,2,2,2,1] => 1155 [3,2,2,2,1,1,1] => 1408 [3,2,2,1,1,1,1,1] => 891 [3,2,1,1,1,1,1,1,1] => 320 [3,1,1,1,1,1,1,1,1,1] => 55 [2,2,2,2,2,2] => 132 [2,2,2,2,2,1,1] => 297 [2,2,2,2,1,1,1,1] => 275 [2,2,2,1,1,1,1,1,1] => 154 [2,2,1,1,1,1,1,1,1,1] => 54 [2,1,1,1,1,1,1,1,1,1,1] => 11 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 [8,5] => 572 [7,6] => 429 [7,5,1] => 2860 [7,4,2] => 6006 [6,6,1] => 1287 [5,5,3] => 3432 [5,4,4] => 2574 [5,4,3,1] => 15015 [5,4,2,2] => 12870 [5,4,2,1,1] => 21450 [5,4,1,1,1,1] => 9009 [5,3,3,2] => 11583 [5,3,3,1,1] => 16016 [5,3,2,2,1] => 21450 [5,3,2,1,1,1] => 20592 [4,4,4,1] => 3432 [4,4,3,2] => 8580 [4,4,3,1,1] => 11583 [4,4,2,2,1] => 12870 [4,3,3,3] => 3432 [4,3,3,2,1] => 15015 [3,3,3,3,1] => 2574 [3,3,3,2,2] => 3432 [3,3,2,2,2,1] => 5148 [3,2,2,2,2,2] => 1287 [2,2,2,2,2,2,1] => 429 [9,5] => 1001 [8,5,1] => 6006 [7,7] => 429 [7,5,2] => 14014 [7,4,3] => 16016 [6,6,2] => 6435 [6,4,4] => 9009 [6,2,2,2,2] => 14014 [5,5,4] => 6006 [5,5,1,1,1,1] => 14014 [5,4,3,2] => 48048 [5,4,3,1,1] => 64064 [5,4,2,2,1] => 68640 [5,4,2,1,1,1] => 63063 [5,3,3,2,1] => 64064 [5,3,2,2,2] => 35035 [5,2,2,2,2,1] => 21021 [4,4,4,2] => 12012 [4,4,3,3] => 12012 [4,4,3,2,1] => 48048 [4,3,2,2,2,1] => 36608 [3,3,3,3,2] => 6006 [3,3,3,3,1,1] => 9009 [3,3,2,2,2,2] => 6435 [2,2,2,2,2,2,2] => 429 [9,5,1] => 11375 [8,5,2] => 32032 [7,5,3] => 45045 [6,6,3] => 21450 [6,5,4] => 30030 [6,5,1,1,1,1] => 63700 [6,3,3,3] => 50050 [6,2,2,2,2,1] => 63700 [5,5,5] => 6006 [5,4,3,2,1] => 292864 [5,4,3,1,1,1] => 210210 [5,3,2,2,2,1] => 162162 [4,4,4,3] => 24024 [4,4,4,1,1,1] => 50050 [3,3,3,3,3] => 6006 [3,3,3,3,2,1] => 30030 [3,3,3,2,2,2] => 21450 [8,8] => 1430 [8,5,3] => 112112 [7,5,3,1] => 416988 [6,6,4] => 51480 [5,5,3,3] => 171600 [5,5,2,2,2] => 250250 [5,4,3,2,1,1] => 1153152 [5,4,2,2,2,1] => 630630 [4,4,4,4] => 24024 [4,4,4,2,2] => 171600 [4,3,3,3,2,1] => 292864 [3,3,3,3,2,2] => 51480 [2,2,2,2,2,2,2,2] => 1430 [8,6,3] => 272272 [6,5,3,3] => 972400 [6,5,2,2,2] => 1361360 [6,4,4,3] => 816816 [6,4,4,1,1,1] => 1591200 [6,3,3,3,2] => 1089088 [6,3,3,3,1,1] => 1591200 [5,5,4,3] => 583440 [5,5,4,1,1,1] => 1089088 [5,5,2,2,2,1] => 1361360 [5,4,3,2,2,1] => 3573570 [5,3,3,3,2,1] => 1633632 [4,4,4,3,2] => 583440 [4,4,4,3,1,1] => 816816 [4,4,4,2,2,1] => 972400 [4,4,4,3,2,1] => 3734016 [5,4,3,3,2,1] => 10720710 [6,3,3,3,2,1] => 6789120 [6,5,2,2,2,1] => 7920640 [5,5,3,3,1,1] => 6534528 [6,5,4,1,1,1] => 6789120 [5,5,3,3,2] => 4594590 [5,5,4,2,2] => 4594590 [6,4,4,2,2] => 6534528 [6,5,4,3] => 3734016 [2,2,2,2,2,2,2,2,2] => 4862 [3,3,3,3,3,3] => 87516 [6,6,6] => 87516 [9,6,3] => 678912 [8,6,4] => 787644 [9,9] => 4862 [5,4,4,3,2,1] => 34918884 [5,5,3,3,2,1] => 31039008 [5,5,4,2,2,1] => 29930472 [6,4,4,2,2,1] => 42325920 [5,5,4,3,1,1] => 26604864 [6,4,4,3,1,1] => 36732852 [6,5,3,3,1,1] => 42325920 [5,5,4,3,2] => 19399380 [6,4,4,3,2] => 26604864 [6,5,3,3,2] => 29930472 [6,5,4,2,2] => 31039008 [6,5,4,3,1] => 34918884 [6,5,4,1,1,1,1] => 23279256 [9,6,4] => 2116296 [8,5,4,2] => 16325712 [8,5,5,1] => 5038800 [5,5,4,3,2,1] => 141892608 [6,4,4,3,2,1] => 193489920 [6,5,3,3,2,1] => 214988800 [6,5,4,2,2,1] => 214988800 [6,5,4,3,1,1] => 193489920 [6,5,4,3,2] => 141892608 [6,5,2,2,2,2,1] => 81477396 [6,5,4,2,1,1,1] => 203693490 [7,5,4,3,1] => 154313250 [8,6,4,2] => 55099278 [10,6,4] => 5038800 [10,7,3] => 3779100 [9,7,4] => 5290740 [9,5,5,1] => 13856700 [6,5,4,3,2,1] => 1100742656 [6,3,3,3,3,2,1] => 203693490 [6,5,3,2,2,2,1] => 733296564 [6,5,4,3,1,1,1] => 814773960 [3,3,3,3,3,3,3] => 1385670 [11,7,3] => 8478750 [4,4,4,4,3,2,1] => 341429088 [6,4,3,3,3,2,1] => 2048574528 [9,6,4,3] => 611080470 [9,6,5,3] => 2007835830 [3,3,3,3,3,3,3,3] => 23371634 [11,7,5,1] => 1699755200 [8,8,8] => 23371634 ----------------------------------------------------------------------------- Created: Sep 14, 2011 at 16:53 by Chris Berg ----------------------------------------------------------------------------- Last Updated: Sep 18, 2022 at 20:49 by Martin Rubey