***************************************************************************** * 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: St000285 ----------------------------------------------------------------------------- Collection: Integer compositions ----------------------------------------------------------------------------- Description: The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def ides_composition(elt): return elt.diagonal_reading_word().inverse().descents_composition() @cached_function def preimages(level): print("computing preimages for level", level) result = dict() for el in ParkingFunctions(level): image = ides_composition(el) result[image] = result.get(image, 0) + 1 return result def statistic(x): return preimages(x.size()).get(x, 0) ----------------------------------------------------------------------------- Statistic values: [1,1] => 2 [2] => 1 [1,1,1] => 5 [1,2] => 5 [2,1] => 5 [3] => 1 [1,1,1,1] => 14 [1,1,2] => 21 [1,2,1] => 31 [1,3] => 9 [2,1,1] => 21 [2,2] => 19 [3,1] => 9 [4] => 1 [1,1,1,1,1] => 42 [1,1,1,2] => 84 [1,1,2,1] => 154 [1,1,3] => 56 [1,2,1,1] => 154 [1,2,2] => 161 [1,3,1] => 91 [1,4] => 14 [2,1,1,1] => 84 [2,1,2] => 126 [2,2,1] => 161 [2,3] => 49 [3,1,1] => 56 [3,2] => 49 [4,1] => 14 [5] => 1 [1,1,1,1,1,1] => 132 [1,1,1,1,2] => 330 [1,1,1,2,1] => 708 [1,1,1,3] => 300 [1,1,2,1,1] => 876 [1,1,2,2] => 1014 [1,1,3,1] => 636 [1,1,4] => 120 [1,2,1,1,1] => 708 [1,2,1,2] => 1182 [1,2,2,1] => 1588 [1,2,3] => 540 [1,3,1,1] => 636 [1,3,2] => 610 [1,4,1] => 204 [1,5] => 20 [2,1,1,1,1] => 330 [2,1,1,2] => 678 [2,1,2,1] => 1182 [2,1,3] => 456 [2,2,1,1] => 1014 [2,2,2] => 1114 [2,3,1] => 610 [2,4] => 104 [3,1,1,1] => 300 [3,1,2] => 456 [3,2,1] => 540 [3,3] => 174 [4,1,1] => 120 [4,2] => 104 [5,1] => 20 [6] => 1 [1,1,1,1,1,1,1] => 429 [1,1,1,1,1,2] => 1287 [1,1,1,1,2,1] => 3135 [1,1,1,1,3] => 1485 [1,1,1,2,1,1] => 4521 [1,1,1,2,2] => 5643 [1,1,1,3,1] => 3795 [1,1,1,4] => 825 [1,1,2,1,1,1] => 4521 [1,1,2,1,2] => 8163 [1,1,2,2,1] => 11355 [1,1,2,3] => 4185 [1,1,3,1,1] => 4929 [1,1,3,2] => 5067 [1,1,4,1] => 1875 [1,1,5] => 225 [1,2,1,1,1,1] => 3135 [1,2,1,1,2] => 7029 [1,2,1,2,1] => 12741 [1,2,1,3] => 5319 [1,2,2,1,1] => 11355 [1,2,2,2] => 13257 [1,2,3,1] => 7545 [1,2,4] => 1443 [1,3,1,1,1] => 3795 [1,3,1,2] => 6201 [1,3,2,1] => 7545 [1,3,3] => 2619 [1,4,1,1] => 1875 [1,4,2] => 1737 [1,5,1] => 393 [1,6] => 27 [2,1,1,1,1,1] => 1287 [2,1,1,1,2] => 3333 [2,1,1,2,1] => 7029 [2,1,1,3] => 3135 [2,1,2,1,1] => 8163 [2,1,2,2] => 9897 [2,1,3,1] => 6201 [2,1,4] => 1275 [2,2,1,1,1] => 5643 [2,2,1,2] => 9897 [2,2,2,1] => 13257 [2,2,3] => 4803 [2,3,1,1] => 5067 [2,3,2] => 5097 [2,4,1] => 1737 [2,5] => 195 [3,1,1,1,1] => 1485 [3,1,1,2] => 3135 [3,1,2,1] => 5319 [3,1,3] => 2157 [3,2,1,1] => 4185 [3,2,2] => 4803 [3,3,1] => 2619 [3,4] => 489 [4,1,1,1] => 825 [4,1,2] => 1275 [4,2,1] => 1443 [4,3] => 489 [5,1,1] => 225 [5,2] => 195 [6,1] => 27 [7] => 1 [1,1,1,1,1,1,1,1] => 1430 [1,1,1,1,1,1,2] => 5005 [1,1,1,1,1,2,1] => 13585 [1,1,1,1,1,3] => 7007 [1,1,1,1,2,1,1] => 22165 [1,1,1,1,2,2] => 29315 [1,1,1,1,3,1] => 20735 [1,1,1,1,4] => 5005 [1,1,1,2,1,1,1] => 25795 [1,1,1,2,1,2] => 49445 [1,1,1,2,2,1] => 70565 [1,1,1,2,3] => 27643 [1,1,1,3,1,1] => 32285 [1,1,1,3,2] => 35035 [1,1,1,4,1] => 13915 [1,1,1,5] => 1925 [1,1,2,1,1,1,1] => 22165 [1,1,2,1,1,2] => 53075 [1,1,2,1,2,1] => 98945 [1,1,2,1,3] => 43813 [1,1,2,2,1,1] => 90365 [1,1,2,2,2] => 110605 [1,1,2,3,1] => 64735 [1,1,2,4] => 13475 [1,1,3,1,1,1] => 32285 [1,1,3,1,2] => 55825 [1,1,3,2,1] => 69355 [1,1,3,3] => 25487 [1,1,4,1,1] => 18535 [1,1,4,2] => 18095 [1,1,5,1] => 4565 [1,1,6] => 385 [1,2,1,1,1,1,1] => 13585 [1,2,1,1,1,2] => 37895 [1,2,1,1,2,1] => 82775 [1,2,1,1,3] => 39193 [1,2,1,2,1,1] => 98945 [1,2,1,2,2] => 125785 [1,2,1,3,1] => 80905 [1,2,1,4] => 18095 [1,2,2,1,1,1] => 70565 [1,2,2,1,2] => 130405 [1,2,2,2,1] => 177325 [1,2,2,3] => 67907 [1,2,3,1,1] => 69355 [1,2,3,2] => 73115 [1,2,4,1] => 26195 [1,2,5] => 3325 [1,3,1,1,1,1] => 20735 [1,3,1,1,2] => 46585 [1,3,1,2,1] => 80905 [1,3,1,3] => 34727 [1,3,2,1,1] => 64735 [1,3,2,2] => 77735 [1,3,3,1] => 43415 [1,3,4] => 8785 [1,4,1,1,1] => 13915 [1,4,1,2] => 22715 [1,4,2,1] => 26195 [1,4,3] => 9373 [1,5,1,1] => 4565 [1,5,2] => 4165 [1,6,1] => 685 [1,7] => 35 [2,1,1,1,1,1,1] => 5005 [2,1,1,1,1,2] => 15587 [2,1,1,1,2,1] => 37895 [2,1,1,1,3] => 18733 [2,1,1,2,1,1] => 53075 [2,1,1,2,2] => 68893 [2,1,1,3,1] => 46585 [2,1,1,4] => 10835 [2,1,2,1,1,1] => 49445 [2,1,2,1,2] => 93313 [2,1,2,2,1] => 130405 [2,1,2,3] => 50567 [2,1,3,1,1] => 55825 [2,1,3,2] => 59807 [2,1,4,1] => 22715 [2,1,5] => 3025 [2,2,1,1,1,1] => 29315 [2,2,1,1,2] => 68893 [2,2,1,2,1] => 125785 [2,2,1,3] => 55187 [2,2,2,1,1] => 110605 [2,2,2,2] => 134627 [2,2,3,1] => 77735 [2,2,4] => 16045 [2,3,1,1,1] => 35035 [2,3,1,2] => 59807 [2,3,2,1] => 73115 [2,3,3] => 26713 [2,4,1,1] => 18095 [2,4,2] => 17473 [2,5,1] => 4165 [2,6] => 335 [3,1,1,1,1,1] => 7007 [3,1,1,1,2] => 18733 [3,1,1,2,1] => 39193 [3,1,1,3] => 18227 [3,1,2,1,1] => 43813 [3,1,2,2] => 55187 [3,1,3,1] => 34727 [3,1,4] => 7645 [3,2,1,1,1] => 27643 [3,2,1,2] => 50567 [3,2,2,1] => 67907 [3,2,3] => 25873 [3,3,1,1] => 25487 [3,3,2] => 26713 [3,4,1] => 9373 [3,5] => 1175 [4,1,1,1,1] => 5005 [4,1,1,2] => 10835 [4,1,2,1] => 18095 [4,1,3] => 7645 [4,2,1,1] => 13475 [4,2,2] => 16045 [4,3,1] => 8785 [4,4] => 1763 [5,1,1,1] => 1925 [5,1,2] => 3025 [5,2,1] => 3325 [5,3] => 1175 [6,1,1] => 385 [6,2] => 335 [7,1] => 35 [8] => 1 [1,1,1,1,1,1,1,1,1] => 4862 [1,1,1,1,1,1,1,2] => 19448 [1,1,1,1,1,1,2,1] => 58058 [1,1,1,1,1,1,3] => 32032 [1,1,1,1,1,2,1,1] => 105248 [1,1,1,1,1,2,2] => 145717 [1,1,1,1,1,3,1] => 107107 [1,1,1,1,1,4] => 28028 [1,1,1,1,2,1,1,1] => 138281 [1,1,1,1,2,1,2] => 277849 [1,1,1,1,2,2,1] => 404404 [1,1,1,1,2,3] => 166166 [1,1,1,1,3,1,1] => 192049 [1,1,1,1,3,2] => 217646 [1,1,1,1,4,1] => 91091 [1,1,1,1,5] => 14014 [1,1,1,2,1,1,1,1] => 138281 [1,1,1,2,1,1,2] => 348634 [1,1,1,2,1,2,1] => 663949 [1,1,1,2,1,3] => 307736 [1,1,1,2,2,1,1] => 616759 [1,1,1,2,2,2] => 783926 [1,1,1,2,3,1] => 468611 [1,1,1,2,4] => 104104 [1,1,1,3,1,1,1] => 229801 [1,1,1,3,1,2] => 415844 [1,1,1,3,2,1] => 525239 [1,1,1,3,3] => 201916 [1,1,1,4,1,1] => 147719 [1,1,1,4,2] => 150436 [1,1,1,5,1] => 41041 [1,1,1,6] => 4004 [1,1,2,1,1,1,1,1] => 105248 [1,1,2,1,1,1,2] => 310882 [1,1,2,1,1,2,1] => 696982 [1,1,2,1,1,3] => 345488 [1,1,2,1,2,1,1] => 849442 [1,1,2,1,2,2] => 1121153 [1,1,2,1,3,1] => 735053 [1,1,2,1,4] => 175252 [1,1,2,2,1,1,1] => 616759 [1,1,2,2,1,2] => 1188671 [1,1,2,2,2,1] => 1635656 [1,1,2,2,3] => 654214 [1,1,2,3,1,1] => 650111 [1,1,2,3,2] => 711634 [1,1,2,4,1] => 264649 [1,1,2,5] => 36806 [1,1,3,1,1,1,1] => 192049 [1,1,3,1,1,2] => 453596 [1,1,3,1,2,1] => 802571 [1,1,3,1,3] => 360184 [1,1,3,2,1,1] => 650111 [1,1,3,2,2] => 809644 [1,1,3,3,1] => 460669 [1,1,3,4] => 99176 [1,1,4,1,1,1] => 147719 [1,1,4,1,2] => 252076 [1,1,4,2,1] => 295141 [1,1,4,3] => 110264 [1,1,5,1,1] => 55561 [1,1,5,2] => 52844 [1,1,6,1] => 9779 [1,1,7] => 616 [1,2,1,1,1,1,1,1] => 58058 [1,2,1,1,1,1,2] => 192907 [1,2,1,1,1,2,1] => 484627 [1,2,1,1,1,3] => 251108 [1,2,1,1,2,1,1] => 696982 [1,2,1,1,2,2] => 939653 [1,2,1,1,3,1] => 647933 [1,2,1,1,4] => 160732 [1,2,1,2,1,1,1] => 663949 [1,2,1,2,1,2] => 1306646 [1,2,1,2,2,1] => 1848011 [1,2,1,2,3] => 748594 [1,2,1,3,1,1] => 802571 [1,2,1,3,2] => 893134 [1,2,1,4,1] => 351769 [1,2,1,5] => 51326 [1,2,2,1,1,1,1] => 404404 [1,2,2,1,1,2] => 996281 [1,2,2,1,2,1] => 1848011 [1,2,2,1,3] => 846604 [1,2,2,2,1,1] => 1635656 [1,2,2,2,2] => 2063809 [1,2,2,3,1] => 1212079 [1,2,2,4] => 266156 [1,2,3,1,1,1] => 525239 [1,2,3,1,2] => 934516 [1,2,3,2,1] => 1155451 [1,2,3,3] => 440594 [1,2,4,1,1] => 295141 [1,2,4,2] => 296054 [1,2,5,1] => 75119 [1,2,6] => 6886 [1,3,1,1,1,1,1] => 107107 [1,3,1,1,1,2] => 302588 [1,3,1,1,2,1] => 647933 [1,3,1,1,3] => 315172 [1,3,1,2,1,1] => 735053 [1,3,1,2,2] => 960652 [1,3,1,3,1] => 615307 [1,3,1,4] => 144188 [1,3,2,1,1,1] => 468611 [1,3,2,1,2] => 893134 [1,3,2,2,1] => 1212079 [1,3,2,3] => 481976 [1,3,3,1,1] => 460669 [1,3,3,2] => 500786 [1,3,4,1] => 181841 [1,3,5] => 24904 [1,4,1,1,1,1] => 91091 [1,4,1,1,2] => 207064 [1,4,1,2,1] => 351769 [1,4,1,3] => 155276 [1,4,2,1,1] => 264649 [1,4,2,2] => 326546 [1,4,3,1] => 181841 [1,4,4] => 38764 [1,5,1,1,1] => 41041 [1,5,1,2] => 67364 [1,5,2,1] => 75119 [1,5,3] => 27676 [1,6,1,1] => 9779 [1,6,2] => 8866 [1,7,1] => 1111 [1,8] => 44 [2,1,1,1,1,1,1,1] => 19448 [2,1,1,1,1,1,2] => 70642 [2,1,1,1,1,2,1] => 192907 [2,1,1,1,1,3] => 103103 [2,1,1,1,2,1,1] => 310882 [2,1,1,1,2,2] => 424853 [2,1,1,1,3,1] => 302588 [2,1,1,1,4] => 77077 [2,1,1,2,1,1,1] => 348634 [2,1,1,2,1,2] => 693836 [2,1,1,2,2,1] => 996281 [2,1,1,2,3] => 406549 [2,1,1,3,1,1] => 453596 [2,1,1,3,2] => 509509 [2,1,1,4,1] => 207064 [2,1,1,5] => 31031 [2,1,2,1,1,1,1] => 277849 [2,1,2,1,1,2] => 693836 [2,1,2,1,2,1] => 1306646 [2,1,2,1,3] => 602569 [2,1,2,2,1,1] => 1188671 [2,1,2,2,2] => 1505944 [2,1,2,3,1] => 893134 [2,1,2,4] => 197351 [2,1,3,1,1,1] => 415844 [2,1,3,1,2] => 746911 [2,1,3,2,1] => 934516 [2,1,3,3] => 357929 [2,1,4,1,1] => 252076 [2,1,4,2] => 254969 [2,1,5,1] => 67364 [2,1,6] => 6391 [2,2,1,1,1,1,1] => 145717 [2,2,1,1,1,2] => 424853 [2,2,1,1,2,1] => 939653 [2,2,1,1,3] => 463177 [2,2,1,2,1,1] => 1121153 [2,2,1,2,2] => 1475452 [2,2,1,3,1] => 960652 [2,2,1,4] => 227843 [2,2,2,1,1,1] => 783926 [2,2,2,1,2] => 1505944 [2,2,2,2,1] => 2063809 [2,2,2,3] => 824021 [2,2,3,1,1] => 809644 [2,2,3,2] => 884411 [2,2,4,1] => 326546 [2,2,5] => 45199 [2,3,1,1,1,1] => 217646 [2,3,1,1,2] => 509509 [2,3,1,2,1] => 893134 [2,3,1,3] => 399311 [2,3,2,1,1] => 711634 [2,3,2,2] => 884411 [2,3,3,1] => 500786 [2,3,4] => 107569 [2,4,1,1,1] => 150436 [2,4,1,2] => 254969 [2,4,2,1] => 296054 [2,4,3] => 110341 [2,5,1,1] => 52844 [2,5,2] => 49951 [2,6,1] => 8866 [2,7] => 539 [3,1,1,1,1,1,1] => 32032 [3,1,1,1,1,2] => 103103 [3,1,1,1,2,1] => 251108 [3,1,1,1,3] => 128557 [3,1,1,2,1,1] => 345488 [3,1,1,2,2] => 463177 [3,1,1,3,1] => 315172 [3,1,1,4] => 77363 [3,1,2,1,1,1] => 307736 [3,1,2,1,2] => 602569 [3,1,2,2,1] => 846604 [3,1,2,3] => 341891 [3,1,3,1,1] => 360184 [3,1,3,2] => 399311 [3,1,4,1] => 155276 [3,1,5] => 22429 [3,2,1,1,1,1] => 166166 [3,2,1,1,2] => 406549 [3,2,1,2,1] => 748594 [3,2,1,3] => 341891 [3,2,2,1,1] => 654214 [3,2,2,2] => 824021 [3,2,3,1] => 481976 [3,2,4] => 105589 [3,3,1,1,1] => 201916 [3,3,1,2] => 357929 [3,3,2,1] => 440594 [3,3,3] => 167761 [3,4,1,1] => 110264 [3,4,2] => 110341 [3,5,1] => 27676 [3,6] => 2519 [4,1,1,1,1,1] => 28028 [4,1,1,1,2] => 77077 [4,1,1,2,1] => 160732 [4,1,1,3] => 77363 [4,1,2,1,1] => 175252 [4,1,2,2] => 227843 [4,1,3,1] => 144188 [4,1,4] => 33517 [4,2,1,1,1] => 104104 [4,2,1,2] => 197351 [4,2,2,1] => 266156 [4,2,3] => 105589 [4,3,1,1] => 99176 [4,3,2] => 107569 [4,4,1] => 38764 [4,5] => 5291 [5,1,1,1,1] => 14014 [5,1,1,2] => 31031 [5,1,2,1] => 51326 [5,1,3] => 22429 [5,2,1,1] => 36806 [5,2,2] => 45199 [5,3,1] => 24904 [5,4] => 5291 [6,1,1,1] => 4004 [6,1,2] => 6391 [6,2,1] => 6886 [6,3] => 2519 [7,1,1] => 616 [7,2] => 539 [8,1] => 44 [9] => 1 ----------------------------------------------------------------------------- Created: Sep 16, 2015 at 08:11 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Mar 07, 2024 at 20:43 by Martin Rubey