*****************************************************************************
* www.FindStat.org - The Combinatorial Statistic Finder *
* *
* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
* *
* 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