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Identifier
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
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Description
The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions.
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)
Created
Sep 16, 2015 at 08:11 by Martin Rubey
Updated
Mar 07, 2024 at 20:43 by Martin Rubey