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Identifier

St000285: Integer compositions ⟶ ℤ

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

>>> Load all 510 entries. <<<

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$F_{2} = q + q^{2}$

$F_{3} = q + 3\ q^{5}$

$F_{4} = q + 2\ q^{9} + q^{14} + q^{19} + 2\ q^{21} + q^{31}$

$F_{5} = q + 2\ q^{14} + q^{42} + 2\ q^{49} + 2\ q^{56} + 2\ q^{84} + q^{91} + q^{126} + 2\ q^{154} + 2\ q^{161}$

$F_{6} = q + 2\ q^{20} + 2\ q^{104} + 2\ q^{120} + q^{132} + q^{174} + q^{204} + 2\ q^{300} + 2\ q^{330} + 2\ q^{456} + 2\ q^{540} + 2\ q^{610} + 2\ q^{636} + q^{678} + 2\ q^{708} + q^{876} + 2\ q^{1014} + q^{1114} + 2\ q^{1182} + q^{1588}$

$F_{7} = q + 2\ q^{27} + 2\ q^{195} + 2\ q^{225} + q^{393} + q^{429} + 2\ q^{489} + 2\ q^{825} + 2\ q^{1275} + 2\ q^{1287} + 2\ q^{1443} + 2\ q^{1485} + 2\ q^{1737} + 2\ q^{1875} + q^{2157} + 2\ q^{2619} + 4\ q^{3135} + q^{3333} + 2\ q^{3795} + 2\ q^{4185} + 2\ q^{4521} + 2\ q^{4803} + q^{4929} + 2\ q^{5067} + q^{5097} + 2\ q^{5319} + 2\ q^{5643} + 2\ q^{6201} + 2\ q^{7029} + 2\ q^{7545} + 2\ q^{8163} + 2\ q^{9897} + 2\ q^{11355} + q^{12741} + 2\ q^{13257}$

$F_{8} = q + 2\ q^{35} + 2\ q^{335} + 2\ q^{385} + q^{685} + 2\ q^{1175} + q^{1430} + q^{1763} + 2\ q^{1925} + 2\ q^{3025} + 2\ q^{3325} + 2\ q^{4165} + 2\ q^{4565} + 4\ q^{5005} + 2\ q^{7007} + 2\ q^{7645} + 2\ q^{8785} + 2\ q^{9373} + 2\ q^{10835} + 2\ q^{13475} + 2\ q^{13585} + 2\ q^{13915} + q^{15587} + 2\ q^{16045} + q^{17473} + 4\ q^{18095} + q^{18227} + q^{18535} + 2\ q^{18733} + 2\ q^{20735} + 2\ q^{22165} + 2\ q^{22715} + 2\ q^{25487} + q^{25795} + q^{25873} + 2\ q^{26195} + 2\ q^{26713} + 2\ q^{27643} + 2\ q^{29315} + 2\ q^{32285} + 2\ q^{34727} + 2\ q^{35035} + 2\ q^{37895} + 2\ q^{39193} + q^{43415} + 2\ q^{43813} + 2\ q^{46585} + 2\ q^{49445} + 2\ q^{50567} + 2\ q^{53075} + 2\ q^{55187} + 2\ q^{55825} + 2\ q^{59807} + 2\ q^{64735} + 2\ q^{67907} + 2\ q^{68893} + 2\ q^{69355} + 2\ q^{70565} + 2\ q^{73115} + 2\ q^{77735} + 2\ q^{80905} + q^{82775} + q^{90365} + q^{93313} + 2\ q^{98945} + 2\ q^{110605} + 2\ q^{125785} + 2\ q^{130405} + q^{134627} + q^{177325}$

$F_{9} = q + 2\ q^{44} + 2\ q^{539} + 2\ q^{616} + q^{1111} + 2\ q^{2519} + 2\ q^{4004} + q^{4862} + 2\ q^{5291} + 2\ q^{6391} + 2\ q^{6886} + 2\ q^{8866} + 2\ q^{9779} + 2\ q^{14014} + 2\ q^{19448} + 2\ q^{22429} + 2\ q^{24904} + 2\ q^{27676} + 2\ q^{28028} + 2\ q^{31031} + 2\ q^{32032} + q^{33517} + 2\ q^{36806} + 2\ q^{38764} + 2\ q^{41041} + 2\ q^{45199} + q^{49951} + 2\ q^{51326} + 2\ q^{52844} + q^{55561} + 2\ q^{58058} + 2\ q^{67364} + q^{70642} + 2\ q^{75119} + 2\ q^{77077} + 2\ q^{77363} + 2\ q^{91091} + 2\ q^{99176} + 2\ q^{103103} + 2\ q^{104104} + 2\ q^{105248} + 2\ q^{105589} + 2\ q^{107107} + 2\ q^{107569} + 2\ q^{110264} + 2\ q^{110341} + q^{128557} + 2\ q^{138281} + 2\ q^{144188} + 2\ q^{145717} + 2\ q^{147719} + 2\ q^{150436} + 2\ q^{155276} + 2\ q^{160732} + 2\ q^{166166} + q^{167761} + 2\ q^{175252} + 2\ q^{181841} + 2\ q^{192049} + 2\ q^{192907} + 2\ q^{197351} + 2\ q^{201916} + 2\ q^{207064} + 2\ q^{217646} + 2\ q^{227843} + q^{229801} + 2\ q^{251108} + 2\ q^{252076} + 2\ q^{254969} + 2\ q^{264649} + 2\ q^{266156} + 2\ q^{277849} + 2\ q^{295141} + 2\ q^{296054} + 2\ q^{302588} + 2\ q^{307736} + 2\ q^{310882} + 2\ q^{315172} + 2\ q^{326546} + 2\ q^{341891} + 2\ q^{345488} + 2\ q^{348634} + 2\ q^{351769} + 2\ q^{357929} + 2\ q^{360184} + 2\ q^{399311} + 2\ q^{404404} + 2\ q^{406549} + 2\ q^{415844} + 2\ q^{424853} + 2\ q^{440594} + 2\ q^{453596} + 2\ q^{460669} + 2\ q^{463177} + 2\ q^{468611} + 2\ q^{481976} + q^{484627} + 2\ q^{500786} + 2\ q^{509509} + 2\ q^{525239} + 2\ q^{602569} + q^{615307} + 2\ q^{616759} + 2\ q^{647933} + 2\ q^{650111} + 2\ q^{654214} + 2\ q^{663949} + 2\ q^{693836} + 2\ q^{696982} + 2\ q^{711634} + 2\ q^{735053} + q^{746911} + 2\ q^{748594} + 2\ q^{783926} + 2\ q^{802571} + 2\ q^{809644} + 2\ q^{824021} + 2\ q^{846604} + q^{849442} + 2\ q^{884411} + 4\ q^{893134} + 2\ q^{934516} + 2\ q^{939653} + 2\ q^{960652} + 2\ q^{996281} + 2\ q^{1121153} + q^{1155451} + 2\ q^{1188671} + 2\ q^{1212079} + 2\ q^{1306646} + q^{1475452} + 2\ q^{1505944} + 2\ q^{1635656} + 2\ q^{1848011} + 2\ q^{2063809}$

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