- FindStat

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Identifier

St000020: Permutations ⟶ ℤ

Values

[1] => 1

[1,2] => 1

[2,1] => 2

[1,2,3] => 1

[1,3,2] => 2

[2,1,3] => 3

[2,3,1] => 4

[3,1,2] => 5

[3,2,1] => 6

[1,2,3,4] => 1

[1,2,4,3] => 2

[1,3,2,4] => 3

[1,3,4,2] => 4

[1,4,2,3] => 5

[1,4,3,2] => 6

[2,1,3,4] => 7

[2,1,4,3] => 8

[2,3,1,4] => 9

[2,3,4,1] => 10

[2,4,1,3] => 11

[2,4,3,1] => 12

[3,1,2,4] => 13

[3,1,4,2] => 14

[3,2,1,4] => 15

[3,2,4,1] => 16

[3,4,1,2] => 17

[3,4,2,1] => 18

[4,1,2,3] => 19

[4,1,3,2] => 20

[4,2,1,3] => 21

[4,2,3,1] => 22

[4,3,1,2] => 23

[4,3,2,1] => 24

[1,2,3,4,5] => 1

[1,2,3,5,4] => 2

[1,2,4,3,5] => 3

[1,2,4,5,3] => 4

[1,2,5,3,4] => 5

[1,2,5,4,3] => 6

[1,3,2,4,5] => 7

[1,3,2,5,4] => 8

[1,3,4,2,5] => 9

[1,3,4,5,2] => 10

[1,3,5,2,4] => 11

[1,3,5,4,2] => 12

[1,4,2,3,5] => 13

[1,4,2,5,3] => 14

[1,4,3,2,5] => 15

[1,4,3,5,2] => 16

[1,4,5,2,3] => 17

[1,4,5,3,2] => 18

[1,5,2,3,4] => 19

[1,5,2,4,3] => 20

[1,5,3,2,4] => 21

[1,5,3,4,2] => 22

[1,5,4,2,3] => 23

[1,5,4,3,2] => 24

[2,1,3,4,5] => 25

[2,1,3,5,4] => 26

[2,1,4,3,5] => 27

[2,1,4,5,3] => 28

[2,1,5,3,4] => 29

[2,1,5,4,3] => 30

[2,3,1,4,5] => 31

[2,3,1,5,4] => 32

[2,3,4,1,5] => 33

[2,3,4,5,1] => 34

[2,3,5,1,4] => 35

[2,3,5,4,1] => 36

[2,4,1,3,5] => 37

[2,4,1,5,3] => 38

[2,4,3,1,5] => 39

[2,4,3,5,1] => 40

[2,4,5,1,3] => 41

[2,4,5,3,1] => 42

[2,5,1,3,4] => 43

[2,5,1,4,3] => 44

[2,5,3,1,4] => 45

[2,5,3,4,1] => 46

[2,5,4,1,3] => 47

[2,5,4,3,1] => 48

[3,1,2,4,5] => 49

[3,1,2,5,4] => 50

[3,1,4,2,5] => 51

[3,1,4,5,2] => 52

[3,1,5,2,4] => 53

[3,1,5,4,2] => 54

[3,2,1,4,5] => 55

[3,2,1,5,4] => 56

[3,2,4,1,5] => 57

[3,2,4,5,1] => 58

[3,2,5,1,4] => 59

[3,2,5,4,1] => 60

[3,4,1,2,5] => 61

[3,4,1,5,2] => 62

[3,4,2,1,5] => 63

[3,4,2,5,1] => 64

[3,4,5,1,2] => 65

[3,4,5,2,1] => 66

[3,5,1,2,4] => 67

[3,5,1,4,2] => 68

>>> Load all 873 entries. <<<

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$F_{1} = q$

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

$F_{3} = q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6}$

$F_{4} = q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + q^{13} + q^{14} + q^{15} + q^{16} + q^{17} + q^{18} + q^{19} + q^{20} + q^{21} + q^{22} + q^{23} + q^{24}$

$F_{5} = q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + q^{13} + q^{14} + q^{15} + q^{16} + q^{17} + q^{18} + q^{19} + q^{20} + q^{21} + q^{22} + q^{23} + q^{24} + q^{25} + q^{26} + q^{27} + q^{28} + q^{29} + q^{30} + q^{31} + q^{32} + q^{33} + q^{34} + q^{35} + q^{36} + q^{37} + q^{38} + q^{39} + q^{40} + q^{41} + q^{42} + q^{43} + q^{44} + q^{45} + q^{46} + q^{47} + q^{48} + q^{49} + q^{50} + q^{51} + q^{52} + q^{53} + q^{54} + q^{55} + q^{56} + q^{57} + q^{58} + q^{59} + q^{60} + q^{61} + q^{62} + q^{63} + q^{64} + q^{65} + q^{66} + q^{67} + q^{68} + q^{69} + q^{70} + q^{71} + q^{72} + q^{73} + q^{74} + q^{75} + q^{76} + q^{77} + q^{78} + q^{79} + q^{80} + q^{81} + q^{82} + q^{83} + q^{84} + q^{85} + q^{86} + q^{87} + q^{88} + q^{89} + q^{90} + q^{91} + q^{92} + q^{93} + q^{94} + q^{95} + q^{96} + q^{97} + q^{98} + q^{99} + q^{100} + q^{101} + q^{102} + q^{103} + q^{104} + q^{105} + q^{106} + q^{107} + q^{108} + q^{109} + q^{110} + q^{111} + q^{112} + q^{113} + q^{114} + q^{115} + q^{116} + q^{117} + q^{118} + q^{119} + q^{120}$

$F_{6} = q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + q^{13} + q^{14} + q^{15} + q^{16} + q^{17} + q^{18} + q^{19} + q^{20} + q^{21} + q^{22} + q^{23} + q^{24} + q^{25} + q^{26} + q^{27} + q^{28} + q^{29} + q^{30} + q^{31} + q^{32} + q^{33} + q^{34} + q^{35} + q^{36} + q^{37} + q^{38} + q^{39} + q^{40} + q^{41} + q^{42} + q^{43} + q^{44} + q^{45} + q^{46} + q^{47} + q^{48} + q^{49} + q^{50} + q^{51} + q^{52} + q^{53} + q^{54} + q^{55} + q^{56} + q^{57} + q^{58} + q^{59} + q^{60} + q^{61} + q^{62} + q^{63} + q^{64} + q^{65} + q^{66} + q^{67} + q^{68} + q^{69} + q^{70} + q^{71} + q^{72} + q^{73} + q^{74} + q^{75} + q^{76} + q^{77} + q^{78} + q^{79} + q^{80} + q^{81} + q^{82} + q^{83} + q^{84} + q^{85} + q^{86} + q^{87} + q^{88} + q^{89} + q^{90} + q^{91} + q^{92} + q^{93} + q^{94} + q^{95} + q^{96} + q^{97} + q^{98} + q^{99} + q^{100} + q^{101} + q^{102} + q^{103} + q^{104} + q^{105} + q^{106} + q^{107} + q^{108} + q^{109} + q^{110} + q^{111} + q^{112} + q^{113} + q^{114} + q^{115} + q^{116} + q^{117} + q^{118} + q^{119} + q^{120} + q^{121} + q^{122} + q^{123} + q^{124} + q^{125} + q^{126} + q^{127} + q^{128} + q^{129} + q^{130} + q^{131} + q^{132} + q^{133} + q^{134} + q^{135} + q^{136} + q^{137} + q^{138} + q^{139} + q^{140} + q^{141} + q^{142} + q^{143} + q^{144} + q^{145} + q^{146} + q^{147} + q^{148} + q^{149} + q^{150} + q^{151} + q^{152} + q^{153} + q^{154} + q^{155} + q^{156} + q^{157} + q^{158} + q^{159} + q^{160} + q^{161} + q^{162} + q^{163} + q^{164} + q^{165} + q^{166} + q^{167} + q^{168} + q^{169} + q^{170} + q^{171} + q^{172} + q^{173} + q^{174} + q^{175} + q^{176} + q^{177} + q^{178} + q^{179} + q^{180} + q^{181} + q^{182} + q^{183} + q^{184} + q^{185} + q^{186} + q^{187} + q^{188} + q^{189} + q^{190} + q^{191} + q^{192} + q^{193} + q^{194} + q^{195} + q^{196} + q^{197} + q^{198} + q^{199} + q^{200} + q^{201} + q^{202} + q^{203} + q^{204} + q^{205} + q^{206} + q^{207} + q^{208} + q^{209} + q^{210} + q^{211} + q^{212} + q^{213} + q^{214} + q^{215} + q^{216} + q^{217} + q^{218} + q^{219} + q^{220} + q^{221} + q^{222} + q^{223} + q^{224} + q^{225} + q^{226} + q^{227} + q^{228} + q^{229} + q^{230} + q^{231} + q^{232} + q^{233} + q^{234} + q^{235} + q^{236} + q^{237} + q^{238} + q^{239} + q^{240} + q^{241} + q^{242} + q^{243} + q^{244} + q^{245} + q^{246} + q^{247} + q^{248} + q^{249} + q^{250} + q^{251} + q^{252} + q^{253} + q^{254} + q^{255} + q^{256} + q^{257} + q^{258} + q^{259} + q^{260} + q^{261} + q^{262} + q^{263} + q^{264} + q^{265} + q^{266} + q^{267} + q^{268} + q^{269} + q^{270} + q^{271} + q^{272} + q^{273} + q^{274} + q^{275} + q^{276} + q^{277} + q^{278} + q^{279} + q^{280} + q^{281} + q^{282} + q^{283} + q^{284} + q^{285} + q^{286} + q^{287} + q^{288} + q^{289} + q^{290} + q^{291} + q^{292} + q^{293} + q^{294} + q^{295} + q^{296} + q^{297} + q^{298} + q^{299} + q^{300} + q^{301} + q^{302} + q^{303} + q^{304} + q^{305} + q^{306} + q^{307} + q^{308} + q^{309} + q^{310} + q^{311} + q^{312} + q^{313} + q^{314} + q^{315} + q^{316} + q^{317} + q^{318} + q^{319} + q^{320} + q^{321} + q^{322} + q^{323} + q^{324} + q^{325} + q^{326} + q^{327} + q^{328} + q^{329} + q^{330} + q^{331} + q^{332} + q^{333} + q^{334} + q^{335} + q^{336} + q^{337} + q^{338} + q^{339} + q^{340} + q^{341} + q^{342} + q^{343} + q^{344} + q^{345} + q^{346} + q^{347} + q^{348} + q^{349} + q^{350} + q^{351} + q^{352} + q^{353} + q^{354} + q^{355} + q^{356} + q^{357} + q^{358} + q^{359} + q^{360} + q^{361} + q^{362} + q^{363} + q^{364} + q^{365} + q^{366} + q^{367} + q^{368} + q^{369} + q^{370} + q^{371} + q^{372} + q^{373} + q^{374} + q^{375} + q^{376} + q^{377} + q^{378} + q^{379} + q^{380} + q^{381} + q^{382} + q^{383} + q^{384} + q^{385} + q^{386} + q^{387} + q^{388} + q^{389} + q^{390} + q^{391} + q^{392} + q^{393} + q^{394} + q^{395} + q^{396} + q^{397} + q^{398} + q^{399} + q^{400} + q^{401} + q^{402} + q^{403} + q^{404} + q^{405} + q^{406} + q^{407} + q^{408} + q^{409} + q^{410} + q^{411} + q^{412} + q^{413} + q^{414} + q^{415} + q^{416} + q^{417} + q^{418} + q^{419} + q^{420} + q^{421} + q^{422} + q^{423} + q^{424} + q^{425} + q^{426} + q^{427} + q^{428} + q^{429} + q^{430} + q^{431} + q^{432} + q^{433} + q^{434} + q^{435} + q^{436} + q^{437} + q^{438} + q^{439} + q^{440} + q^{441} + q^{442} + q^{443} + q^{444} + q^{445} + q^{446} + q^{447} + q^{448} + q^{449} + q^{450} + q^{451} + q^{452} + q^{453} + q^{454} + q^{455} + q^{456} + q^{457} + q^{458} + q^{459} + q^{460} + q^{461} + q^{462} + q^{463} + q^{464} + q^{465} + q^{466} + q^{467} + q^{468} + q^{469} + q^{470} + q^{471} + q^{472} + q^{473} + q^{474} + q^{475} + q^{476} + q^{477} + q^{478} + q^{479} + q^{480} + q^{481} + q^{482} + q^{483} + q^{484} + q^{485} + q^{486} + q^{487} + q^{488} + q^{489} + q^{490} + q^{491} + q^{492} + q^{493} + q^{494} + q^{495} + q^{496} + q^{497} + q^{498} + q^{499} + q^{500} + q^{501} + q^{502} + q^{503} + q^{504} + q^{505} + q^{506} + q^{507} + q^{508} + q^{509} + q^{510} + q^{511} + q^{512} + q^{513} + q^{514} + q^{515} + q^{516} + q^{517} + q^{518} + q^{519} + q^{520} + q^{521} + q^{522} + q^{523} + q^{524} + q^{525} + q^{526} + q^{527} + q^{528} + q^{529} + q^{530} + q^{531} + q^{532} + q^{533} + q^{534} + q^{535} + q^{536} + q^{537} + q^{538} + q^{539} + q^{540} + q^{541} + q^{542} + q^{543} + q^{544} + q^{545} + q^{546} + q^{547} + q^{548} + q^{549} + q^{550} + q^{551} + q^{552} + q^{553} + q^{554} + q^{555} + q^{556} + q^{557} + q^{558} + q^{559} + q^{560} + q^{561} + q^{562} + q^{563} + q^{564} + q^{565} + q^{566} + q^{567} + q^{568} + q^{569} + q^{570} + q^{571} + q^{572} + q^{573} + q^{574} + q^{575} + q^{576} + q^{577} + q^{578} + q^{579} + q^{580} + q^{581} + q^{582} + q^{583} + q^{584} + q^{585} + q^{586} + q^{587} + q^{588} + q^{589} + q^{590} + q^{591} + q^{592} + q^{593} + q^{594} + q^{595} + q^{596} + q^{597} + q^{598} + q^{599} + q^{600} + q^{601} + q^{602} + q^{603} + q^{604} + q^{605} + q^{606} + q^{607} + q^{608} + q^{609} + q^{610} + q^{611} + q^{612} + q^{613} + q^{614} + q^{615} + q^{616} + q^{617} + q^{618} + q^{619} + q^{620} + q^{621} + q^{622} + q^{623} + q^{624} + q^{625} + q^{626} + q^{627} + q^{628} + q^{629} + q^{630} + q^{631} + q^{632} + q^{633} + q^{634} + q^{635} + q^{636} + q^{637} + q^{638} + q^{639} + q^{640} + q^{641} + q^{642} + q^{643} + q^{644} + q^{645} + q^{646} + q^{647} + q^{648} + q^{649} + q^{650} + q^{651} + q^{652} + q^{653} + q^{654} + q^{655} + q^{656} + q^{657} + q^{658} + q^{659} + q^{660} + q^{661} + q^{662} + q^{663} + q^{664} + q^{665} + q^{666} + q^{667} + q^{668} + q^{669} + q^{670} + q^{671} + q^{672} + q^{673} + q^{674} + q^{675} + q^{676} + q^{677} + q^{678} + q^{679} + q^{680} + q^{681} + q^{682} + q^{683} + q^{684} + q^{685} + q^{686} + q^{687} + q^{688} + q^{689} + q^{690} + q^{691} + q^{692} + q^{693} + q^{694} + q^{695} + q^{696} + q^{697} + q^{698} + q^{699} + q^{700} + q^{701} + q^{702} + q^{703} + q^{704} + q^{705} + q^{706} + q^{707} + q^{708} + q^{709} + q^{710} + q^{711} + q^{712} + q^{713} + q^{714} + q^{715} + q^{716} + q^{717} + q^{718} + q^{719} + q^{720}$

Description

The rank of the permutation.
This is its position among all permutations of the same size ordered lexicographically.
This can be computed using the Lehmer code of a permutation:
$$\text{rank}(\sigma) = 1 +\sum_{i=1}^{n-1} L(\sigma)_i (n − i)!,$$
where $L(\sigma)_i$ is the $i$-th entry of the Lehmer code of $\sigma$.

Code

def statistic(x):
    return x.rank()+1

Created

Oct 01, 2011 at 20:56 by Chris Berg

Updated

Jun 28, 2022 at 16:25 by Nadia Lafreniere