- FindStat

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Identifier

St001482: Permutations ⟶ ℤ

Values

[1] => 1

[1,2] => 1

[2,1] => 2

[1,2,3] => 3

[1,3,2] => 4

[2,1,3] => 6

[2,3,1] => 10

[3,1,2] => 12

[3,2,1] => 15

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

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

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

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

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

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

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

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

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

[2,3,4,1] => 90

[2,4,1,3] => 84

[2,4,3,1] => 108

[3,1,2,4] => 72

[3,1,4,2] => 96

[3,2,1,4] => 90

[3,2,4,1] => 135

[3,4,1,2] => 168

[3,4,2,1] => 189

[4,1,2,3] => 140

[4,1,3,2] => 160

[4,2,1,3] => 168

[4,2,3,1] => 216

[4,3,1,2] => 224

[4,3,2,1] => 252

[1,2,3,4,5] => 180

[1,2,3,5,4] => 198

[1,2,4,3,5] => 210

[1,2,4,5,3] => 252

[1,2,5,3,4] => 264

[1,2,5,4,3] => 288

[1,3,2,4,5] => 240

[1,3,2,5,4] => 264

[1,3,4,2,5] => 320

[1,3,4,5,2] => 416

[1,3,5,2,4] => 396

[1,3,5,4,2] => 468

[1,4,2,3,5] => 350

[1,4,2,5,3] => 420

[1,4,3,2,5] => 400

[1,4,3,5,2] => 520

[1,4,5,2,3] => 600

[1,4,5,3,2] => 650

[1,5,2,3,4] => 528

[1,5,2,4,3] => 576

[1,5,3,2,4] => 594

[1,5,3,4,2] => 702

[1,5,4,2,3] => 720

[1,5,4,3,2] => 780

[2,1,3,4,5] => 360

[2,1,3,5,4] => 396

[2,1,4,3,5] => 420

[2,1,4,5,3] => 504

[2,1,5,3,4] => 528

[2,1,5,4,3] => 576

[2,3,1,4,5] => 600

[2,3,1,5,4] => 660

[2,3,4,1,5] => 900

[2,3,4,5,1] => 1260

[2,3,5,1,4] => 1100

[2,3,5,4,1] => 1400

[2,4,1,3,5] => 840

[2,4,1,5,3] => 1008

[2,4,3,1,5] => 1080

[2,4,3,5,1] => 1512

[2,4,5,1,3] => 1584

[2,4,5,3,1] => 1848

[2,5,1,3,4] => 1232

[2,5,1,4,3] => 1344

[2,5,3,1,4] => 1540

[2,5,3,4,1] => 1960

[2,5,4,1,3] => 1848

[2,5,4,3,1] => 2156

[3,1,2,4,5] => 720

[3,1,2,5,4] => 792

[3,1,4,2,5] => 960

[3,1,4,5,2] => 1248

[3,1,5,2,4] => 1188

[3,1,5,4,2] => 1404

[3,2,1,4,5] => 900

[3,2,1,5,4] => 990

[3,2,4,1,5] => 1350

[3,2,4,5,1] => 1890

[3,2,5,1,4] => 1650

[3,2,5,4,1] => 2100

[3,4,1,2,5] => 1680

[3,4,1,5,2] => 2184

[3,4,2,1,5] => 1890

[3,4,2,5,1] => 2646

[3,4,5,1,2] => 3276

[3,4,5,2,1] => 3528

[3,5,1,2,4] => 2376

[3,5,1,4,2] => 2808

>>> Load all 1200 entries. <<<

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$F_{1} = q$

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

$F_{3} = q^{3} + q^{4} + q^{6} + q^{10} + q^{12} + q^{15}$

$F_{4} = q^{18} + q^{21} + q^{24} + q^{32} + q^{35} + q^{36} + q^{40} + q^{42} + q^{60} + q^{72} + q^{84} + 2\ q^{90} + q^{96} + q^{108} + q^{135} + q^{140} + q^{160} + 2\ q^{168} + q^{189} + q^{216} + q^{224} + q^{252}$

$F_{5} = q^{180} + q^{198} + q^{210} + q^{240} + q^{252} + 2\ q^{264} + q^{288} + q^{320} + q^{350} + q^{360} + 2\ q^{396} + q^{400} + q^{416} + 2\ q^{420} + q^{468} + q^{504} + q^{520} + 2\ q^{528} + 2\ q^{576} + q^{594} + 2\ q^{600} + q^{650} + q^{660} + q^{702} + 2\ q^{720} + q^{780} + q^{792} + q^{840} + 2\ q^{900} + q^{960} + q^{990} + q^{1008} + q^{1080} + q^{1100} + q^{1188} + q^{1232} + q^{1248} + q^{1260} + q^{1344} + q^{1350} + 2\ q^{1400} + q^{1404} + q^{1512} + q^{1540} + q^{1584} + q^{1600} + q^{1650} + 3\ q^{1680} + 2\ q^{1848} + 2\ q^{1890} + q^{1960} + q^{2016} + q^{2080} + q^{2100} + q^{2156} + q^{2160} + q^{2184} + q^{2240} + q^{2376} + q^{2400} + q^{2520} + q^{2600} + 2\ q^{2640} + q^{2646} + q^{2808} + q^{2880} + q^{2912} + q^{2970} + q^{3024} + q^{3080} + q^{3168} + q^{3276} + 2\ q^{3360} + q^{3510} + 2\ q^{3528} + q^{3600} + q^{3696} + q^{3744} + q^{3850} + q^{3900} + q^{3960} + q^{4032} + q^{4320} + q^{4368} + q^{4400} + q^{4620} + 2\ q^{4680} + q^{4704} + q^{4752} + q^{4900} + q^{5390} + q^{5400} + q^{5544} + q^{5600} + q^{5616} + q^{5850} + q^{5940} + q^{6048} + q^{6240} + q^{6720} + q^{6930} + q^{7020} + q^{7560}$

$F_{6} = q^{2700} + q^{2880} + q^{2970} + q^{3150} + q^{3360} + q^{3366} + q^{3456} + q^{3600} + q^{3672} + q^{3780} + q^{3840} + 2\ q^{3960} + q^{4320} + q^{4368} + 2\ q^{4488} + q^{4536} + q^{4608} + q^{4800} + q^{4896} + q^{4914} + q^{5120} + 2\ q^{5184} + q^{5250} + q^{5400} + q^{5508} + q^{5600} + q^{5616} + q^{5712} + q^{5760} + 2\ q^{5940} + q^{6000} + q^{6048} + q^{6240} + 2\ q^{6300} + q^{6318} + q^{6400} + q^{6426} + q^{6720} + 2\ q^{6732} + q^{6804} + q^{6912} + q^{7020} + q^{7168} + q^{7280} + q^{7344} + 2\ q^{7560} + q^{7680} + q^{7800} + q^{7904} + 2\ q^{7920} + q^{8160} + q^{8190} + q^{8512} + 2\ q^{8640} + q^{8736} + q^{8892} + q^{8910} + 2\ q^{8960} + 2\ q^{8976} + 2\ q^{9000} + q^{9072} + q^{9180} + q^{9600} + q^{9750} + q^{9828} + q^{9880} + q^{9900} + q^{10098} + q^{10200} + q^{10260} + 3\ q^{10368} + q^{10530} + 2\ q^{10640} + 3\ q^{10800} + q^{11016} + q^{11220} + q^{11232} + q^{11400} + 2\ q^{11424} + q^{11440} + 2\ q^{11520} + q^{11700} + q^{11880} + 3\ q^{12096} + q^{12240} + q^{12320} + q^{12350} + q^{12600} + q^{12636} + 2\ q^{12852} + q^{12870} + q^{12960} + q^{13104} + q^{13338} + 2\ q^{13440} + q^{13464} + 2\ q^{13500} + q^{13608} + q^{13770} + q^{13824} + q^{14280} + 4\ q^{14400} + q^{14630} + q^{14688} + q^{14742} + q^{14820} + q^{14850} + q^{14994} + q^{15120} + q^{15200} + q^{15360} + q^{15390} + q^{15680} + q^{15840} + q^{15876} + q^{16016} + q^{16200} + q^{16500} + q^{16720} + q^{16830} + q^{17136} + q^{17248} + 3\ q^{17280} + q^{17472} + q^{17820} + q^{17850} + q^{18018} + 2\ q^{18144} + q^{18240} + 2\ q^{18360} + q^{18480} + q^{18620} + q^{18700} + q^{18720} + q^{18900} + q^{19656} + q^{19950} + q^{19992} + q^{20160} + q^{20196} + q^{20250} + q^{20482} + q^{20520} + q^{20736} + q^{20944} + 2\ q^{21000} + q^{21060} + q^{21120} + q^{21168} + q^{21420} + q^{21504} + 2\ q^{21600} + q^{21888} + q^{22176} + q^{22400} + q^{22440} + q^{22680} + q^{23040} + q^{23100} + q^{23408} + q^{23712} + q^{23760} + q^{23940} + q^{24000} + 2\ q^{24192} + q^{24480} + q^{24750} + 4\ q^{25200} + 2\ q^{25536} + q^{25600} + q^{25920} + q^{26180} + q^{26400} + q^{26656} + q^{26676} + 3\ q^{26880} + q^{27000} + q^{27200} + q^{27540} + q^{27648} + 2\ q^{27720} + q^{28000} + q^{28050} + q^{28224} + 2\ q^{28350} + q^{28512} + q^{29120} + q^{29376} + q^{29400} + q^{29920} + 2\ q^{29952} + 4\ q^{30240} + q^{30600} + q^{30780} + q^{31200} + q^{31500} + q^{31680} + q^{31920} + q^{32000} + q^{32340} + 3\ q^{32400} + 2\ q^{32760} + q^{33000} + q^{33264} + q^{33600} + q^{33660} + 2\ q^{33696} + q^{33792} + q^{34200} + q^{34272} + 2\ q^{34560} + q^{34944} + q^{35200} + q^{35640} + 2\ q^{35840} + q^{35904} + q^{36000} + 2\ q^{36288} + q^{36960} + q^{37632} + 2\ q^{37800} + q^{38080} + q^{39000} + q^{39200} + q^{39312} + q^{39520} + 3\ q^{39600} + q^{39690} + q^{39936} + q^{40320} + 2\ q^{40392} + q^{40500} + q^{40800} + q^{41496} + q^{42000} + q^{42120} + q^{42240} + q^{42560} + q^{43120} + 3\ q^{43200} + q^{43316} + q^{43680} + q^{44064} + q^{44550} + q^{44688} + q^{44800} + 4\ q^{44880} + q^{44928} + 2\ q^{45360} + q^{45760} + q^{45864} + q^{46080} + q^{46200} + q^{47124} + q^{47520} + q^{47872} + q^{48000} + q^{48960} + q^{49140} + q^{49280} + q^{49400} + q^{49500} + 2\ q^{49504} + q^{50176} + 2\ q^{50400} + q^{50490} + q^{51480} + 3\ q^{51840} + 2\ q^{52360} + q^{52416} + q^{52650} + 2\ q^{52800} + 3\ q^{52920} + q^{53352} + q^{54000} + 2\ q^{55080} + q^{55328} + q^{55440} + q^{55692} + q^{56160} + q^{56320} + q^{56576} + q^{56700} + 2\ q^{57024} + q^{57120} + 2\ q^{57600} + q^{57750} + q^{58240} + q^{58500} + q^{58520} + q^{58752} + q^{59400} + q^{59584} + q^{59904} + 6\ q^{60480} + q^{60588} + q^{60800} + q^{61152} + q^{61560} + q^{61880} + q^{62244} + q^{63360} + q^{63648} + 3\ q^{64800} + 2\ q^{65280} + q^{65450} + 2\ q^{65520} + q^{66000} + q^{66560} + q^{66640} + q^{66690} + q^{66880} + q^{67200} + q^{67320} + q^{67392} + 2\ q^{68850} + q^{69120} + q^{69300} + 2\ q^{70200} + 4\ q^{70560} + q^{70720} + q^{71136} + 2\ q^{71280} + q^{71820} + q^{72576} + q^{72618} + q^{73500} + q^{73920} + q^{74100} + q^{74800} + q^{75600} + q^{76800} + 2\ q^{76950} + q^{77112} + q^{77760} + 2\ q^{78624} + 2\ q^{80640} + 2\ q^{80784} + q^{80850} + q^{81000} + q^{81536} + q^{81900} + q^{82944} + q^{82992} + 2\ q^{83160} + q^{84000} + 2\ q^{84240} + q^{85536} + 3\ q^{85680} + q^{86184} + 2\ q^{86400} + q^{87360} + q^{87750} + 2\ q^{88128} + q^{88452} + 2\ q^{88920} + 2\ q^{89100} + q^{89760} + q^{89856} + q^{89964} + 3\ q^{90720} + q^{91200} + q^{91392} + q^{91520} + 2\ q^{91800} + 2\ q^{93366} + q^{93600} + 2\ q^{94080} + q^{95040} + q^{95200} + q^{95256} + q^{95760} + q^{96096} + q^{96390} + q^{96824} + q^{97200} + q^{97920} + q^{98000} + q^{98280} + q^{98496} + q^{98560} + q^{99840} + q^{100800} + q^{101088} + q^{101376} + 2\ q^{102600} + 2\ q^{102816} + q^{102960} + q^{103488} + 4\ q^{103680} + q^{103950} + 2\ q^{105300} + q^{106704} + q^{106920} + q^{107100} + q^{107520} + q^{107712} + q^{107730} + q^{107800} + q^{108108} + q^{108290} + q^{108800} + q^{108864} + q^{109200} + 2\ q^{109440} + q^{110160} + q^{110880} + q^{111150} + q^{111720} + 2\ q^{112000} + q^{112320} + q^{113400} + q^{114048} + q^{114660} + 2\ q^{114912} + q^{115200} + q^{116480} + q^{117040} + q^{117810} + q^{118560} + q^{119700} + q^{119808} + q^{119952} + q^{120042} + 5\ q^{120960} + 2\ q^{121176} + q^{122892} + q^{123760} + q^{124488} + q^{124800} + q^{126360} + 2\ q^{126720} + q^{127008} + q^{128000} + q^{128520} + q^{129276} + q^{129600} + q^{130900} + q^{131040} + q^{133056} + q^{133380} + q^{133760} + q^{134400} + q^{134640} + q^{134784} + q^{136080} + q^{136800} + q^{137280} + q^{137700} + q^{138240} + q^{138320} + q^{138600} + q^{139230} + q^{140448} + q^{142560} + q^{142800} + q^{143616} + 2\ q^{143640} + q^{144000} + q^{145152} + q^{145600} + q^{146880} + q^{147744} + q^{147840} + q^{148512} + q^{149760} + q^{151200} + q^{151470} + q^{152320} + q^{153216} + q^{153900} + q^{154440} + q^{154700} + 2\ q^{155520} + q^{156000} + q^{157080} + 2\ q^{157248} + q^{158400} + q^{159600} + q^{161568} + q^{163200} + 2\ q^{164160} + q^{165240} + q^{166320} + 2\ q^{168300} + 2\ q^{168480} + q^{168960} + q^{169728} + q^{170170} + 4\ q^{172800} + q^{174720} + q^{175560} + q^{176256} + q^{177840} + q^{178200} + q^{179200} + q^{180180} + q^{182400} + 2\ q^{183600} + 2\ q^{184680} + q^{186732} + 2\ q^{187200} + q^{188100} + q^{188496} + 3\ q^{190080} + q^{190944} + q^{191520} + q^{192780} + 3\ q^{194400} + q^{194480} + q^{195840} + q^{196350} + q^{199584} + q^{199680} + 2\ q^{200640} + q^{201600} + q^{201960} + q^{204000} + q^{204204} + 2\ q^{205200} + q^{205632} + q^{206550} + q^{207480} + q^{209440} + q^{210600} + q^{212160} + q^{215460} + 2\ q^{216000} + q^{216216} + q^{218790} + q^{219450} + q^{220320} + q^{225720} + q^{228096} + q^{228800} + q^{229500} + q^{230850} + q^{233376} + q^{234000} + q^{235620} + q^{237600} + q^{240084} + q^{240768} + q^{241920} + q^{243000} + q^{243100} + q^{244800} + q^{246400} + q^{250800} + q^{251328} + q^{252450} + q^{252720} + q^{258552} + q^{259200} + q^{262548} + 2\ q^{263340} + q^{266760} + q^{272160} + q^{273600} + q^{274560} + q^{275400} + q^{277200} + q^{280500} + q^{280800} + q^{282150} + q^{285120} + q^{291720} + q^{295680} + q^{297000} + q^{300960} + q^{302940} + q^{306000} + q^{307800} + q^{316008} + q^{324000} + q^{332640} + q^{336600} + q^{338580} + q^{356400}$

Description

The product of the prefix sums of a permutation.
This is, given a permutation $\pi$ of length $n$,
$$ \prod_{i=1}^{n-1}(\pi(1)+\dots+\pi(i)).$$

References

[1] Halder, A. Sum over permutations involving sign MathOverflow:344474

Code

def statistic(pi):
    pi = list(pi)
    return prod(sum(pi[:i]) for i in range(1,len(pi)))

Created

Oct 24, 2019 at 09:18 by Christian Stump

Updated

Oct 24, 2019 at 09:18 by Christian Stump