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Identifier

St001293: Permutations ⟶ ℤ

Values

[1] => 1

[1,2] => 9

[2,1] => 8

[1,2,3] => 55

[1,3,2] => 54

[2,1,3] => 50

[2,3,1] => 47

[3,1,2] => 47

[3,2,1] => 45

[1,2,3,4] => 875

[1,2,4,3] => 870

[1,3,2,4] => 861

[1,3,4,2] => 848

[1,4,2,3] => 848

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

[2,1,3,4] => 805

[2,1,4,3] => 800

[2,3,1,4] => 763

[2,3,4,1] => 736

[2,4,1,3] => 750

[2,4,3,1] => 728

[3,1,2,4] => 763

[3,1,4,2] => 750

[3,2,1,4] => 735

[3,2,4,1] => 708

[3,4,1,2] => 700

[3,4,2,1] => 686

[4,1,2,3] => 736

[4,1,3,2] => 728

[4,2,1,3] => 708

[4,2,3,1] => 686

[4,3,1,2] => 686

[4,3,2,1] => 672

[1,2,3,4,5] => 2877

[1,2,3,5,4] => 2870

[1,2,4,3,5] => 2862

[1,2,4,5,3] => 2845

[1,2,5,3,4] => 2845

[1,2,5,4,3] => 2835

[1,3,2,4,5] => 2835

[1,3,2,5,4] => 2828

[1,3,4,2,5] => 2796

[1,3,4,5,2] => 2764

[1,3,5,2,4] => 2779

[1,3,5,4,2] => 2754

[1,4,2,3,5] => 2796

[1,4,2,5,3] => 2779

[1,4,3,2,5] => 2772

[1,4,3,5,2] => 2740

[1,4,5,2,3] => 2730

[1,4,5,3,2] => 2715

[1,5,2,3,4] => 2764

[1,5,2,4,3] => 2754

[1,5,3,2,4] => 2740

[1,5,3,4,2] => 2715

[1,5,4,2,3] => 2715

[1,5,4,3,2] => 2700

[2,1,3,4,5] => 2667

[2,1,3,5,4] => 2660

[2,1,4,3,5] => 2652

[2,1,4,5,3] => 2635

[2,1,5,3,4] => 2635

[2,1,5,4,3] => 2625

[2,3,1,4,5] => 2541

[2,3,1,5,4] => 2534

[2,3,4,1,5] => 2460

[2,3,4,5,1] => 2404

[2,3,5,1,4] => 2443

[2,3,5,4,1] => 2394

[2,4,1,3,5] => 2502

[2,4,1,5,3] => 2485

[2,4,3,1,5] => 2436

[2,4,3,5,1] => 2380

[2,4,5,1,3] => 2394

[2,4,5,3,1] => 2355

[2,5,1,3,4] => 2470

[2,5,1,4,3] => 2460

[2,5,3,1,4] => 2404

[2,5,3,4,1] => 2355

[2,5,4,1,3] => 2379

[2,5,4,3,1] => 2340

[3,1,2,4,5] => 2541

[3,1,2,5,4] => 2534

[3,1,4,2,5] => 2502

[3,1,4,5,2] => 2470

[3,1,5,2,4] => 2485

[3,1,5,4,2] => 2460

[3,2,1,4,5] => 2457

[3,2,1,5,4] => 2450

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

[3,2,4,5,1] => 2320

[3,2,5,1,4] => 2359

[3,2,5,4,1] => 2310

[3,4,1,2,5] => 2352

[3,4,1,5,2] => 2320

[3,4,2,1,5] => 2310

[3,4,2,5,1] => 2254

[3,4,5,1,2] => 2229

[3,4,5,2,1] => 2205

[3,5,1,2,4] => 2320

[3,5,1,4,2] => 2295

>>> Load all 1200 entries. <<<

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$F_{1} = q$

$F_{2} = q^{8} + q^{9}$

$F_{3} = q^{45} + 2\ q^{47} + q^{50} + q^{54} + q^{55}$

$F_{4} = q^{672} + 3\ q^{686} + q^{700} + 2\ q^{708} + 2\ q^{728} + q^{735} + 2\ q^{736} + 2\ q^{750} + 2\ q^{763} + q^{800} + q^{805} + q^{840} + 2\ q^{848} + q^{861} + q^{870} + q^{875}$

$F_{5} = q^{2100} + 4\ q^{2124} + 3\ q^{2148} + 3\ q^{2163} + 2\ q^{2187} + 3\ q^{2190} + 3\ q^{2205} + 2\ q^{2212} + 2\ q^{2214} + 6\ q^{2229} + 4\ q^{2254} + 2\ q^{2268} + 2\ q^{2278} + q^{2295} + 5\ q^{2310} + 4\ q^{2320} + 2\ q^{2340} + q^{2352} + 4\ q^{2355} + 2\ q^{2359} + 2\ q^{2376} + 2\ q^{2379} + 2\ q^{2380} + 4\ q^{2394} + 4\ q^{2404} + 2\ q^{2436} + 2\ q^{2443} + q^{2450} + q^{2457} + 4\ q^{2460} + 2\ q^{2470} + 2\ q^{2485} + 2\ q^{2502} + 2\ q^{2534} + 2\ q^{2541} + q^{2625} + 2\ q^{2635} + q^{2652} + q^{2660} + q^{2667} + q^{2700} + 3\ q^{2715} + q^{2730} + 2\ q^{2740} + 2\ q^{2754} + 2\ q^{2764} + q^{2772} + 2\ q^{2779} + 2\ q^{2796} + q^{2828} + 2\ q^{2835} + 2\ q^{2845} + q^{2862} + q^{2870} + q^{2877}$

$F_{6} = q^{23760} + 5\ q^{23925} + 6\ q^{24090} + 4\ q^{24200} + q^{24255} + 4\ q^{24354} + 6\ q^{24365} + 4\ q^{24464} + 6\ q^{24519} + 3\ q^{24552} + 12\ q^{24629} + q^{24640} + 2\ q^{24717} + 6\ q^{24794} + 6\ q^{24816} + 4\ q^{24904} + q^{24948} + 2\ q^{24960} + 8\ q^{24981} + 4\ q^{25058} + 4\ q^{25168} + 6\ q^{25224} + 10\ q^{25245} + 2\ q^{25256} + 4\ q^{25333} + 6\ q^{25355} + 2\ q^{25389} + 6\ q^{25410} + 2\ q^{25488} + 14\ q^{25520} + 3\ q^{25542} + 7\ q^{25630} + 8\ q^{25653} + 4\ q^{25674} + 9\ q^{25707} + 3\ q^{25795} + 5\ q^{25872} + 2\ q^{25928} + 3\ q^{25938} + 4\ q^{25949} + 4\ q^{25950} + 8\ q^{25982} + 6\ q^{26059} + 2\ q^{26082} + 3\ q^{26103} + 6\ q^{26115} + 10\ q^{26136} + 4\ q^{26192} + 6\ q^{26246} + 3\ q^{26334} + 2\ q^{26367} + 8\ q^{26379} + 2\ q^{26390} + 3\ q^{26400} + 6\ q^{26411} + 6\ q^{26544} + 3\ q^{26565} + 6\ q^{26598} + 2\ q^{26642} + 8\ q^{26654} + 2\ q^{26664} + 6\ q^{26829} + 2\ q^{26950} + 4\ q^{27006} + 3\ q^{27027} + 4\ q^{27060} + 4\ q^{27083} + 4\ q^{27104} + 6\ q^{27137} + 6\ q^{27170} + q^{27216} + 2\ q^{27258} + 6\ q^{27270} + 2\ q^{27280} + 3\ q^{27324} + 2\ q^{27335} + 4\ q^{27357} + 2\ q^{27368} + 2\ q^{27380} + 2\ q^{27412} + 8\ q^{27445} + 4\ q^{27489} + 6\ q^{27522} + 4\ q^{27545} + q^{27555} + 2\ q^{27600} + 8\ q^{27632} + 3\ q^{27678} + 7\ q^{27720} + 4\ q^{27732} + 6\ q^{27765} + 4\ q^{27797} + 4\ q^{27830} + 2\ q^{27874} + 2\ q^{27930} + 2\ q^{27951} + 8\ q^{27984} + 2\ q^{28028} + 4\ q^{28040} + 2\ q^{28050} + 8\ q^{28061} + 2\ q^{28084} + q^{28140} + 2\ q^{28161} + q^{28182} + 4\ q^{28194} + 4\ q^{28215} + 2\ q^{28248} + 2\ q^{28259} + 4\ q^{28304} + 4\ q^{28336} + 2\ q^{28392} + 2\ q^{28404} + 4\ q^{28413} + 2\ q^{28446} + 8\ q^{28469} + 2\ q^{28479} + 2\ q^{28490} + 4\ q^{28644} + 4\ q^{28656} + 4\ q^{28754} + 2\ q^{28875} + 6\ q^{28952} + 2\ q^{29008} + 2\ q^{29062} + 2\ q^{29064} + 4\ q^{29085} + 3\ q^{29106} + 4\ q^{29139} + 2\ q^{29162} + 2\ q^{29183} + 4\ q^{29195} + q^{29260} + q^{29295} + 2\ q^{29328} + q^{29337} + 4\ q^{29370} + 2\ q^{29382} + 2\ q^{29414} + 2\ q^{29470} + 2\ q^{29480} + 2\ q^{29547} + 2\ q^{29645} + 2\ q^{29790} + 2\ q^{29832} + 2\ q^{30030} + 4\ q^{30086} + 2\ q^{30184} + 2\ q^{30219} + 2\ q^{30261} + q^{30800} + 3\ q^{30877} + q^{30954} + 2\ q^{31010} + 2\ q^{31064} + 2\ q^{31120} + 2\ q^{31185} + 2\ q^{31197} + 6\ q^{31295} + 3\ q^{31405} + q^{31416} + q^{31440} + 2\ q^{31472} + 4\ q^{31482} + 4\ q^{31570} + 2\ q^{31592} + q^{31605} + 4\ q^{31647} + 2\ q^{31680} + 2\ q^{31725} + 6\ q^{31757} + 4\ q^{31890} + q^{31944} + 2\ q^{32000} + q^{32010} + q^{32032} + 4\ q^{32109} + 4\ q^{32165} + 3\ q^{32175} + 4\ q^{32186} + 2\ q^{32296} + 2\ q^{32319} + q^{32340} + 2\ q^{32352} + 4\ q^{32373} + 4\ q^{32429} + 2\ q^{32450} + 2\ q^{32604} + 2\ q^{32616} + 2\ q^{32648} + 2\ q^{32704} + 2\ q^{32714} + q^{32760} + 2\ q^{32781} + q^{32802} + 2\ q^{32879} + 2\ q^{33024} + 2\ q^{33066} + q^{33110} + 3\ q^{33187} + 2\ q^{33264} + 4\ q^{33320} + 2\ q^{33374} + q^{33418} + 2\ q^{33430} + q^{33453} + 2\ q^{33495} + 2\ q^{33507} + 2\ q^{33605} + q^{33726} + q^{33750} + 2\ q^{33782} + q^{33792} + q^{33880} + q^{33915} + q^{33957}$

Description

The sum of all $1/(i+\pi(i))$ for a permutation $\pi$ times the lcm of all possible values among permutations of the same length.
It turns out that the least common multiple of all values is [1], so this statistic is given by
$$ k \cdot \sum_{i=1}^n \frac{1}{i+\pi(i)}$$
where $\pi$ is a permutation of length $n$ and $k = \operatorname{lcm}\{ 1,\dots,2n\}$.

References

[1] a(n) = lcm1, 2, ..., 2*n. OEIS:A099996

Code

def statistic(pi):
    n = len(pi)
    return sum( 1/(i+pi(i)) for i in [1,n] ) * lcm(i for i in [1 .. 2*n] )

Created

Nov 19, 2018 at 11:44 by Christian Stump

Updated

Nov 19, 2018 at 11:44 by Christian Stump