The dinner table problem: the rectangular case

Transcription

1 The ie table poblem: the ectagula case axiv:math/009v [mathco] Jul 00 Itouctio Robeto Tauaso Dipatimeto i Matematica Uivesità i Roma To Vegata 00 Roma, Italy tauaso@matuiomait Decembe, 0 Assume that people ae seate aou a table a we wat to eumeate the umbe of ways that they ca be pemute such that eighbos ae o moe eighbos afte the eaagemet Of couse the aswe epes o the topology of the table: if the table is a cicle the it is easy to chec by a simple compute pogam that the pemutatios that veify this popety ae, if it is a log ba a all the pesos seat alog oe sie the they ae Futhemoe, if it is a ectagula table with two sies the the eaagemets ae 9 The fist two cases ae espectively escibe by sequeces A09 a A00 of the O-Lie Ecyclopeia of Itege Sequeces [], o the othe ha the ectagula case oes ot appea i the liteatue Hee is a vali eaagemet fo = : Fist ie whose associate pemutatio is π = Seco ie I woul lie to wamly tha Alessao Nicolosi a Giogio Mieov fo awig my attetio to this poblem

2 Fo a geeic umbe of pesos the equie popety ca be establishe moe fomally i this way: πi + πi fo i It is iteestig to ote that this eaagemet poblem aou a table has also aothe emaable itepetatio Cosie igs to be place o a boa, oe i each ow a colum, i such a way that they ae o-attacig with espect to these iffeet topologies of the boa: if we eumeate the ways o a tooial boa we fi the sequece A09, fo a egula boa we have A00, a fially if we ivie the boa i the mai fou quaats we ae cosieig the ew sequece ow ecoe as A0 Hee is the igs isplacemet that coespos to the pemutatio π itouce befoe: ¼ ¼ ¼Â¼ à ¼ ¼ ¼ ¼ ¼ ¼ à ¼ ¼Â¼ ¼ ¼ ¼Â¼ ¼ ¼ ¼ ¼ à ¼ à ¼ ¼ ¼ ¼ ¼ ¼ Close fomulas fo the fist two sequeces ae ow: fo A09 it is see [] a,0 = =0 c! + c c c=0 a fo A00 it is see [] a, = =0 c! c c c=0 emembe that = 0 if a = I this pape we popose to stuy the sequece a, efie fo as follows: a, eotes the total umbe of pemutatios π of {,,, } such that πi + πi fo i The table below povies some umeical values:

3 a,0 a, a, a, We will show that the followig fomula hols fo : 0 a, = c c! =0 c =0 =0 c =0 c c i= l i, = l i, c i= l i, = l i, whee = N = { i : i mo }, =, c = c a i J q, L = q, [l,, l c ] = l i J! J J J ={,,c} i J l i q, L I the last sectio, we will pove that the pobability that a pemutatio belogs to the set eumeate by a, tes always to e as goes to ifiity A moe pecise expasio eveals how the limit pobability epes o : a,! = e + + O Asymptotic aalysis: cases = 0 a = Popositio The followig asymptotic expasios hol: a,0 e! O,

4 a a,! e 0 + O, Poof The expasio of a,0 /! is cotaie i [] a it was obtaie fom a ecuece elatio by the metho of uetemie coefficiets Howeve, sice we ae paticulaly iteeste i, we will explicitely eive the coefficiet of / As egas a, /!, we give the etaile poof oly fo the coefficiets of / a / the othes ca be compute i a simila way Sice a,! = =0 c=0 c c! +c c a as goes to ifiity the esultig seies is uifomly coveget with espect to, the we ca cosie each tem oe by oe Moeove, it suffices to aalyse the cases whe c is equal to, a because the atioal fuctio +c / / c Fo c = [ [ ] + [ ] ] + [ ] I a simila way, fo c = + + a fo c = + Hece a,!! =0 = + = +!! Taig the sums we fi that a, e ! e

5 Note that the same stategy ca be applie also to a,0 /! Fo example the coefficiet of / ca be easily calculate: fo c = the fomula gives + + a theefoe a,0!! =0 e + + =! e The fomula fo Theoem Fo a, = 0 =0 c =0 =0 c =0 c c! c c i= l i, = l i, c i= l i, = l i, whee = N = { i : i mo }, =, c = c a i J q, L = q, [l,, l c ] = l i J! J J J ={,,c} i J l i q, L Rema Note that fo = the above fomula coicies with the oe ue to Robbis: a, = c c! q, [l,, l c ], c whee l + +l c= l i =0 c=0 q, [l,, l c ] = = l + +l c= l i l + +l c= l i J={,,c} c l + +l c= l i i J l i J! J c! = c c c!

6 Poof of the Theoem Fo i =,,, let T i, be the set of pemutatios of {,,, } such that i a i + ae -cosecutive: T i, = { π S : π i + π i = } fo i = +,, we cosie T i, as a empty set The by the Iclusio-Exclusio Piciple a, = I T i, I {,,,} assumig the covetio that whe the itesectio is mae ove a empty set of iices the it is the whole set of pemutatios S A -compoet of a set of iices I is a maximal subset of -cosecutive iteges, a we eote by I the umbe of -compoets of I So the above fomula ca be ewitte as a, = =0 c =0 =0 c =0 i I whee N = { i : i mo } a = N T i, i I I I N, I =, I =c Rema I oe to bette illustate the iea of the poof, which is ispie by the oe of Robbis i [], we give a example of how a pemutatio π which belogs to the above itesectio of T i, s ca be selecte Assume that =, =, =, c =, =, c = N a N ae espectively the o umbes a the eve umbes betwee a Now we choose I a I : let l i, be the compoet sizes each compoet fixes actually l i, + umbes a let j i, be the sizes of the gaps betwee cosecutive compoets The the choise of I a I is equivalet to select a itegal solutio of l, =, l i, l, + l, =, l i, j 0, + j, =, j i, 0 j 0, + j, + j, =, j i, 0 Fo example taig l, =, l, =, l, =, j 0, =, j, =, j 0, = 0, j, = a j, = 0, we select the set of iices I = {, } a I = {,, 0} Hee is the coespoig table aagemet: j 0, l, + j, l, + j, l, +

7 Now we eistibute the thee compoets selectig a patitio J J = {,, }, say J = {, } a J = {} This meas that the fist two compoets {} a {, } will go to the o seats a the thi compoet {, 0} will go to the eve seats The we ecie the compoet isplacemets a oietatios: fo example i the o seats we place fist {, } evese a the {} a i the eve seats we place {, 0} evese To etemie the compoet positios we ee the ew gap sizes a theefoe we solve the followig two equatios { j 0, + j, + j, =, j i, 0 j 0, + j, =, j i, 0 If we tae j 0, = 0, j, =, j, = 0, j 0, =, j, = a we fill the empty places with the emaiig umbes,, 9 a, we obtai the followig table eaagemet: j, j 0, j, We go bac to the poof Each I is etemie by a itegal solutio of { l, + l, + + l c, =, l i, j 0, + j, + + j c, = c, j i, 0 whee l,,,l c, ae the sizes of each compoet a j 0,,, j c, ae the sizes of the gaps The coutig of the umbe of itegal solutios of this systems yiels the followig facto i the fomula c c i= l i, = l i, c i= l i, = l i, Now, give I,,I, we select a pemutatio π i I I T i, : we eistibute the c = c compoets selectig a patitio J,,J of the set {,,c} we allow that J ca be empty, fo each set J of the patitio, we etemie the sizes of the ew gaps solvig j 0, + j, + + j J, = i J l i J, j i, 0, we choose the oe of compoets a thei oietatio i J! J ways

8 we fill the empty places with the emaiig umbes i c! ways Taig ito accout all these effects we obtai c! J J ={,,c} i J l i i J l i J! J J Fially, sice c = J, it is easy to get the equie fomula Asymptotic aalysis: the geeal case Theoem Fo 0 a,! = e + + O Poof The cases = 0 a = have bee aleay iscusse, so we assume that By the geeal fomula a,! = polyomial i of egee c polyomial i of egee + c Sice c, the atioal fuctios go to zeo faste tha / as goes to ifiity, uless eithe + c = c o + c = c + I the fist case c = a l i, =, i the seco case thee is oe a oly oe iex such that c + = a l i, = fo some i By simmety, we ca assume that that paticula iex = a multiply the coispoig tem by : a,! that is a,! + / =0 / =0 + +!! / =0 / =0 / =0 / =0! / { }} { q, [,,]+! / / { }} { q, [,,, ], { }} { q, [,, ] / = { }} { q, [,,, ]!! / / + /! / / /

9 We stat cosieig the seco tem Sice { }} { q, [,,, ] =,, / /!! 0 =,, / / / + / + 0 = 0 = =,, the it becomes / =0 / =0 / = /! /! e / = e Now we cosie the fist tem Sice { }} { q, [,, ] /,,! = 0 =,, / / 0 = 0 =,,,,

10 the it becomes / =0 / =0 + /! that is / =0 / =0 + + /! =0 =0 Rimiig that =, it is equivalet to e + / / + / +! =+ a taig the limit of the emaiig sums we obtai e + + that is e + Fially, puttig all togethe we fi that a, e + + e! + / +! /! + = e + Refeeces [] B Aspvall a F M Liag The ie table poblem Techical Repot STAN-CS- 0-9, Compute Sciece Depatmet, Stafo Uivesity, Stafo, Califoia, 90 [] D P Robbis The pobability that eighbos emai eighbos afte aom eaagemets, Ame Math Mothly, 90, [] N J A Sloae The O-Lie Ecyclopeia of Itege Sequeces, o the web at 0