THE DINNER TABLE PROBLEM: THE RECTANGULAR CASE. Roberto Tauraso Dipartimento di Matematica, Università di Roma Tor Vergata, Roma, Italy

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1 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 006, #A THE DINNER TABLE PROBLEM: THE RECTANGULAR CASE Robeto Tauaso Dipatimeto di Matematia, Uivesità di Roma To Vegata, 00 Roma, Italy tauaso@matuiomait Reeived: 9//05, Revised: 4/5/06, Aepted: 4/7/06, Published: 4//06 Abstat Coside people who ae seated adomly at a etagula table with / ad / seats alog the two opposite sides, fo two dies What is the pobability that eighbos at the fist die ae o loge eighbos at the seod oe? We give a expliit fomula ad show that its asymptoti behavio as goes to ifiity is e + 4/ it is ow that it is e 4/ fo a oud table A moe geeal pemutatio poblem is also osideed Itodutio Assume that 8 people ae seated aoud a table ad we wat to eumeate the umbe of ways that they a be pemuted suh that eighbos ae o loge eighbos afte the eaagemet Of ouse the aswe depeds o the topology of the table: if the table is a ile, the it is easy to he by a simple ompute pogam that the umbe of pemutatios that satisfy this popety ae 8 If it is a log ba ad all people sit alog oe side, the thee ae 54 Futhemoe, if it is a etagula table with two sides the the eaagemets umbe 95 The fist two ases ae espetively desibed by sequees A089 ad A00464 of the O-Lie Eylopedia of Itege Sequees [8] O the othe had, the etagula ase does ot appea i the liteatue ad eetly the oespodig sequee has bee labeled as A08 Hee is a valid eaagemet fo 8: Fist die Seod die

2 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 006, #A whose assoiated pemutatio is π Fo a geei umbe of pesos the equied popety a be established moe fomally i this way: πi + πi fo i It is iteestig to ote that this eaagemet poblem aoud a table also has aothe emaable itepetatio Coside igs to be plaed o a boad, oe i eah ow ad olum, i suh a way that they ae o-attaig with espet to these diffeet topologies of the boad: if we eumeate ¼ ¼ ¼Â¼ the ways o a tooidal boad we fid the sequee A089, fo a egula boad we have A00464, ad fially if we divide the boad i the à ¼ ¼ ¼ mai fou quadats we ae osideig the ew sequee Hee is the 8 igs displaemet that oespods to the pemutatio ¼ ¼ ¼ à π itodued befoe: ¼ ¼Â¼ ¼ 8 ¼ ¼Â¼ ¼ 6 ¼ ¼ ¼ à 4 ¼ à ¼ ¼ ¼ ¼ ¼ ¼ Closed fomulas fo the fist two sequees ae ow: fo A089 it is see [] a,0! ad fo A00464 it is see [],[4],[5],[6],[7] a,! whee 0 if ad + I this pape we study the sequee a,d defied fo d as follows: a,d deotes the total umbe of pemutatios π of {,,, } suh that πi + d πi d fo i d

3 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 006, #A Note that if a pemutatio π has this popety the π also has the same popety The table below povides some umeial values: a,0 a, a, a, We will show that the followig fomula holds fo d : d d d d a,d! 0 d 0 i,, d i,d d,d q,d L whee N { i : i mod d}, is the miimum of ad, d, d, L [l,,, l d,d] o [l,, l ] afte eidexig, ad i J q,d L J! J J Jd {,,} i J Eve if the above fomula seems vey ompliated, it is quite maageable to attempt a asymptoti aalysis I the last setio, we pove that the pobability that a pemutatio belogs to the set eumeated by a,d always teds to e as goes to ifiity A moe peise expasio will eveal how the limitig pobability depeds o d: a,d! e 4d + + O I would lie to wamly tha Alessado Niolosi ad Giogio Mieov fo dawig my attetio to this poblem

4 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 006, #A 4 Asymptoti Aalysis: Cases d 0 ad d Popositio The followig asymptoti expasios hold: a,0 e 4! O 5 6, ad a,! e O 5 6 Poof The expasio of a,0 /! is otaied i [] ad it was obtaied fom a euee elatio by the method of udetemied oeffiiets With egad to a, /!, we give the detailed poof oly fo the oeffiiets of / ad / the othes a be omputed i a simila way Sie a,!!! ad, by the Chu-Vademode idetity see, fo example, p69 i [], 0!!!,!! the alteatig sum of a, /! is domiated fo ay by the oveget seies + /! e Theefoe, by uifom ovegee, we a study the asymptotis of a, /! tem by tem Moeove!!!! + whee s s + is the fallig fatoial, ad it suffies to aalyze the ases whe is equal to, ad beause fo + the atioal futio + / / Sie the fallig fatoial s is the geeatig futio fo the Stilig umbes of the fist id [ s ] see, fo example, p49 i [] s [ ] s s, 0 fo we obtai [ [ ] + [ ] ] + [ ]

5 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 006, #A 5 I a simila way, fo, + +, ad fo, + Hee, a,! Sie, fo s 0,! + + +!!! s s s! s e, taig the sums we fid that a, e ! e Note that the same stategy a also be applied to a,0 /! Fo example, the oeffiiet of / a be easily alulated i this way: fo the fomula gives + + ad theefoe a,0!! e + e 4!

6 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 006, #A 6 The Fomula fo d Theoem Fo d d a,d 0 d d d d 0! whee N { i : i mod d}, [l,, l ] afte eidexig, ad q,d L J Jd {,,} i J d, i,, d i,d d,d q,d L d, L [l,,, l d,d] o i J J! J Rema Note that fo d the above fomula oiides with the oe we gave i the itodutio: a,! q, [l,, l ] beause l + +l l + +l!!! q, [l,, l ] l + +l l + +l J{,,} i J l i J! J! It is iteestig to ote that the geeatig futio of the sequee a,,! fx + s0 s x x s!, + x

7 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 006, #A 7 whih is due to to L Calitz see [6], a be easily poved by a vey simila agumet: fx + s0 + s0 s!x s + x x s! x s s s Hee, fo, the oeffiiet of x is equal to [x ]fx s + x l l!! l + +l l + +l a, Poof of Theoem st pat Fo i,, d, let T i,d be the set of pemutatios of {,,, } suh that i ad i+d ae d-oseutive, that is, the distae betwee i ad i+d i the list π, π,, π is equal to d: T i,d { π S : π i + d π i d } fo i d+,, we oside T i,d as a empty set The, by the Ilusio-Exlusio Piiple, a,d I {,,,} I T i,d, i I assumig the ovetio that whe the itesetio is made ove a empty set of idies the it is the whole set of pemutatios S A d-ompoet of a set of idies I is a maximal subset of d-oseutive iteges, ad we deote by I the umbe of d-ompoets of I So the above fomula a be ewitte as a,d d T i,d i I I d I N, I d 0 d d d d 0 T i,d i I I d I N, I, I whee N { i : i mod d} ad N Rema I ode to bette illustate the idea of the poof, whih is ispied by the oe of Robbis i [7], we give a example of how a pemutatio π that belogs to the above

8 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 006, #A 8 itesetio of T i,d s a be seleted Assume that, d,,,, N ad N ae espetively the odd umbes ad the eve umbes betwee ad Now we hoose I ad I : let, be the size of the ith-ompoet i N eah ompoet fixes, + umbes ad let j i, be the size of the gap betwee the ith-ompoet ad the i+st-ompoet i N The the hoie of I ad I is equivalet to seletig a itegal solutio of l,,, l, + l,,, 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, ad j, 0, we selet the set of idies I {, 5} ad I {, 8, 0} Hee is the oespodig table aagemet: j 0, l, + j, l, + j, l, + Now we edistibute the thee ompoets seletig a patitio J J {,, }, say J {, } ad J {} This meas that the fist two ompoets {} ad {, 5} will go to the odd seats ad the thid ompoet {8, 0} will go to the eve seats The we deide the ompoet displaemets ad oietatios: fo example, i the odd seats we plae fist {, 5} evesed ad the {}, ad i the eve seats we plae {8, 0} evesed To detemie the ompoet positios we eed the ew gap sizes, ad theefoe we solve the two equatios { j 0, + j, + j, l, l, J, j i, 0 j 0, + j, l, J, j i, 0 whee j i, is the size of the ew gap betwee the ith-ompoet ad the i+st-ompoet i N If we tae j 0, 0, j,, j, 0, j 0,, j, ad we fill the empty plaes with the emaiig umbes, 6, 9 ad, we obtai the followig table eaagemet:

9 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 006, #A 9 j, j 0, j, Poof of Theoem d pat Followig the otatios itodued i the pevious ema, eah I, fo d, is detemied by a itegal solutio of { l, + l, + + l,,, j 0, + j, + + j,, j i, 0 whee l,,, l, ae the ompoet sizes ad j 0,,,j, ae the gap sizes i N The outig of the itegal solutios of these d systems yields the followig fato i the fomula i,, d i,d d Now, give I,,I d, we selet a pemutatio π i I I d T i,d followig these steps:,d We edistibute the d ompoets seletig a patitio J,,J d of the set of idies {,,} we allow J to be empty Fo eah set J of the patitio, we detemie the sizes of the ew gaps solvig j 0, + j, + + j J, i J J, j i, 0 This a be doe i i J J ways Fo eah set J of the patitio, we hoose the ode of the oespodig J ompoets ad thei oietatio i J! J ways We fill the empty plaes with the emaiig umbes i! ways Taig ito aout all these effets, we obtai i J! J! J J J Jd {,,} i J Fially, sie d J, the d J ad we get the desied fomula

10 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 006, #A 0 4 Asymptoti Aalysis: The Geeal Case Theoem Fo d 0 a,d! e 4d + + O Poof The ases d 0 ad d have bee aleady disussed, so we assume that d Fom the begiig of the poof of Theoem we have a,d! d! T i,d i I I d I N, I Moeove, sie eah I a be seleted i ways, eah idex i I a be odeed i ways ad the emaiig umbes a be aaged i! ways, we have 0! T i,d i I I d!! I N, I,, d,, d d!! This meas that the alteatig sum of a,d /! is domiated, fo ay, by the oveget seies + +! ed Theefoe, by uifom ovegee, we a study the asymptotis of a,d /! tem by tem By Theoem, ad sie d d J, eah tem has the followig fom:!! i J J! ost J + ad theefoe goes to zeo faste tha / as goes to ifiity, uless eithe + o + I the fist ase, fo ay,,d emembe that ad all ompoet sizes ae equal to I the seod ase, the same situatio holds with two exeptios: 0 0 fo some idex 0 ad oe of the ompoets i N 0 has size equal to By symmety, we a assume that this patiula idex 0 is equal to d ad multiply

11 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 006, #A the oespodig tem by d: that is a,d! a,d! +d /d /d +d +!! /d /d /d /d d 0! /d { }} { q,d [,,]+! /d /d d { }} { d q,d [,,, ], d { }} { q,d [,,] /d d { }} { q,d [,,, ]! d We stat by osideig the seod tem Sie { }} { q,d [,,, ] d d,, d d! /d /d + /d d d /d d! /d /dd /d d /d! d d d +! 0 d d,, d d /d /d /d d + /d d + 0 d d d d d,,, d d 0 the seod tem is /d /d d 0 /d d d /d! /d d d! e /d d e

12 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 006, #A Now we oside the fist tem Sie { }} { q,d [,, ] d /d,, d! 0 d,, d /d /d 0 d d 0 d d,, d,, d d d d d + d 0 d d d d + dd + d, d the fist tem is /d /d + d /d! d ; d that is, /d /d + + d Reallig that d, we have + d + + /d! d + d + d d + d d + Moeove + d + d d d d

13 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 006, #A Theefoe, the fist tem is equivalet to e + + d /d d Taig the sums we obtai e + d /d d /d +! d d d + d + d d /d +! /d! d + d that is, e + 4d 6 Fially, puttig eveythig togethe we fid that a,d e + 4d 6 + e! e + d dd ; 4d Refeees [] M Abamso ad W Mose, Combiatios, suessios, ad the -igs poblem, Math Magazie, 9 966, 69 7 [] B Aspvall ad F M Liag The die table poblem Tehial Repot STAN-CS-80-89, Compute Siee Depatmet, Stafod Uivesity, Stafod, Califoia, 980 [] R L Gaham, D E Kuth ad O Patashi, Coete Mathematis, Addiso-Wesley, 990 [4] I Kaplasy, Symboli solutio of etai poblems i pemutatios, Bull Ame Math So, , [5] I Kaplasy, The asymptoti distibutio of us of oseutive elemets, A Math Statist, 6 945, 00 0 [6] J Rioda A euee fo pemutatios without isig o fallig suessios, A Math Statist, 6 965, [7] D P Robbis The pobability that eighbos emai eighbos afte adom eaagemets, Ame Math Mothly, , 4 [8] N J A Sloae The O-Lie Eylopedia of Itege Sequees, o the web at