A generalization of Eulerian numbers via rook placements

Transcription

1 A geeralizatio of Euleria umbers via rook placemets Esther Baaia Steve Butler Christopher Cox Jeffrey Davis Jacob Ladgraf Scarlitte Poce Abstract We cosider a geeralizatio of Euleria umbers which cout the umber of placemets of c rooks o a board where there are exactly c rooks i each row ad each colum, ad exactly k rooks below the mai diagoal. The stadard Euleria umbers correspod to the case c =. We show that for ay c the resultig umbers are symmetric ad give geeratig fuctios of these umbers for small values of k. Itroductio Rook placemets o boards have a woderful ad rich history i combiatorics (see, e.g., Butler, Ca, Haglud ad Remmel []). Traditioally the rooks are placed i a o-attackig fashio (i.e., at most oe rook i each row ad colum) ad the combiatorial aspects come from cosiderig variatios o the board shapes. Istead of varyig the board, we could also chage the restrictios o how may rooks are allowed i each row ad each colum. If we have a square board ad the umber of rooks i each colum ad row is fixed, the this correspods to coutig o-egative matrices with fixed row ad colum sums (c.f. A68, A5, A579, etc., i the OEIS [6]). I this paper, we will look at this latter case of placig multiple rooks i each row ad colum more closely. We begi i Sectio by explorig the coectios betwee these rook placemets ad jugglig patters. I Sectio we look at Euleria umbers (which correspod to the umber of o-attackig rook placemets o a College of St. Beedict, Collegeville, MN 56, USA embaaia@csbsju.edu Iowa State Uiversity, Ames, IA 5, USA butler@iastate.edu Caregie Mello Uiversity, Pittsburgh, PA 5, USA cocox@adrew.cmu.edu Uiversity of South Carolia, Columbia, SC 98, USA davisj56@ .sc.edu Michiga State Uiversity, East Lasig, MI 88, USA ladgr@msu.edu Califoria State Uiversity, Moterey Bay, Seaside, CA 9955, USA scpoce@csumb.edu

2 board with a fixed umber of rooks below the mai diagoal) ad i Sectio geeralize to the case i which c rooks are placed i each row ad each colum. I Sectio 5, we provide geeratig fuctios for special cases of these geeralized Euleria umbers. We ed with cocludig remarks ad ope problems i Sectio 6. Miimal jugglig patters ad rook placemets Jugglig patters ca be described by a siteswap sequece listig the throws that the patter requires, i.e., t t... t where at time s i (mod ) we throw the ball so that it will lad t i beats i the future. A sequece of throws ca be juggled if ad oly if there are o collisios, i.e., two balls ladig at the same time, which is equivalet to +t, +t,..., +t beig distict modulo. Oe well kow property of siteswap sequeces is that the average of the throws is the umber of balls eeded to juggle the patter (see [, 5]). A miimal jugglig patter is a valid jugglig patter t t... t with t i. These form the basic buildig blocks of jugglig patters sice all jugglig patters of period arise by startig from some miimal jugglig patter ad addig multiples of to the various throws (such additios do ot affect modular coditios). More about this approach is foud i Buhler, Eisebud, Graham ad Wright []. This aturally leads to the problem of eumeratig miimal jugglig patters. This is doe by relatig such patters to rook placemets o a square board. I particular we will cosider the board B, with labels o each cell (i, j) give by the followig rule: { j i if j i, + j i if j < i. We ca iterpret the rows of B as the throwig times (modulo ) ad the colums of B as the ladig times (modulo ). The label of the cell (i, j) is the the smallest possible throw required to throw at time i ad lad at time j. Give a miimal jugglig patter t t... t we form a rook placemet by placig a rook i row i o the cell labeled t i for i (ote that this forces the rook to be placed i the colum correspodig to the ladig time modulo ). Sice ladig times are uique modulo o two rooks will be i the same colum, so this forms a o-attackig rook placemet with rooks. Coversely, give a oattackig rook placemet with rooks we ca form a miimal jugglig patter by readig off the cell labels of the covered square startig at the first row ad readig dow. This establishes the bijective relatioship betwee miimal jugglig patters ad o-attackig rook patters o B. A example of this is show i Figure for the miimal jugglig patter. We ca extract iformatio about the miimal jugglig patter by properties of the rook placemets, icludig, for example, the umber of balls.

3 Figure : A o-attackig rook placemet o B 5 correspodig to the miimal jugglig patter. Propositio. The umber of rooks below the mai diagoal i a o-attackig rook placemet o B is the same as the umber of balls ecessary to juggle the correspodig miimal jugglig patter. Proof. Suppose there are k rooks below the mai diagoal i a placemet of oattackig rooks o B. The whe we sum the labels of all the cells covered by a rook, i.e. we sum the throw heights for the jugglig sequece, we have t l = k + j i = k. l= j= Sice the average of the throws is the umber of balls eeded for the sequece, the claim follows. Note that i Figure there are three rooks below the mai diagoal ad that the jugglig patter requires three balls to juggle.. Multiplex Jugglig ad c-rook placemets A atural variatio i jugglig is to allow multiple balls to be caught ad throw at a time. This is kow as multiplex jugglig, ad we will see that may of the basic ideas geeralize well to this settig. We will let c deote a had capacity, i.e. at each beat we make c throws (allowig some of the throws to be, which happes whe the umber of actual balls throw is less tha c). Siteswap sequeces of period ow correspod to a sequece of sets, T T... T, where each T i is a (multi-)set of the form {t i,, t i,,..., t i,c }, deoted i shorthad otatio as [t i, t i,... t i,c ]. A multiplex jugglig sequece is valid if ad oly if the jugglig modular coditio is satisfied. Namely, every l appears exactly c times i the multiset i= {t i,j + i (mod )} i. j c I other words, o more tha c balls lad at each time. As i stadard jugglig patters, the umber of balls b eeded to juggle the patter relates to a average. I A throw idicates a ball is ot ladig.

4 particular, i= c t i,j = b. j= We say a multiplex jugglig sequece is a miimal multiplex jugglig sequece if ad oly if t i,j for all throws t i,j. There is a relatioship betwee period, had capacity c multiplex jugglig sequeces ad placemets of rooks o B. This is doe by geeralizig from oattackig rook placemets to c-rook placemets, placemets of c rooks with exactly c rooks i every row ad colum, where multiple rooks are allowed i cells. There is a bijectio betwee miimal multiplex jugglig patters of period with had capacity c ad c-rook placemets o B. I particular, for each i we place c rooks i the i-th row correspodig to t i,,..., t i,c. Coversely, give a c-rook placemet we ca form a miimal multiplex jugglig sequece by lettig T i deote the cells covered by the rooks i row i (with appropriate multiplicity). A example of this is show i Figure for the miimal multiplex jugglig patter [][][][][]. Figure : A -rook placemet o B 5 correspodig to the miimal multiplex jugglig patter [][][][][] By the same argumet used for Propositio we have the followig. Propositio. The umber of rooks below the mai diagoal i a c-rook placemet o B is the same as the umber of balls ecessary to juggle the correspodig miimal multiplex jugglig patter. For example, the multiplex jugglig patter i Figure requires four balls to juggle. Euleria umbers The Euleria umbers, deoted k, are usually defied as the umber of permutatios of [], π = π π... π, with k ascets (π i < π i+ ), or equivaletly the umber of permutatios with k descets (π i > π i+ ). There is a bijectio betwee permutatios of [] with k descets ad permutatios with k drops (i > π i ), so that k also couts permutatios of [] with k drops (see []). Give a board, with rows ad

5 colums labeled,,...,, we ca use our permutatio to form a o-attackig rook placemet by placig rooks at positios (i, π i ). A drop i the permutatio correspods to a rook below the mai diagoal, so we will call ay rook below the mai diagoal a drop. By Propositio, the umber of drops i a o-attackig rook placemet equals the umber of balls ecessary for the correspodig jugglig patter. Therefore, k also couts the umber miimal jugglig patters of period usig k balls. k k= k= k= k= k= k=5 k=6 = = = = = = = Table : The Euleria umbers for 7. The Euleria umbers have may ice properties, some of which ca be see i Table. For example, they are symmetric, i.e. k = k. This ca be show by otig if we start with a permutatio with k ascets ad reverse the permutatio, we ow have k ascets (i.e., ascets go to descets ad vice-versa; ad there are cosecutive pairs). We will give a differet proof of this symmetry i the ext sectio usig rook placemets. Aother well kow property of the Euleria umbers is a recurrece relatio. Propositio. The Euleria umbers satisfy k = ( k) k + (k + ) k. This recurrece is agai prove usig permutatios ad ascets. Here, we provide a alterate proof usig rook placemets ad drops. Proof. Start by cosiderig a o-attackig rook placemet o a ( ) ( ) board with k drops. Add a -th row ad -th colum, ad place a rook i positio (, ). The ewly added rook is ot below the diagoal ad so we have ot created ay ew drops. We ca ow create oe additioal drop by takig ay rook (other tha the oe just added) which is o or above the mai diagoal, say i positio (i, j), move that rook to positio (, j) ad move the rook i positio (, ) to positio (i, ). This moves the rook i the j-th colum below the mai diagoal creatig a ew drop. Sice o other rook moves we ow have precisely k drops ad a o-attackig rook placemet. Note that there are ( ) (k ) = k ways we could have chose which rook to move, so that i total this gives ( k) k boards of size with k drops. 5

6 Now, cosider a o-attackig rook placemet o a ( ) ( ) board with k drops. Add a -th row ad -th colum, ad place a rook i positio (, ). As before we switch, but ow oly switch with a rook which is below the mai diagoal (i.e., a drop). This will ot chage the umber of drops, so the result is a oattackig rook placemet o a board with k drops. There are k rooks we ca choose to switch with, or alteratively, we ca leave the -th rook i positio (, ); thus, there are k + ways to build the desired rook placemet, so that i total this gives (k + ) k boards of size with k drops. Fially, we ote that each board with k drops is formed uiquely from oe of these operatios. This ca be see by takig such a board ad the otig the locatio of the rook(s) i the last row ad i the last colum. Suppose these are i positios (i, ) ad (, j), respectively. We the move these rooks to positios (i, j) ad (, ). This ca at most decrease the umber of drops by oe (i.e, movig the rook i the last colum does ot affect the umber of drops). Now removig the last row ad colum gives a ( ) ( ) board havig a o-attackig rook placemet with either k or k drops. Geeralized Euleria umbers The geeralized Euleria umbers, deoted, are the umber of c-rook placemets k c o the board with k drops. Just as the Euleria umbers cout the umber of miimal jugglig patters of period with k balls, the geeralized Euleria umbers cout the umber of miimal multiplex jugglig patters of period with k balls ad had capacity c. Notice that the geeralized Euleria umbers reduce to the Euleria umbers whe c =. I Table we give some of the geeralized Euleria umbers for c = ad. These umbers appear to satisfy a symmetry property similar to Euleria umbers. We will give two proofs of this symmetry, oe i terms of rook placemets ad the other usig miimal multiplex jugglig patters. Theorem. Let, k ad c be o-egative itegers. The k c = c( ) k Proof. We costruct a bijectio betwee the rook placemets with k rooks below the mai diagoal ad those with c( ) k rooks below the diagoal. Cosider a rook placemet with c rooks i every row ad colum, ad k rooks below the diagoal. Now, shift every rook oe space to the right cyclically. Let us cosider the umber of rooks which are strictly above the mai diagoal. All c rooks i the last colum were shifted to the first colum. So, oe of these rooks are above the mai diagoal. All of the k rooks that were iitially below the mai diagoal are ow either o or still below the mai diagoal. c. 6

7 k k k= k= k= k= k= k=5 k=6 k=7 k=8 = = = = 7 7 = k= k= k= k= k= k=5 k=6 k=7 k=8 k=9 = = = = Table : Small values of the geeralized Euleria umbers for c = ad. All other rooks will be above the diagoal. Sice there are c rooks o the board total, there are c c k = c( ) k rooks above the diagoal after this shift. Fially, we switch the rows ad colums of the board. This flips the rook placemet across the mai diagoal. After this trasformatio, there are ow c( ) k rooks below the mai diagoal. This compositio of trasformatios is ivertible by switchig rows ad colums the shiftig every rook left oe space. Thus, the trasformatio gives a bijectio, completig the proof. Before we ca give the secod proof, we must first establish some basic properties of (multiplex) jugglig sequeces. Lemma 5. If T T... T satisfies the jugglig modular coditios with had capacity c, ad α Z with gcd(α, ) =, the (αt α )(αt α )... (αt α ), where αt i := {αt i,,..., αt i,c }, ad the subscripts are take modulo, also satisfies the jugglig modular coditios. Proof. We have A = {αt iα,j + i} i j c = {α ( t iα,j + iα ) } i j c = {α ( t i,j + i ) } i, j c where we use that gcd(α, ) = so that α is ivertible modulo ad as i rages betwee ad, the so does i := iα. Sice {t i,j + i} i has c occurreces each j c of through the scalig by α ad takig terms modulo we also have that A will have c occurreces each of through. 7

8 Lemma 6. If T T... T satisfies the jugglig modular coditios of had capacity c, ad β Z, the (T + β)(t + β)... (T + β), where T i + β := [t i, + β, t i, + β,..., t i,c + β], still satisfies the jugglig modular coditios. Proof. The multiset A = {(t i,j + β) + i} i j c is foud by takig {t i,j + i} i j c shiftig each elemet by β. Sice T T... T satisfy the jugglig modular coditios the so also must A. Jugglig proof of Theorem. We show there is a bijectio betwee the miimal multiplex jugglig sequeces usig k balls ad those usig c( ) k balls for a fixed legth ad had capacity c. So let T T... T be a valid miimal multiplex jugglig sequece with k balls ad had capacity c. By Lemma 5 ad Lemma 6, if we scale each T i by (reversig the idexig) ad add the the resultig sets still satisfy the modular jugglig coditios. I particular we have that the followig satisfies the modular jugglig coditio: ( T )( T )... ( T ). We also ote the resultig throws all lie betwee ad so that this is ideed a miimal jugglig patter. The umber of balls i the ew jugglig sequece is i= c j= ( ( ) ( ) ti,j = c( ) i= c ) t i,j = c( ) k. Fially, we ote that this operatio is its ow iverse, ad thus gives the desired bijectio. j= ad 5 Geeralized Euleria umbers for small k We ow look at determiig the values of the geeralized Euleria umbers for k c small k. This depeds of course o both ad c. However, for a fixed k there are oly fiitely may c that eed to be cosidered. This is a cosequece of the followig lemma. Lemma 7. For c k we have k c = k Proof. It will suffice to establish the followig claim. k. Claim. Every c-rook placemet with k drops has at least c k rooks i every etry o the mai diagoal. 8

9 We proceed to establish this by usig iductio o k + c. For k + c =, the oly possible case is k = ad c = for which there is oly oe placemet, amely oe rook i each cell o the mai diagoal. Now assume that we have established the claim for all k, c with k + c < l, ad let k + c = l. Let S be a c-rook placemet with k drops. We ca iterpret the rook placemet as a icidece relatioship of a regular bipartite graph. By Hall s Marriage Theorem, we kow we ca fid a perfect matchig i this bipartite graph which correspods to T, a -rook placemet cotaied i S. Suppose there are i drops i T. The, S T is a (c )-rook placemet with k i drops. Sice (c )+(k i) < c+k = l, by our iductio hypothesis, there are at least c k +i rooks o each etry o the mai diagoal i S T, ad hece also i S. If i, we are doe. If i =, the T is agai the uique -rook placemet where every rook is o the mai diagoal, so S still has at least c k rooks o each etry o the mai diagoal. This ca also be established i terms of miimal multiplex jugglig patters. Jugglig proof of Lemma 7. If there are k balls, the at each step we ca throw at most k balls, i.e., each T i has at least c k etries of. It follows that i the correspodig c-rook placemet each row has at least c k rooks o the diagoal. We will be lookig at the geeralized Euleria umbers for k =,,. By k c Lemma 7 this reduces dow to oly six cases to cosider, amely,,,,, ad. Sice k = k, the, ad have bee previously determied (see A95, A6 ad A98, respectively, i the OEIS [6]). So that leaves, ad ad i Table we give the geeratig fuctio for these three sequeces. I the remaider of this sectio we will demostrate the techiques used to determie the geeratig fuctios by workig through the case for 5. Placig rooks i a geeric rook placemet We break the problem of coutig c-rook placemets ito several sub-problems accordig to the way the rooks below the mai diagoal are placed relative to oe aother (i.e., relative placemets istead of absolute placemets). Give some geeric placemet of the k rooks below the mai diagoal we ca determie the umber of ways to place the remaiig rooks o or above the mai diagoal. We the combie the results over all possible geeric placemets. We will carefully work through the rook placemet show i Figure which cosists of two rooks below the mai diagoal ad where both rooks are i the same colum ad differet rows. Here a, b, c ad d are the umber of rows betwee the various trasitio poits (a trasitio poit to passig a rook, or rooks, i a row or a colum as we move alog the mai diagoal).. 9

10 x = x + x + 7x + 67x 5 + 6x x x 8 + = x x x x 5 + 5x 6 ( x) ( x) ( 5x + 5x ) x = x + x + 9x x x x 8 + = x + x x x 6 676x x 8 668x 9 +97x 6x + 65x ( x) ( x) ( 5x + 5x ) ( 8x + x ) x = x + x + 5x + 68x x x 7 + = x 7x + 9x 6x 5 + 8x 6 555x x 8 x x 558x + 5x 7x ( x) ( x) ( 5x + 5x ) ( x + 7x x ) Table : Geeratig fuctios for some of the geeralized Euleria umbers. We place the remaiig rooks oe row at a time startig from the bottom ad goig to the top. For each ew row, the way we place rooks will deped o all of the choices we have made previously. However, it suffices to kow oly what is happeig locally. I particular, we oly eed to kow how may colums ca have rooks placed ito them, as well as the respective umbers that ca go ito those colums. We ca represet these by a partitio of what we will call the excess (the total umber of rooks that ca still be placed i the colums after the row has had its rooks placed). As we move oe row up the board we will gai a ew colum (from the diagoal) ad the excess will chage i oe of several ways. There are o rooks below or to the left of the ew diagoal cell. Iitially we ow have a ew colum that ca take up to c rooks, ad we place c rooks i the row. The excess remais uchaged. There are τ rooks below the ew diagoal cell. Iitially we have the ew colum, but that ca oly take up to c τ rooks (i.e., τ rooks have already goe ito the colum), ad we still have to place c rooks i the row. The excess decreases by τ. There are σ rooks to the left of the ew diagoal cell. Iitially we have the ew colum that ca take up to c rooks, ad we place c σ rooks i the row (i.e.,

11 d c b a Figure : A -rook placemet with two rooks below the mai diagoal where both rooks are i the same colum ad differet rows. σ rooks have already goe ito the row). The excess icreases by σ. We ote that it is possible for the last two situatios to occur simultaeously. I goig from row to row we will trasitio from partitios of the old excess to partitios of the ew excess. We illustrate this with a example i which case the excesses are both. We idicate a colum which ca still have r rooks placed ito it by r, the udereath look at all possible ways we ca place rooks ito those colums, ad fially ote the resultig set of colums cotributig to the ew excess. This ca be modeled by a trasitio matrix where the colums of the trasitio matrix correspod to the excess of the origial row ad the rows of the trasitio matrix correspod to the partitios of the excess of the ew row. ( ). Repeatig this for all possible situatios that might arise for trasitioig betwee excesses,, or, we get the trasitio matrices i the followig table.

12 Trasitio from Trasitio to ( ) () () ( ) () () ( ) ( ) ( ) We ow start below the bottom row (i possible way) ad we move up from row to row ad multiply o the left by the trasitio that we perform betwee the two rows. At ay poit we stop, the resultig vector will deote the umber of ways to fill up the board to that row such with a particular excess. I particular, if we carry this procedure all the way to the top we will get a matrix whose etry is the umber of ways to fill i the rooks o ad above the mai diagoal. For Figure, where we have of rus of a, b, c ad d rows as well as three other trasitios to make, the resultig product that gives our cout is as follows () ( d ) ( ) c ( ) () b ()() a. Fially, for this geeric rook placemet we sum over all possible choices of a, b, c ad d that gives a board, i.e., () ( d ) ( ) c ( ) () b ()() a. a+b+c+d= I order to help evaluate this sum, we will add i a extra parameter x that keeps track of how may of each trasitio we made, or viewed aother way the power of x correspods to the umber of rows we have. Therefore whe coutig the umber of placemets o a board, we are iterested i the coefficiet of x of the expressio (x) d x ( ) ( ) c ( ) x x x (x) b x(x) a. x x a,b,c,d This sum ca be decomposed as a combiatio of geometric sums givig (x) d x ( ) ( ) c ( ) x x x (x) b x(x) a x x a,b,c,d ( ) ( ) ( ( ) c ) ( ) ( x x = x x d x x d c b (x) b)( ) x a a = x x ( ) ( ( ) ) ( ) x x I x x x x

13 x = ( x) ( x) ( ) ( ( ) ) ( ) x x 5x + 5x x x x ( x) = ( x) ( x)( 5x + 5x ). This is the geeratig fuctio for oe of the geeric ways to place rooks. We ca ow repeat this procedure for every way i which we ca place rooks below the mai diagoal ad add the idividual geeratig fuctios together. All the seve geeric cases, with their correspodig geeratig fuctios, are show i Figure. Addig the idividual geeratig fuctios together the gives us the overall geeratig fuctio that was give i Table. This same process works for determiig the geeratig fuctio of for ay k c fixed k ad c. The mai challege lies i that the umber of geeric cases that have to be cosidered grows drastically as we icrease c ad k. This is demostrated i Table. It is possible to automate this process, which was used for determiig the geeratig fuctios for k = give i Table. c= c= c= c= c=5 c=6 c=7 k= k= 7 k= k= k= k= k= Table : The umber of geeric c-rook placemets with k rooks below the mai diagoal. 6 Coclusio The geeralized Euleria umbers are a atural extesio of the Euleria umbers, at least i regards to the iterpretatio comig from rook placemets. We have also see that these umbers exhibit a symmetry similar to that of the Euleria umbers. It would be iterestig to kow which other properties ad relatioships ivolvig Euleria umbers geeralize. Some atural cadidates to try ad geeralize iclude the followig. Is there a geeralizatio of the recurrece i Propositio for Euleria umbers to geeralized Euleria umbers? Related to this, is there a simple geeratig fuctio for the geeralized Euleria umbers?

14 x x ( x) ( x) ( x) ( x) ( 5x + 5x ) x ( x) ( x) x (5 7x) ( x) ( x) ( 5x + 5x ) x (5 7x) ( x) ( x) ( 5x + 5x ) x ( x) ( x) ( x)( 5x + 5x ) x ( x) ( x) ( x)( 5x + 5x ) Figure : All geeric -rook placemets ad correspodig geeratig fuctios. Is there a geeralizatio of Worpitzky s idetity, x = ( x+k ) k k, to geeralized Euleria umbers? Worpitzky s idetity is used i coutig the umber of jugglig patters (see []), so a geeralizatio might be useful i coutig multiplex jugglig patters.

15 Is there a geeralizatio of the idetity of Chug, Graham ad Kuth [], ( a+b ) k ( k a = a+b ) k k b (This uses the covetio =.) k More iformatio about the Euleria umbers ad various idetities ad relatioships that could be cosidered are give i Graham, Kuth, Patashik [, Sectio 6.]. We also ote the origial motivatio for ivestigatig these umbers was lookig ito the mathematics of multiplex jugglig. There is a close coectio betwee the mathematics of jugglig ad the mathematics of rook placemets. We hope to see this relatioship stregtheed i future work. k Refereces [] Fred Butler, Mahir Ca, Him Haglud, ad Jeffrey Remmel, Rook Theory Notes, mauscript. Available olie at [] Fa Chug, Ro Graham, ad Doald Kuth, A symmetrical Euleria idetity, Joural of Combiatorics 7 (), 9 8. [] Ro Graham, Doald Kuth, ad Ore Patashik, Cocrete Mathematics: A Foudatio for Computer Sciece (d ed.), Addiso-Wesley Logma Publishig Co., Ic., Bosto, 99. [] Joe Buhler, David Eisebud, Ro Graham, ad Coli Wright, Jugglig Drops ad Descets, America Math Mothly (99) [5] Burkhard Polster, The Mathematics of Jugglig, Spriger, New York,. [6] Neil Sloae, The O-Lie Ecyclopedia of Iteger Sequeces. Available olie at oeis.org. 5