Pascal s Labeling and Path Counting. October 4, Construct Pascal s labeling of lattice points of Euclidean plane.

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1 Pascal s Labelig ad Path Coutig Ko-Wei Lih Istitute of Mathematics Academia Siica Naag, Taipei 5, Taiwa mawlih@siica.edu.tw Daphe Der-Fe Liu Departmet of Mathematics Califoria State Uiversity, Los Ageles Los Ageles, CA 932, U. S. A. dliu@calstatela.edu October 4, 23 The purpose of this expository paper is to achieve the followig objectives i a selfcotaied maer.. Costruct Pascal s labelig of lattice poits of Euclidea plae. 2. Show that labels are equal to biomial coefficiets. 3. Show that biomial coefficiets or their absolute values cout certai lattice paths. 4. Demostrate that Pascal s labelig maes it easier to discover ad prove idetities ivolvig biomial coefficiets. Pascal s labelig We first draw a Cartesia coordiate mesh o the two dimesioal Euclidea plae so that each itersectio poit has iteger coordiates. We call these itersectio poits lattice poits. We wat to label lattice poits i a orderly fashio so that each label is determied by two of its eighborig labels. We proceed with the followig rule. If the label give to a lattice poit with coordiates (, is deoted by [, ], the we stipulate that [, ] [,]+[, ]. ( This idetity says that ay label is equal to the sum of the labels immediately to its left ad to its lower left. I additio to the above rule, we eed some iitial labels to get the process started. Sice ay oe term of ( is completely determied by the other two, it is easy to see that, oce labels o the horizotal ad the upper vertical axes are fixed, labels of all lattice poits ca be computed by meas of (. We could choose ay real umbers as iitial labels. But we are labelig poits with iteger coordiates, it seems atural that we use

2 Figure : Pascal s labelig itegers as iitial labels. The simplest choice would be maig everythig. The we get a whole plae of s which is a bit dull. To mae thigs more iterestig, we label all lattice poits o the horizotal axis by ad all lattice poits o the upper vertical axis by. The outcome of the labelig process is illustrated i Figure. We call it Pascal s labelig of the plae. 2 ENE-paths ad the biomial coefficiet 2. Coutig ENE-paths What are these labels? Do they have special meaigs? Sice they are all oegative umbers i the first quadrat, do they cout some sort of objects? I order to aswer these questios, we first pay our attetio oly to the first quadrat. Let us cosider paths startig at the origi that move accordig to the followig rules. A path ca stay at the origi without goig aywhere. It is a trivial path that starts at the origi ad eds at the origi. For a o-trivial path, there are two ids of admissible steps goig out of ay lattice poit: either it moves to the right eighborig lattice poit (called a E-step or it moves to the orth-east eighborig lattice poit (called a NE-step. Ay path that is determied by these rules is called a ENE-path. A ENE-path from the origi to the poit (6, 3 is illustrated i Figure 2. 2

3 4 3 (6, Figure 2: A ENE-path A ENE-path termiatig at a poit (, comes either from a ENE-path termiatig at the poit (, or from a ENE-path termiatig at the poit (,. So the umber of ENE-paths termiatig at the poit (, satisfies precisely the same idetity (. How may ENE-paths termiate at the origi? Oe, the trivial oe. How may ENE-paths termiate at a poit o the upper vertical axis? Noe, the first step of a ENE-path ca ever go vertical. How may ENE-paths termiate at a poit o the positive horizotal axis? Oe, every step of that path must move horizotally. The umbers of ENE-paths termiatig at the oegative axes are precisely the iitial labels give by Pascal s labelig. With the same iitial umbers ad the same defiig formula, the obvious coclusio is as follows. The label give to a lattice poit (, of the first quadrat i Pascal s labelig couts the umber of ENE-paths termiatig at that poit. I this way, we have give a combiatorial iterpretatio to those labels, well, at least i the first quadrat. Of course, this iterpretatio ca be exteded i a trivial way to the fourth quadrat filled with s because oe of the ENE-paths are goig there. 2.2 Biomial coefficiets as labels Now we have uderstood what those labels really mea. How are we goig to compute them oce a poit (, is give. Of course, we ca start with iitial labels ad wor our way to (, by meas of ( i every step. That surely guaratees us to get the correct label. But it is too cumbersome whe a poit is far away from the origi. Is there a compact formula to compute those labels i a straightforward maer? Let us tae a closer loo ito how a ENE-path is determied. Suppose that a poit (, is give, where both ad are positive itegers. A ENE-path from the origi 3

4 is goig to move steps to reach that poit. We may use a sequece (a,a 2,...,a of legth to record its movemet. Each a i is either or idicatig that the i-thstepisa E-step or a NE-step, respectively. We call such a sequece the sigature sequece of that ENE-path. For example, i Figure 2 the path has sigature sequece (,,,,,. Sice each ENE-path climbs up exactly uits to reach (,, there are exactly s amog the a i s. Coversely, each sequece of s ad s completely determies a ENE-path termiatig at (,. So, if we wat to cout how may ENE-paths termiatig at the poit (,, we might as well cout such sequeces istead. How do we cout sigature sequeces? We start with a sequece of empty positios. We pic a positio to put the first there. There are obviously choices for this first positio. Whe we cotiue to pic aother positio to place the secod, there are oly empty positios left to choose. Totally there are ( ways to place the first two s. If we eep doig this way, the each time we reduce the possible choice for empty positios by oe. The fial cout will be ( ( 2 ( + possible choices for placig s. However, all these possible choices will ot produce distict sequeces. The reaso is because, oce the positios for placig s are fixed, the sequece produced is the same o matter i what order we fill i those s. Sice there are! ( ( 2 2 ways of fixig a orderig of these s, there are oly ( ( 2 ( +/! distict legitimate sigature sequeces. We use a special otatio ( to deote this umber ad coclude that the formula ( ( ( +! computes the umber of ENE-paths reachig the poit (,. So far this otatio ( oly maes sese whe both ad are positive itegers. We ow that there is a trivial ENE-path from the origi to itself. So we mae the covetio ( (. Moreover, we observe that the umerator of eve maes sese whe is ay real umber. We just go ahead to exted the defiitio of ( by the followig formula, i which r deotes a arbitrary real umber. ( r r(r (r +, iteger >;!, ;, iteger <. (2 For istace, whe r 5, 2 ( ( These umbers ( r s are called biomial coefficiets because they appear as coefficiets i the expasio of (x + y r whe r is a oegative iteger or x/y <. If r ad are positive itegers, the formula ( expressed i terms of biomial coefficiets loos lie 4

5 the followig. ( r ( r + ( r. (3 We call this formula the basic recursio. It is called a recursio because it computes the value o the left-had side with the help of earlier values o the right had side, which are supposed to have bee computed i previous stages. Does the basic recursio hold i the other three quadrats of the plae? Let us verify it by maipulatig the defiitio. ( ( r r (r (r (r (r + + +! (! (r (r (r (r + +!! (r (r +(r +! r(r (r +! ( r. The above verificatio wors for 2. The cases for ca be checed without ay difficulty. Now the biomial coefficiets satisfy recursio (3 for all permissible argumets ad they coicides with our labels o the axes, hece they are idetical with our labels o all lattice poits. As a matter of fact, ay poit (r, (ot ecessarily a lattice poit o ay horizotal lie i Figure ca have a label ( r. 3 Symmetries 3. Symmetries i the first quadrat We have succeeded i iterpretig Pascal s labelig as marig poits with biomial coefficiets. However, a casual ispectio of Figure will reveal some regularities ad symmetries. Such pheomea should lead to idetities ivolvig biomial coefficiets. We first observe that all labels alog the mai 45 diagoal lie i the first quadrat are s. This certaily holds i geeral sice a o-trivial ENE-path must tae every step i the ortheast directio to reach a poit o the mai diagoal. This offers a very simple proof for the idetity (, iteger. The mai reaso that we called our labelig of the plae Pascal s labelig is because the part below the mai diagoal is merely a differet depictio of the followig array of 5

6 umbers, commoly ow as Pascal s triagle Pascal s triagle obscures the possibility of extedig biomial coefficiets to egative argumets ad maes it hard to visualize symmetries, except the obvious left-right symmetry with respect to the cetral vertical lie. We should shed the historical habit of exhibitig biomial coefficiets i Pascal s triagle ad start usig Pascal s labelig more extesively. Whe the left-right symmetry of Pascal s triagle is trasplated ito our Pascal s labelig of the plae, it shows that, i the first quadrat, the label foud at uits below the mai diagoal coicides with the label foud at uits above the horizotal axis. We should be able to establish the followig idetity: ( (, iteger, iteger. (4 To prove the above is simple sice we ca switch the s ad s of the sigature sequece of a ENE-path to the poit (, to get the sigature sequece of a ENE-path to the poit (,, ad vise versa. The idetity holds trivially if is egative. The above idetity also expresses a orderly correspodece betwee lies. Whe we fix the oegative ad let vary, all labels ( are arraged alog a horizotal lie through the poit (,. Sice every label ( has a fixed differece betwee the upper ad the lower labels, all these labels are arraged alog a 45 diagoal lie through the poit (,. Therefore idetity (4 matches up the correspodig labels o those two lies. 3.2 WN-paths ad symmetries i the secod quadrat Sice a large portio of biomial coefficiets i the secod quadrat are egative, we may have the impressio that they do ot cout objects. However, if we cosider the absolute values of those biomial coefficiets, we may obtai a appropriate combiatorial iterpretatio. Let us call the secod quadrat with the oegative vertical axis removed the reduced secod quadrat. We see that i this reduced secod quadrat the absolute values of biomial coefficiets form a Pascal s triagle with the tip of the triagle placed at the poit (,. 6

7 4 ( 6, Figure 3: A WN-path How are we goig to give a path-coutig iterpretatio of these absolute values? The Pascal s triagle i the reduced secod quadrat ca be regarded as a fa-out of the Pascal s triagle i the first quadrat situated betwee the mai diagoal lie ad the horizotal axis. If we are prepared to flip the path-coutig apparatus from the first quadrat to the secod quadrat, we have to chage the orietatio of a path properly. It is ot hard to recogize that all NE-steps should correspod to upward goig N-steps ad E-steps should correspod to leftward goig W-steps. More formally, we say that a path is a WN-path if it starts at the poit (, ad each of its movemet falls ito oe of the followig two types: either it moves to the left eighborig lattice poit (called a W-step or it moves to the orth eighborig lattice poit (called a N-step. A WN-path is illustrated i Figure 3. Whe <ad are itegers, we use { } to deote the umber of WN-paths from the poit (, to the poit (,. Clearly, the followig recursio is satisfied. { } { } { } +. (5 Now we cosider the term ( { }.Weseethat { } { } ( +( { } { } ( ( { } (, by (5. This shows that ( { } satisfies the basic recursio (3, too. As to the boudary values, we see that ( { } ( ( { } ( ad (. For itegers <ad, we see that ( { } { } ( (, or equivaletly, (. (6 7

8 The above discussio leads us to the followig coclusio. The absolute value of the label give to a lattice poit (, of the reduced secod quadrat i Pascal s labelig couts the umber of WN-paths termiatig at that poit. The parity of gives the parity of that label. Agai, this coclusio ca be exteded trivially to the third quadrat filled with s because o WN-paths are goig there. Armed with this iterpretatio, we are able to offer path-coutig proofs. For example, ow we ca establish a idetity aalogous to (4. Note that the poits (, ad(, are symmetric about the lie determied by the equatio x + y + whe <ad. By iterchagig W-steps with N-steps, a WN-path to the poit (, is coverted ito a WN-path to the poit (,, ad vice versa. Hece { } { }, iteger <, iteger. (7 Whe we fix a oegative ad let vary, idetity (7 reveals a symmetry betwee a horizotal lie ad a vertical lie both i the secod quadrat. Sice { } ( { ( ad } ( (, we have the followig equivalet form for (7 i terms of biomial coefficiets. ( ( (, iteger <, iteger. (8 3.3 Symmetries betwee the first two quadrats We observe that there are equal umbers of s to the left ad to the right of the poit P (, o the horizotal lie L 2 through the poit (,. The two lattice poits o L at equal distace to the poit P have labels of either idetical or opposite sigs, depedig o the parity of. We may say that this is a siged symmetry. Suppose that the poit to the right of P has coordiates (,. The the symmetric poit will have coordiates (x, such that x, i.e., x. Our observatio leads us to the 2 2 followig idetity. ( ( (, itegers, iteger. (9 Let us prove it. This idetity is trivially true whe <. So assume that. The term o the left-had side couts the umber of ENE-paths to the poit (,. If we chage every NE-step to a step goig vertically, the each such path is coverted ito a path from the origi to the poit (, usig oly E-steps ad N-steps. We flip this path to the secod quadrat ad traslate it horizotally to the left by oe uit. We fially obtai a WN-path to the poit (,. The whole process ca be reversed to get the origial ENE-path bac. Cosequetly, we have the equality ( { }. 8

9 This together with (6 imply (9. By combiig idetities (4 ad (9, we get the followig idetity. ( ( (, iteger, iteger. ( Whe we fix a oegative ad let vary, idetity ( reveals a siged symmetry betwee a horizotal lie i the first quadrat ad a vertical lie i the secod quadrat. Actually, we ca derive more from (9. If we replace r i the defiig formula (2 by avariablex, we get the followig polyomial of degree. ( x x(x (x +, iteger >.! is a polyomial of degree. Accordig to idetity (9, these two Similarly, ( ( x polyomials have idetical values whe x rus through all oegative itegers. It follows that they are idetical polyomials ad idetity (9 ca be geeralized to the followig form for ay real umber r. ( r ( r (, iteger. ( This useful proof techique of extedig idetities from itegers to reals is called the polyomial argumet. 4 Idetities by recursio 4. Alteratig sums Each ENE-path of legth will reach a uique lattice poit o the vertical lie segmet betwee the poit (, ad (,. Therefore the sum ( ( + ( + + represets the total umber of ENE-paths of legth. Sice there are two choices for such a path to cotiue at ay itermediate poit, the total umber should be 2.Cosequetly,we have proved the followig idetity. ( 2, iteger. (2 Although the idex rages over all itegers i the above idetity, there are actually fiitely may ozero summads. If we cosider WN-paths istead, the each such path of legth will reach a uique lattice poit o the 45 lie segmet coectig the two poits (, ad (,. Agai, sice there are two choices for a WN-path to cotiue at ay itermediate poit, the total umber is 2. Cosequetly, we have the followig idetity. { } + 2, iteger. 9

10 I terms of biomial coefficiets, it becomes ( + ( 2, iteger. Of course, this ca also be obtaied from (2 by a applicatio of (9. Now istead of summig up all terms { } + over, we cosider the sum of their correspodig biomial coefficiets ( +. By (9 agai, we are actually looig for the sum ( (. For >, we may use our basic recursio to do cacellatios as follows. ( ( ( + +( ( ( ( ( ( ( ( +( ( ( +( +. However, the sum should be if. If we adopt the covetio that, the the followig idetity is established. ( (, iteger. Exactly the same method used i the above proof ca establish the followig slightly more geeral result. ( ( r r ( (, iteger. Wheever the basic recursio is employed i a derivatio, the proof ca usually be writte as a mathematical iductio. Nevertheless, a proof by iductio is less iformative tha a direct derivatio because the former ofte leaves us woderig how the solutio was obtaied i the first place. 4.2 Sums alog horizotal or diagoal lies Suppose that we start at a poit ( +,m+. Oe applicatio of the basic recursio maes its label ( + m+ equal to the sum of oe label to the left ad oe label to the lower

11 left. If we apply the basic recursio agai to the left label, the the same pheomeo occurs. We ca eep iteratig the basic recursio to compute the sum of labels alog a horizotal lie as follows. ( ( ( + + m + m + m ( ( ( + + m + m m ( ( ( ( m + m m m ( j m + + j (. m Whe adj, we have the followig formula for summatio o the upper idex. ( ( +, itegers m,. (3 m m + O the other had, if we eep iteratig the basic recursio o the lower left labels, we get a sum alog the 45 diagoal lie. This summig process will be widig dow to the lower half plae of s. I the followig formula, there is o eed to put a upper boud o the summig idex. The summatio is actually computed over fiitely may ozero terms. Or equivaletly, i ( i m + i i ( i m i ( +, itegers m,. m + ( + m, itegers m,. This idetity ca be writte i yet aother equivalet form. We replace m i by a ew variable ad reame m as r. The we have the followig idetity which holds for all reals r by the polyomial argumet. ( ( r + r + m +, iteger m. (4 m m 4.3 Fiboacci umbers revealed Now suppose that we are summig up labels from the upper left to the lower right. We choose to restrict the summatio to the first ad the fourth quadrats so that there are oly fiitely may ozero terms o such a ati-diagoal lie. Let F (, iteger.

12 E(r + s, C(r, D(r + s, O A(r, B(r + s, Figure 4: First Vadermode s covolutio It is easy to see that F F. The terms F s satisfy the recursio F F + F 2 for iteger 2 (5 sice F ( + ( ( ( + ( + ( ( F + F 2. ( 2 With the two iitial values ad the recursio (5, the terms F s are precisely the wellow Fiboacci umbers,, 2, 3, 5, 8, 3, 2, 34, 55,... 5 Vadermode s covolutios 5. Basic covolutios I this subsectio, we use the method of coutig ENE-paths to derive two importat idetities commoly ow as Vadermode s covolutios. Give positive itegers r, s, ad, the umber of ENE-paths from the origi to the poit E(r + s, i Figure 4 is equal to ( r+s. Each such ENE-path itersects the thic vertical lie segmet through the poit A(r, at exactly oe poit C(r,, where r. Afterward, the path moves from poit C to poit E i E- or NE-steps. That portio of the path correspods to a uique ENE-path from the origi to the poit with 2

13 E(r + s, m + F (r, m + C(r, m D(r + s, m O A(r, B(r + s, Figure 5: Secod Vadermode s covolutio coordiates (s,. The umber of possible ENE-paths before the itersectio is equal to ( ( r ad the umber of possible cotiuatios is equal to s.cosequetly,wehave the followig first Vadermode s covolutio. ( ( ( r s r + s, iteger. (6 Sice there are oly fiitely may ozero terms i the above sum, the polyomial argumet ca be applied to show that the idetity holds for all reals r ad s. Suppose that i additio to a vertical referece segmet we simultaeously fix a horizotal referece segmet i the above coutig process as illustrated i Figure 5. The same reasoig wors except that the parameter measures the displacemet of the crossig poit F to the horizotal referece segmet istead of the displacemet to the horizotal axis. We thus have the secod form of Vadermode s covolutio, which also holds for all reals r ad s. ( r m + ( s ( r + s, itegers m,. (7 m + The followig idetity is a easy corollary of the above. ( ( ( p s p + s, iteger p, itegers m,. (8 m + + p m + We simply replace ( ( p m+ by p p m, the apply (7. Sice there are oly fiitely may ozero terms to add up, this idetity holds for all reals s by the polyomial argumet. 5.2 Coutig by two stages Vadermode s covolutios were derived by breaig dow the coutig of ENE-paths ito two parts. This method ca be further applied to the followig situatio. Suppose 3

14 (p + q +,m+ + (p +,m+ (p, m (, (p +, (p + q +, Figure 6: Diagram for idetity (9 that m,, p, q are oegative itegers such that q. I Figure 6, we cosider ENEpaths from the origi to the poit (p+q +,m++. Such a path must use a NE-step to cross the bad betwee the horizotal lies through (,mad(,m+. However, this NE-step caot occur to the right of the vertical lie through (p +,, for otherwise such a path ca have at most q NE-steps above that bad, ad hece it fails to reach its destiatio uder the assumptio q. Let that NE-step move from (p, m to (p +,m+,where p. It follows that such a ENE-path is determied by a ENE-path from the origi to (p, m ad a ENE-path from (p +,m+to (p + q +,m+ +. The latter part correspods to a uique ENE-path from (, (p + (p +,m+ (m+ to (p+q+ (p +,m++ (m+ (q+,, ad vice versa. Therefore we have the followig sum of products. p ( ( p q + m ( p + q + m + +, itegers p, m, itegers q. A equivalet form ca be obtaied if we use as the summig idex. ( ( ( p + q p + q + itegers p, m,, m m + + itegers q. p The above method ca be adapted to the reduced secod quadrat. Let p, q < ad m, be itegers. I Figure 7, we cosider all WN-paths to the poit (p + q +,m+ +. For ay such WN-path, there is a uique N-step goig across the bad betwee the horizotal lies through (,ad(,+. Let that N-step move from (q +, to(q +, +, where p + q + q +. The such a WN-path is determied by a WN-path from (, to (q +, ad a WN-path from (q +, + to (p + q +,m+ +. The latter part correspods to a uique WN-path from (, (q+ (q++,+ (+ to (p+q+ (q++,m++ (+ (p, m, ad vice versa. 4 (9

15 (p + q +,m+ + (q +, + (q +, (p + q +, (q, (, Figure 7: Diagram for idetity (2 I terms of biomial coefficiets, we thus have the followig sum of products i the reduced secod quadrat. The mius sig o the right had side comes from the fact ( m ( ( m+ ( m++. p+ q ( ( ( p q + p + q +, m m + + itegers p, q <, itegers m,. The followig equivalet form ca be obtaied if we use as the summig idex. ( ( ( p + q p + q + itegers p, q <,, m m + + itegers m,. q+ p Vadermode s covolutios sum up products of labels o two vertical lies as the idex rus through all itegers. The idetities (9 ad (2 give sums of products of labels o two horizotal lies withi appropriate rages. As a matter of fact, usig idetities (4, (8 ad (9, we ca derive idetities (9 ad (2 from (7. The reader is ecouraged to wor out the details as exercises ad also try his/her sills o the followig idetities of similar type. Exercise. q+ p ( ( p + q m ( p + q +, m + + itegers p<,m, itegers q. (2 Exercise 2. ( ( p + q + m p ( q p + m ( m, m + + itegers p<,m, itegers q. 5

16 5.3 Further sums of products Usig ow symmetries or siged symmetries, it is possible to trasform labels o other types of lies to labels o two vertical lies so that Vadermode s covolutios become applicable. For example, we wat to compute the followig sum of products. S ( p m + ( s + (, iteger p, itegers m,. We first otice that each product is uless m + p. Now suppose that s is a iteger ad s m. It follows that s +. We may first use idetity ( to obtai ( ( s + ( s+. s + Now the origial products ca be trasformed ito products of labels o vertical lies. We will reach the fial aswer by applicatios of idetities (8, (, ad (4. S ( ( p ( s m + s + ( ( ( p s m + s + ( p ( s p m + s ( ( p m s m p + s m ( s m ( p+m. p We have proved the followig idetity for all itegers s m. Hece it holds for all reals s by the polyomial argumet. ( ( ( p s + s m ( ( p+m, iteger p, itegers m,. (2 m + p Our fial example is to compute the followig sum. S 2 ( ( p s (, itegers p, m,. m p We first use idetity ( to obtai ( ( p m ( p m. m p m 6

17 This time Vadermode s covolutio (7 applies. S 2 ( ( m s ( p m p m ( ( s m ( p+m + ( s m ( p+m. p m p m We ca also coclude that the followig idetity holds for all reals s by the polyomial argumet. ( ( ( p s s m ( ( p+m, itegers p, m,. (22 m p m p Remar. Idetities (3, (4, (, (3, (4, ad (6 appear i Table 74 ad idetities (7, (8, (9, (2, ad (22 appear i Table 69 of the boo by Graham, Kuth, ad Patashi []. Refereces [] Roald L. Graham, Doald E. Kuth, ad Ore Patashi, Cocrete Mathematics, A foudatio for computer sciece, Secod editio, Addiso-Wesley, Readig, MA, 994, p