RISES, LEVELS, DROPS AND + SIGNS IN COMPOSITIONS: EXTENSIONS OF A PAPER BY ALLADI AND HOGGATT S. Heubach Departmet of Mathematics, Califoria State Uiversity Los Ageles 55 State Uiversity Drive, Los Ageles, CA 9003-804 sheubac@calstatela.edu P. Z. Chi Departmet of Mathematics, Humboldt State Uiversity, Arcata, CA 955 phyllis@math.humboldt.edu R. P. Grimaldi Departmet of Mathematics, Rose-Hulma Istitute of Techology Terre Haute, IN 4803-3999 ralph.grimaldi@rose-hulma.edu (Submitted March 00, Revised Jauary 00). INTRODUCTION A compositio of cosists of a ordered sequece of positive itegers whose sum is. A palidromic compositio (or palidrome) is oe for which the sequece reads the same forwards ad backwards. We derive results for the umber of + sigs, summads, levels (a summad followed by itself), rises (a summad followed by a larger oe), ad drops (a summad followed by a smaller oe) for both compositios ad palidromes of. This geeralizes a paper by Alladi ad Hoggatt [], where summads were restricted to be oly s ad s. Some results by Alladi ad Hoggatt ca be geeralized to compositios with summads of all possible sizes, but the coectios with the Fiboacci sequece are specific to compositios with s ad s. However, we will establish a coectio to the Jacobsthal sequece [8], which arises i may cotexts: tiligs of a 3 x board [], meets betwee subsets of a lattice [3], ad alteratig sig matrices [4], to ame just a few. Alladi ad Hoggatt also derived results about the umber of times a
particular summad occurs i all compositios ad palidromes of, respectively. Geeralizatios of these results are give i []. I Sectio we itroduce the otatio that will be used, methods to geerate compositios ad palidromes, as well as some easy results o the total umbers of compositios ad palidromes, the umbers of + sigs ad the umbers of summads for both compositios ad palidromes. We also derive the umber of palidromes ito i parts, which form a elarged Pascal s triagle. Sectio 3 cotais the harder ad more iterestig results o the umbers of levels, rises ad drops for compositios, as well as iterestig coectios betwee these quatities. I Sectio 4 we derive the correspodig results for palidromes. Ulike the case of compositios, we ow have to distiguish betwee odd ad eve. The fial sectio cotais geeratig fuctios for all quatities of iterest.. NOTATION AND GENERAL RESULTS We start with some otatio ad geeral results. Let C, P = the umber of compositios ad palidromes of, respectively C +, P + = the umber of + sigs i all compositios ad palidromes of, respectively C, s P s = the umber of summads i all compositios ad palidromes of, respectively C (x) = the umber of compositios of edig i x C (x, y) = the umber of compositios of edig i x + y r, l, d = the umber of rises, levels, ad drops i all compositios of, respectively r, l, d = the umber of rises, levels, ad drops i all palidromes of, respectively. We ow look at ways of creatig compositios ad palidromes of. Compositios of + ca be created from those of by either appedig + to the right
ed of the compositio or by icreasig the rightmost summad by. This process is reversible ad creates o duplicates, hece creates all compositios of +. To create all palidromes of, combie a middle summad of size m (with the same parity as, 0 m ) with a compositio of m o the left ad its mirror image o the right. Agai, the process is reversible ad creates o duplicates (see Lemma of []). We will refer to these two methods as the Compositio Creatio Method (CCM) ad the Palidrome Creatio Method (PCM), respectively. Figure illustrates the PCM. 6 4 5 3 3 33 33 Figure : Creatig palidromes of =6ad = We ca ow state some basic results for the umber of compositios, palidromes, + sigs ad summads. Theorem. C = for, C 0 :=.. P k = P k+ = k for k 0. 3. C + =( ) for, C + 0 := 0. 4. P + k+ = kk for k 0, P + k =(k )k for k, P + 0 := 0. 5. C s =( +),for, C s 0 :=. 3
6. P s k+ =(k +) k for k 0, P s k =(k +) k for k, P s 0 :=. Proof:. The umber of compositios of ito i parts is ( ) i (see Sectio.4 i [5]). Thus, for, C = ( ) =. i. Usig the PCM as illustrated i Figure, it is easy to see that P k = P k+ = C i =+(++ + k )= k. i=0 3. A compositio of with i summads has i + sigs. Thus, the umber of + sigs ca be obtaied by summig accordig to the umber of summads i the compositio: ( ) C + ( )! = (i ) = (i ) i i= (i )!( i)! ( ) = ( ) =( ). () i= i 4. The umber of + sigs i a palidrome of k+ is twice the umber of + sigs i the associated compositio, plus two + sigs coectig the two compositios with the middle summad. P + k+ = = (C i +C i + )= ( i +(i ) i ) (i +) i = k k, where the last equality is easily proved by iductio. For palidromes of k, thesame reasoig applies, except that there is oly oe + sig whe a compositio of k is combied with its mirror image. Thus, P + k = k (C i +C i + )+(C k +C k + )= (C i +C i + ) C k = k k k =(k ) k. 4
5. & 6. The umber of summads i a compositio or palidrome is oe more tha the umber of + sigs, ad the results follows by substitutig the previous results ito C S = C + + C ad P S = P + + P. Part 4 of Theorem 4 could have bee proved similarly to part, usig the umber of palidromes of ito i parts, deoted by P i. These umbers exhibit a iterestig patter which will be proved i Lemma. 0 0 0 0 3 0 3 0 3 3 3 3.... Figure : Palidromes with i parts Lemma P j j k =0ad Pk = P j k = P j k = ( ) k j for j =,..., k, k. Proof: The first equality follows from the fact that a palidrome of a odd umber has to have a odd umber of summads. For the other cases we will iterpret the palidrome as a tilig where cuts are placed to create the parts. Sice we wat to create a palidrome, we look oly at oe of the two halves of the tilig ad fiish the other half as the mirror image. If =k, to create j parts we select (j ) = j positios out of the possible (k ) = k cuttig positios. 5
If =k, the we eed to distiguish betwee palidromes havig a odd or eve umber of summads. If the umber of summads is j, the there caot be a cut directly i the middle, so oly k = k cuttig positios are available, out of which we select (j ) = j. If the umber of summads is j, the the umber of palidromes correspods to the umber of compositios of k, withhalftheumber of summads (=j), which equals ( k j ) 3. LEVELS, RISES AND DROPS FOR COMPOSITIONS We ow tur our attetio to the harder ad more iterestig results for the umbers of levels, rises ad drops i all compositios of. Theorem 3. l = 36 ((3 +) +8( ) ) for ad l 0 =0.. r = d = 9 ((3 5) ( ) ) for 3 ad r 0 = r = r =0. Proof:. I order to obtai a recursio for the umber of levels i the compositios of, we look at the right ed of the compositios, as this is where the CCM creates chages. Applyig the CCM, the levels i the compositios of + are twice those i the compositios of, modified by ay chages i the umber of levels that occur at the right ed. If a is added, a additioal level is created i all the compositios of that ed i, i.e., a total of C () = C additioal levels. If the rightmost summad is icreased by, oe level is lost if the compositio of eds i x + x, ad oe additioal level is created if the compositio of eds i x +(x ). Thus, l k+ = l k + C k C k (x, x)+ C k (x, x ) x= x= 6
= l k + k C k x + C k (x ) x= x= = l k + k ( k 3 + k 5 + + +)+( k 4 + +) = l k +( k k 3 + k 4 +) = l k + k, 3 while l k = l k + k C k C k (x, x)+ C k (x, x ) x= x= = l k + k 3 ( k 4 + k 6 + + +)+( k 5 + + +) = l k +( k 3 k 4 + k 5 + ) + = l k + k +. 3 Altogether, for all, l =l + +( ) 3. () The homogeeous ad particular solutios, l (h) ad l (p), respectively, are give by l (h) = c ad l (p) = A ( ) + B. Substitutig l (p) ito Eq. () ad comparig the coefficiets for powers of ad -, respectively, yields A = ad B =. Substitutig l 9 = l (h) + l (p) = c + 9 ( ) + ito Eq. () ad usig the iitial coditio l =yieldsc =, givig the 36 equatio for l for 3. (Actually, the formula also holds for ).. It is easy to see that r = d, sice for each opalidromic compositio there is oe which has the summads i reverse order. For palidromic compositios, the symmetry matches each rise i the first half with a drop i the secod half ad vice versa. Sice C + = r + l + d, it follows that r = C+ l.
Table shows values for the quatities of iterest. I Theorem 4 we will establish the patters suggested i this table. 3 4 5 6 8 9 0 C + 0 4 3 80 9 448 04 304 50 64 l 0 6 4 34 8 8 398 88 934 40 r = d 0 0 3 9 3 5 35 33 593 35 Table : Values for C +, l ad r Theorem 4. r + = r + l ad more geerally, r = i= l i for 3.. C + = r + r +. 3. C + =4 (l + l )=4 (r r ). 4. l r = a,wherea is the th term of the Jacobsthal sequece. Proof:. The first equatio follows by substitutig the formulas of Theorem 3 for r ad l ad collectig terms. The geeral formula follows by iductio.. This follows from part, sice C + = r + l + d ad r = d. 3. The first equality follows by substitutig the formula i Theorem 3 for l ad l. The secod equality follows from part. 4. The sequece of values for f = l r is give by,, 3, 5,,, 43,... This sequece satisfies several recurrece relatios, for example f =f +( ) or f = f, both of which ca be verified by substitutig the formulas give i Theorem 3. These recursios defie the Jacobsthal sequece (A00045 i [8]), ad compariso of the iitial values shows that f = a. 8
4. LEVELS, RISES AND DROPS FOR PALINDROMES We ow look at the umbers of levels, rises ad drops for palidromes. Ulike the case for compositios, there is o sigle formula for the umber of levels, rises ad drops, respectively. Here we have to distiguish betwee odd ad eve values of, as well as look at the remaider of k whe divided by 3. Theorem 5 For k,. l k = 9 ( )k + ( k 53 + ) k 6 3 + lk+ = 9 ( )k + ( k + ) k 63 3 +. r k = d k = 9 ( )k ( k 58 ) k 63 3 + 6 k 0 mod (3) k mod (3) 4 k mod (3) 4 k 0 mod (3) 6 k mod (3) k mod (3) r k+ = d k+ = 9 ( )k ( k ) k 63 3 + 3 k 0 mod (3) k mod (3) k mod (3) k 0 mod (3) 3 k mod (3) k mod (3) Proof: We use the PCM, where a middle summad m =l or m =l + (l 0) is combied with a compositio of k l ad its mirror image, to create a palidrome of = k or =k +, respectively. The umber of levels i the palidrome is twice the umber of levels of the compositio, plus ay additioal levels created whe the compositios are joied with the middle summad. We will first look at the case where (ad thus m) iseve. Ifl = m =0, a compositio of k is joied with its mirror image, ad we get oly oe additioal level. If l>0, the we get two additioal levels for a compositio edig i m, for m =l k l. Thus, k/3 lk = l k l + C k + l=0 l= 9 C k l (l) =s + k + s. (3)
Sice l 0 = l = 0, the first summad reduces to s = 8 { (3i +) i +8( ) i} = i + i i + 4 ( ) i i= 9 i= 3 i= 9 i=0 = 9 (k ) + ( ) d x i + 3 dx i= x = 9 (( )k +) = 9 k + { (k +) k k+} + 3 9 ( )k = ( k 9 ( )k + 3 ) k. (4) 9 To compute s,otethatc (i) =C (i ) =... = C i+ () = C i+ = i for i<ad C () =. The latter case oly occurs whe k =3l. Letk := 3j + r, where r =,, 3. (This somewhat ucovetioal defiitio allows for a uified proof.) Thus, with I A deotig the idicator fuctio of A, k/3 j s = C k l (l) = 3j+r l l + I {r=3} l= l= j ( = r ) 3 j l + I{r=3} = r (3 ) j + I {r=3} l= = k r { k +6 k 0mod(3) + I {r=3} = k r k r mod (3), for r =,. Combiig Equatios (3), (4) ad (5) ad simplifyig gives the result for l k. (5) For =k +, we make a similar argumet. Agai, each palidrome has twice the umber of levels of the associated compositio, ad we get two additioal levels wheever the compositio eds i m, form =l + k l. Thus, lk+ = l=0 l k l + (k )/3 l=0 C k l (l +)=:s + s 3. With a argumet similar to that for s,we derive k+ 4 k 0mod(3) s 3 = k+ +6 k mod(3) k+ k mod(3) Combiig Equatios (4) ad (6) ad simplifyig gives the result for l k+. Fially, the results for r ad d follow from the fact that r = d = P + l. (6) 0
5. GENERATING FUNCTIONS Let G a (x) = k=0 a k x k be the geeratig fuctio of the sequece {a } 0.We will give the geeratig fuctios for all the quatities of iterest. Theorem 6. G C (x) = x x ad G P (x) = +x x.. G C + (x) = x ( x) ad G P + (x) = x +x 3 +x 4 ( x ). 3. G C s (x) = 3x+3x ( x) ad G P s (x) = +x x +x 4 ( x ). 4. G l (x) = x ( x) (+x)( x) ad G r (x) =G d (x) = x 3 (+x)( x). 5. G l (x) = x (+3x+4x +x 3 x 4 4x 5 6x 6 ) (+x )(+x+x )( x ) ad G r (x) =G d (x) = x4 (+3x+4x +4x 3 +4x 4 ) (+x )(+x+x )( x ). Proof:. &. The geeratig fuctios for {C } 0, {P } 0 ad {C + } 0 are straightforward usig the defiitio ad the formulas of Theorem. We derive G P + (x), as it eeds to take ito accout the two differet formulas for odd ad eve. From Theorem, we get G P + (x) = = P + k xk + P + k xk k= k= (k ) k x k + (k ) k x k () k= k= Separatig each sum i Eq. () ito terms with ad without a factor of k, ad recombiig like terms across sums leads to G P + (x) = +x 4 = +x 4 4xk(x ) k (x + x ) (x ) k k= k= d ( ) dx x x+x x = x +x 3 +x 4 ( x ). 3. Sice C s = C + C +, G Cs (x) =G C (x)+g C + (x); likewise for G P s (x).
4. The geeratig fuctio for l ca be easily computed usig Mathematica or Maple, usig either the recursive or the explicit descriptio. The relevat Mathematica commads are <<DiscreteMath RSolve GeeratigFuctio[{a[+]==a[]+(/3)*^(-)+(-/3)*(-)^(-), a[0]==0,a[]==0},a[],,z][[,]] PowerSum[((/36) + (/))*^ + (/9)*(-)^,{z,,}] ( Furthermore, G r (x) =G d (x) = GC + (x) G l (x) ),sicer = d = C+ l. 5. I this case we have six differet formulas for l, depedig o the remaider of with respect to 6. Let G i (x) deote the geeratig fuctio of { l6k+i } k=0. The, usig the defiitio of the geeratig fuctio ad separatig the sum accordig to the remaider (similar to the computatio i part ), we get G l (x) =G 0 (x 6 )+x G (x 6 )+x G (x 6 )+ + x 5 G 5 (x 6 ). The fuctios G i (x) ad the resultig geeratig fuctio (x) are derived usig G l the followig Mathematica commads: <<DiscreteMath RSolve g0[z_]=powersum[(/6)((6()+53)* ^(3)+08+8(-)^()),{z,,}] g[z_]=powersum[(/63)((63+)* ^(3)-36+4(-)^),{z,,}] g[z_]=powersum[(/63)((6+95)* ^(3)-8-4(-)^),{z,,0}] g3[z_]=powersum[(/63)((6+86)* ^(3)+54-4(-)^),{z,,0}] g4[z_]=powersum[(/63)((5+4)* ^(3)-36+4(-)^),{z,,0}] g5[z_]=powersum[(/63)((5+56)* ^(3)-8+4(-)^),{z,,0}] gefu[z]:= g0[z^6]+z g[z^6]+z^ g[z^6]+z^3 g3[z^6]+z^4 g4[z^6]+z^5 g5[z^6] Fially, G r (x) =G d (x) = ( GP + (x) G l (x) ),sice r = d = P + l.
ACKNOWLEDGEMENTS The authors would like to thak the aoymous referee for his thorough readig ad for helpful suggestios which have led to a improved paper. Refereces [] K. Alladi & V.E. Hoggatt, Jr. Compositios with Oes ad Twos. Fiboacci Quarterly 3.3 (95): 33-39. [] P. Z. Chi, R. P. Grimaldi & S. Heubach. The Frequecy of Summads of a Particular Size i Palidromic Compositios. To appear i Ars Combiatoria. [3] D. E. Dayki, D. J. Kleitma & D. B. West. Number of Meets betwee two Subsets of a Lattice. Joural of Combiatorial Theory, A6 (99): 35-56. [4] D. D. Frey & J. A. Sellers. Jacobsthal Numbers ad Alteratig Sig Matrices. Joural of Iteger Sequeces, 3 (000): #00..3. [5] R. P. Grimaldi. Discrete ad Combiatorial Mathematics, 4 th Editio. Addiso- Wesley Logma, Ic., 999. [6] R. P. Grimaldi. Compositios with Odd Summads. Cogressus Numeratium 4 (000): 3-. [] S. Heubach. Tilig a m-by- Area with Squares of Size up to k-by-k with m 5. Cogressus Numeratium 40 (999): 43-64. [8] Sloae s Olie Iteger Sequeces. http://www.research.att.com/ jas/sequeces AMS Classificatio Number: 05A99 3