A bijective proof is given for the following theorem: The number of compositions of n into parts congruent to a (mod b) equals the number of compositions of n + b - a into parts congruent to b (mod a) that are greater than or equal to b. The bijection is then shown to preserve palindromicity

Diffenderfer, James D.

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Georgia Southern University: Digital Commons@Georgia Southern

Georgia Southern UniversityDigital Commons@Georgia SouthernMathematical Sciences Faculty Publications Mathematical Sciences, Department of2014Building a Better Bijection between Classes ofCompositionsJames D. DiffenderferGeorgia Southern University, jdiffenderfer@georgiasouthern.eduFollow this and additional works at: https://digitalcommons.georgiasouthern.edu/math-sci-facpubsPart of the Mathematics CommonsThis article is brought to you for free and open access by the Mathematical Sciences, Department of at Digital Commons@Georgia Southern. It hasbeen accepted for inclusion in Mathematical Sciences Faculty Publications by an authorized administrator of Digital Commons@Georgia Southern.For more information, please contact digitalcommons@georgiasouthern.edu.Recommended CitationDiffenderfer, James D.. 2014. "Building a Better Bijection between Classes of Compositions." Integers, 14: A44.https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/293#A44 INTEGERS 14 (2014)BUILDING A BETTER BIJECTION BETWEEN CLASSES OFCOMPOSITIONSJames D. Di↵enderferDept. of Mathematical Sciences, Georgia Southern University, Statesboro, Georgiajdiffenderfer@georgiasouthern.eduReceived: 1/19/14, Accepted: 5/27/14, Published: 8/27/14AbstractA bijective proof is given for the following theorem: The number of compositions ofn into parts congruent to a (mod b) equals the number of compositions of n+ b  ainto parts congruent to b (mod a) that are greater than or equal to b. The bijectionis then shown to preserve palindromicity.1. IntroductionA composition of an integer n is a representation of n as a sum of strictly positiveintegers called parts. A composition, µ, with parts x1, x2, . . . , xm is represented byµ : x1 + x2 + . . .+ xm. The aim of this paper is to provide a bijective proof of thefollowing theorem:Theorem 1.1. Suppose a  b. The number of compositions of n into parts congru-ent to a (mod b) equals the number of compositions of n+b a into parts congruentto b (mod a) where each part is greater than or equal to b.The search for a bijective proof of this theorem was first inspired by Sills [4]who produced a bijective proof of the restricted case when a = 1 and b = 2. Thisbijection was then generalized to establish a bijection for the stronger case whena = 1 and b was any positive integer [1]. While Theorem 1.1 can be easily provenusing generating functions, at the time of [1] a bijective proof seemed elusive. Aswe shall see, this is possibly because the bijection presented in this paper doesnot follow from the previous bijections and is, perhaps, even more elegant in itssimplicity. In the spirit of [1], after establishing the bijective proof of Theorem 1.1we will show that the bijective map preserves palindromicity and thereby prove thatthe two classes are also equal for palindromic compositions.2. Building a Better BijectionBijective Proof of Theorem 1.1. Let µ : x1 + x2 + x3 + · · · + xm be a compositionof n with parts congruent to a (mod b). Since each xi ⌘ a (mod b) let xi = a+ bki,INTEGERS: 14 (2014) 2where ki   0 for 1  i  m. We can now think of the composition asa+ bk1 + a+ bk2 + a+ bk3 + · · ·+ a+ bkm 1 + a+ bkm.Let {si}`i=1 be the subsequence of the sequence of positive integers such that ksi 6= 0,for `  m. Thus, our composition more accurately looks likea+ · · ·+ a| {z }s1+bks1 + a+ · · ·+ a| {z }s2 s1+bks2 + · · ·+ a+ · · ·+ a| {z }s` s` 1+bks` + a+ · · ·+ a| {z }m s`.We can now combine these groups of a’s into one part which gives usas1 + bks1 + a(s2   s1) + bks2 + · · ·+ a(s`   s` 1) + bks` + a(m  s`).Since we are mapping this to a composition of n+ b  a we now place b in the leftmost position and add ( a) to as1 resulting in the compositionb+ a(s1   1) + bks1 + a(s2   s1) + bks2 + · · ·+ a(s`   s` 1) + bks` + a(m  s`).At this point we take one b from each bksi and adjoin it with the multiple of a tothe right to form a new part. Letting y0 = b + a(s1   1), y` = b + a(m   s`), andyi = b+ a(si+1   si) for 1  i < `, we have the compositiony0 + b(ks1   1) + y1 + b(ks2   1) + y2 + · · ·+ b(ks`   1) + y`.We now split the remaining multiples of b up into sums yieldingµ0 : y0 + b+ · · ·+ b| {z }ks1 1+y1 + b+ · · ·+ b| {z }ks2 1+y2 + · · ·+ b+ · · ·+ b| {z }ks` 1+y`. (1.1.1)which is a composition of n+ b  a in which every part is greater than or equal tob and each part is congruent to b (mod a).To map this composition back to the original one, split each yi up into b + ariand combine adjacent b’s into the part bksi , for 0  i  `. Add ( b) to the leftmost b in the composition and then add a to ar0. Finally, separate the multiplesof a into sums of a’s and combine one a that is left adjacent with bksi to form thepart xi = a+ bksi .3. Preserving PalindromicityA palindromic composition of n is a composition in which the parts are ordered suchthat they are read the same forward and backwards. Thus, µ : x1 + x2 + . . .+ xmis a palindromic composition i↵ xm j+1 = xj , for 1  j  m.INTEGERS: 14 (2014) 3Corollary 3.1. The number of palindromic compositions of n into parts congruentto a (mod b) equals the number of palindromic compositions of n+ b  a into partscongruent to b (mod a) that are greater than or equal to b.Proof. Let µ : x1 + x2 + · · · + xm be a palindromic composition of n with partscongruent to a (mod b). In the bijection, we let xi = a+bki. Since µ is palindromic,xm j+1 = xj , a+ bkm j+1 = a+ bkj , km j+1 = kj .Thus, when we first rewrite the composition asa+ bk1 + a+ bk2 + a+ bk3 + · · ·+ a+ bkm 1 + a+ bkmit is not palindromic because of the leftmost a. However, later in the bijection weadd ( a) to this a which would cause this to be a palindromic composition. Saywe have added ( a) to the first a to make the compositionbk1 + a+ bk2 + a+ bk3 + · · ·+ a+ bkm 1 + a+ bkm.Once again, letting {si}`i=1 be the subsequence of integers such that ksi 6= 0, we seethat the previous composition is, more accurately,a+ · · ·+ a| {z }s1 1+bks1 + a+ · · ·+ a| {z }s2 s1+bks2 + · · ·+ a+ · · ·+ a| {z }s` s` 1+bks` + a+ · · ·+ a| {z }m s`.Since we know this to be palindromic, we find that s1   1 = m  s` and s` j+1  s` j = sj+1   sj , for 1  j < `, and that ks` i+1 = ksi , for 1  i  `. This impliesthat y` j+1 = yj in the composition µ0 given in (1.1.1), thereby proving the mappreserves palindromicity.4. ConclusionClearly the bijection presented here is much more simplistic as it requires no outsidemachinery, such as binary representation [3, Sec. IV, Ch. 1, p. 151] which wasemployed in the less general bijections. Corollary 3.1 can also be established usinggenerating functions for palindromic compositions found in [2, p. 350-351].References[1] J. Di↵enderfer, A bijection between two classes of restricted compositions, Fibonacci Quart.50(4) (2012): 360 - 365, 2012.[2] V. E. Hoggatt and M. Bicknell, Palindromic compositions, Fibonacci Quart. 13 (1975): 350-356, 1975.[3] P. A. MacMahon, Combinatory Analysis, vol. 1, Cambridge: at the University Press, 1915.[4] A. Sills, Compositions, partitions and Fibonacci numbers, Fibonacci Quart. 49 (2011): 348-354, 2011.

Building a Better Bijection between Classes of Compositions

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Building a Better Bijection between Classes of Compositions

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