#A52 INTEGERS 10 (2010), COMBINATORIAL INTERPRETATIONS OF BINOMIAL COEFFICIENT ANALOGUES RELATED TO LUCAS SEQUENCES

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1 #A5 INTEGERS 10 (010), COMBINATORIAL INTERPRETATIONS OF BINOMIAL COEFFICIENT ANALOGUES RELATED TO LUCAS SEQUENCES Bruce E Sagan 1 Departent of Matheatics, Michigan State University, East Lansing, MI sagan@athsuedu Carla D Savage Departent of Coputer Science, North Carolina State University, Raleigh, NC savage@cayleycscncsuedu Received: 6/6/10, Accepted: 8/1/10, Published: 11/15/10 Abstract Let s and t be variables Define polynoials {n} in s, t by {0} = 0, {1} = 1, and {n} = s {n 1} + t {n } for n If s, t are integers then the corresponding sequence of integers is called a Lucas sequence Define an analogue of the binoial coefficients by { } n = {n}! {}! {n }! where {n}! = {1} {} {n} It is easy to see that { n } is a polynoial in s and t The purpose of this note is to give two cobinatorial interpretations for this polynoial in ters of statistics on integer partitions inside a (n ) rectangle When s = t = 1 we obtain cobinatorial interpretations of the fibonoial coefficients which are sipler than any that have previously appeared in the literature 1 Introduction Given variables s, t we define the corresponding sequence of Lucas polynoials, {n}, by {0} = 0, {1} = 1, and for n : {n} = s {n 1} + t {n } (1) When s, t are integers, the corresponding integer sequence is called a Lucas sequence [9, 10, 11] These sequences have any interesting nuber-theoretic and cobinatorial properties Define the lucanoials for 0 n by { n } = {n}! {}! {n }! 1 Wor partially done while a Progra Officer at NSF The views expressed are not necessarily those of the NSF Partially supported by NSA grant H ()

2 INTEGERS: 10 (010) 698 where {n}! = {1} {} {n} It is not hard to show that { n } is a polynoial in s and t (This follows fro Proposition below) The purpose of this note is to give two siple cobinatorial interpretations of the lucanoials They are based on statistics associated with integer partitions λ inside a (n ) rectangle More specifically, we will show that { n } is the generating function for certain tilings of such λ and their copleents with doinos and onoinos Various specializations of the paraeters s and t are of interest When s = t = 1, {n} becoes the nth Fibonacci nuber, and { n } is nown as a fibonoial coefficient Gessel and Viennot [7] gave an interpretation of the fibonoials in ters of nonintersecting lattice paths and ased for a sipler one Benjain and Plott [] gave another interpretation in ters of tilings, but it is not as straightforward as ours When s = l and t = 1 we get two new interpretations of the l-noial coefficients of Loehr and Savage [8] This case is of interest because of its connection with the Lecture Hall Partition Theore introduced by Bousquet-Mélou and (Kio) Erisson [4, 5, 6] Finally, letting s = q + 1 and t = q one gets new interpretations for the classical q-binoial coefficients For ore inforation about these iportant polynoials, see the text of Andrews [1] We should ention that it is possible to derive our results fro q-binoial coefficient identities using algebraic anipulations, analogous to what is done for l-noial coefficients in Section 33 of [8] However, we wish to deonstrate how these results follow fro siple, cobinatorial arguents Recursions In this section we will present the recurrence relations we will need for our cobinatorial interpretations To obtain these results, we will use a tiling odel that is often useful when dealing with Lucas sequences See the boo of Benjain and Quinn [3] for ore details Suppose we have n squares arranged in a 1 n rectangle We nuber the squares 1,, n fro left to right and also nuber the vertical edges of the squares 0,, n left to right A linear tiling, T, is a covering of the rectangle with disjoint doinos (covering two squares) and onoinos (covering one square) Let L n be the set of all such T The tilings in L 3 are drawn in Figure 1 where a dot in a square represents a onoino while two dots connected by a horizontal line represent a doino Let the weight of a tiling be w(t ) = s t d where and d are the nuber of onoinos and doinos in T, respectively We will use the sae weight for all other types of tilings considered below Since the last tile in any tiling ust be

3 INTEGERS: 10 (010) 699 L 3 : {4} = s 3 + st + st Figure 1: The tilings in L 3 and corresponding weights a onoino or doino, the initial conditions and recursion (1) iediately give {n + 1} = T L n w(t ) See Figure 1 for an illustration of the case n = 3 For our second cobinatorial interpretation, we will need another sequence of polynoials closely related to the {n} Define n using recursion (1) but with the initial conditions 0 = and 1 = s If s = t = 1 then n is the nth Lucas nuber A cobinatorial interpretation for these polynoials is obtained via another type of tiling In a circular tiling of the 1 n rectangle, the edges labeled 0 and and n are identified so that it is possible to have a doino crossing this edge and covering the first and last squares Such a doino, if it exists, will be called the circular doino of the tiling Let C n be the set of circular tilings of a 1 n rectangle So L n C n is the subset of all circular tilings with no circular doino For exaple, C 3 consists of the tilings in L 3 displayed previously together with Now for n 1 we have n = T C n w(t ) (3) Indeed, to show that the su satisfies (1) first note that we already have a weightpreserving bijection for the linear tilings involved And if T C n has a circular edge, then reoval of the tile covering square n 1 will tae care of the reainder In order to ae (3) also hold for C 0, we give the epty tiling ɛ of the 1 0 box weight w(ɛ) = Bear in ind that ɛ considered as an eleent of L 0 still has w(ɛ) = 1 Context will always ae it clear which weight we are using We start with two recursions for the Lucas polynoials These are well nown for Lucas sequences; see [3, p 38, Identity 73] for (4) and [3, p 46, Identity 94] or [10, p 01, Equation 49] for (5) Also, the proofs in the integer case generalize to variable s and t without difficulty But we will provide a deonstration for copleteness and to ephasize the siplicity of the cobinatorics involved

4 INTEGERS: 10 (010) 700 Lea 1 For 1 and n 0 we have { + n} = {n + 1} {} + t { 1} {n} (4) For, n 0 we have { + n} = n {} + {n} (5) Proof For the first identity, the left-hand side is the generating function for T L +n 1 The second and first ters on the right correspond to those tilings which do or do not have a doino crossing the edge labeled n, respectively To illustrate, if = n = then the tilings in Figure 1 are counted by { + n} The first two tilings do not have a doino crossing the edge labeled and so are counted by {n + 1} {} The edge of the third tiling does cross that edge and is counted by t { 1} {n} Multiply the second equation by and consider two copies of L +n 1 In each tiling in the first copy distinguish the edge labeled 1, and do the sae for the edge labeled in the second copy The set of tilings in both copies where a doino does not cross the distinguished edge accounts for the ters corresponding to linear pairs in n {} + {n} If a doino crosses the distinguished edge 1 in a tiling T, then consider the restriction T of T to the first squares as a circular tiling by shifting it so that the doino between squares 1 and becoes the circular edge Also let T be the restriction of T to the last n 1 squares considered as a linear tiling The pairs (T, T ) account for the reaining ters in {n} A siilar bijection using the tilings with a doino crossing the distinguished edge accounts for the rest of the ters in n {}, copleting the proof We can use the recurrence relations in the previous lea to produce recursions for the lucanoials Proposition For, n 1 we have { } + n { } { } + n 1 + n 1 = {n + 1} + t { 1} 1 n 1 = n { + n 1 1 } + { + n 1 n 1 }

5 INTEGERS: 10 (010) 701 Proof Given, n and any polynoials p and q such that { + n} = p {}+q {n}, straightforward algebraic anipulation of the definition of lucanoials yields { } { } { } + n + n 1 + n 1 = p + q 1 n 1 Cobining this observation with Lea 1, we are done 3 The Cobinatorial Interpretations Our cobinatorial interpretations of { } +n will involve integer partitions A partition is a wealy decreasing sequence λ = (λ 1, λ,, λ ) of nonnegative integers The λ i, 1 i, are called parts and note that we are allowing zero as a part The Ferrers diagra of λ, also denoted λ, is an array of left-justified rows of boxes with λ i boxes in row i We say that λ is contained in an n rectangle, written λ n, if it has parts and each part is at ost n In this case, λ deterines another partition λ whose parts are the lengths of the coluns of the copleent of λ in n The first diagra in Figure shows λ = (3,,, 0, 0) contained in a 5 4 rectangle with copleent λ = (5, 4,, ) A linear tiling of λ is a covering of its Ferrers diagra with disjoint doinos and onoinos obtained by linearly tiling each λ i The set of such tilings is denoted L λ Note that if λ n then T L λ gives a tiling of each of its rows, while T L λ gives a tiling of each colun of its copleent We will also need L n which is the set of all tilings in L n which do not begin with a onoino This is equivalent to beginning with a doino if n, and for n < yields L 0 = {ɛ} and L 1 = We define L λ siilarly The second diagra in Figure shows a tiling in L λ L λ In a circular tiling of λ we use circular tilings on Figure : A partition λ contained in 5 4 and a tiling

6 INTEGERS: 10 (010) 70 each λ i So if λ i = 0 then it will get the epty tiling which has weight The notation C λ is self-explanatory If one views the tiling T in Figure as an eleent of L λ L λ then it has weight w(t ) = s6 t 7 But as an eleent of C λ C λ it has w(t ) = 4s 6 t 7 As usual, context will clarify which weight to use We are now in a position to state and prove our two cobinatorial interpretations for the lucanoials The first has the nice property that it is ultiplicity free The second is pleasing because it displays the natural syetry of { } +n Theore 3 For, n 0 we have { } + n = λ n T L λ L λ w(t ), (6) and { } + n +n = λ n T C λ C λ w(t ) (7) Proof We will show that the right-hand side of (6) satisfies the first recursion in Proposition as the initial conditions are easy to verify Given λ n there are two cases If λ 1 = n then the generating function for tilings of the first row of λ is {n + 1}, and { } +n 1 1 counts the ways to fill the rest of the rectangle If λ1 < n then λ 1 = The generating function for L is t { 1} and { } +n 1 n 1 taes care of the rest Proving that both sides of (7) is siilar using the fact that if we let f(, n) = +n{ } +n then, by the second recursion in Proposition, we have f(, n) = n f( 1, n) + f(, n 1) This copletes the proof We end by noting that it would be interesting to find analogues for { n } of various nown identities for ordinary binoial coefficients It ight then be possible to use the previous theore to provide cobinatorial proofs One approach would be to apply algebraic anipulations to the corresponding results for q-binoial coefficients [ ] n For exaple, this ethod was used in [8] to give an analog of the q Chu-Vanderonde suation for s = l, t = 1 For general s and t it is easy to see that where { } [ n = y (n ) n ] x/y x = s + s + 4t and y = s s + 4t

7 INTEGERS: 10 (010) 703 Unfortunately, this approach tends to introduce algebraic functions of s and t for which a cobinatorial interpretation is unclear References [1] Andrews, G E, The Theory of Partitions, Cabridge Matheatical Library Cabridge University Press, Cabridge, 1998 Reprint of the 1976 original [] Benjain, A T, and Plott, S S, A cobinatorial approach to Fibonoial coefficients, Fibonacci Quart 46/47 (008/09), 7 9 [3] Benjain, A T, and Quinn, J J, Proofs That Really Count, Vol 7 of The Dolciani Matheatical Expositions, Matheatical Association of Aerica, Washington, DC, 003 [4] Bousquet-Mélou, M, and Erisson, K, Lecture hall partitions, Raanujan J 1 (1997), [5] Bousquet-Mélou, M, and Erisson, K, Lecture hall partitions II, Raanujan J 1 (1997), [6] Bousquet-Mélou, M, and Erisson, K, A refineent of the lecture hall theore J Cobin Theory Ser A 86 (1999), [7] Gessel, I, and Viennot, G, Binoial deterinants, paths, and hoo length forulae Adv in Math 58 (1985), [8] Loehr, N, and Savage, C, Generalizing the cobinatorics of binoial coefficients via l- noials, Integers, to appear [9] Lucas, E, Theorie des Fonctions Nueriques Sipleent Periodiques, Aer J Math 1 (1878), [10] Lucas, E, Theorie des Fonctions Nueriques Sipleent Periodiques [Continued], Aer J Math 1 (1878), [11] Lucas, E, Theorie des Fonctions Nueriques Sipleent Periodiques Aer J Math 1 (1878), 89 31