Parity Theorems for Statistics on Domino Arrangements

Parity Theorems for Statistics on Domino Arrangements Mar A Shattuc Mathematics Department University of Tennessee Knoxville, TN 37996-1300 shattuc@mathutedu Carl G Wagner Mathematics Department University of Tennessee Knoxville, TN 37996-1300 wagner@mathutedu Submitted: Oct 1, 004; Accepted: Jan 1, 005; Published: Jun 14, 005 MR Subject Classifications: 11B39, 05A15 Abstract We study special values of Carlitz s -Fibonacci and -Lucas polynomials F n (,t) and L n (,t) Brief algebraic and detailed combinatorial treatments are presented, the latter based on the fact that these polynomials are bivariate generating functions for a pair of statistics defined, respectively, on linear and circular domino arrangements 1 Introduction In what follows, N and P denote, respectively, the nonnegative and the positive integers If is an indeterminate, then n := 1 + + + n 1 if n P, 0! := 1, n! := 1 n if n P, and ( ) n n!, if 0 n; :=! (n )! (11) 0, if <0or0 n< A useful variation of (11) is the well nown formula [10, p9] ( ) n = 0d 0+1d 1 + +d = p(, n, t) t, (1) t>0 d 0 +d 1 + +d =n d i N where p(, n, t) denotes the number of partitions of the integer t with at most n parts, each no larger than This paper elucidates certain features of the -Fibonacci polynomials F n (, t) := 066 n/ ( n ) t, n N, (13) the electronic journal of combinatorics 1 (005), #N10 1

and the -Lucas polynomials L n (, t) := 066 n/ n ( ) n t, n P (14) (n ) Note that F n (1, 1) = F n, where F 0 = F 1 = 1 and F n = F n 1 + F n, n (this parameterization of the Fibonacci numbers, also employed by Wilf [1], results here in a notation with mnemonic features superior to that of the classical parameterization), and L n (1, 1) = L n,wherel 1 =1,L =3,andL n = L n 1 + L n, n 3 Our aim here is to present both algebraic and combinatorial treatments of F n (1, 1), F n ( 1,t), L n (1, 1), and L n ( 1,t) Our algebraic proofs mae freuent use of the identity [11, pp 01 0] n>0 ( ) n x n = x, N (15) (1 x)(1 x) (1 x) Our combinatorial proofs use the fact that F n (, t) andl n (, t) are generating functions for a pair of statistics defined, respectively, on linear and circular arrangements of nonoverlapping dominos, and embody the following general strategy: Let (Γ n ) be a seuence of finite discrete structures, with Γ n = G n Each statistic s : Γ n N gives rise to a -generalization of G n, in the form of the generating function G n () := γ Γ n s(γ) = {γ Γ n : s(γ) =} Of course, G n (1) = G n On the other hand, G n ( 1) = Γ (0) n Γ (1) n where Γ (i) n := {γ Γ n : s(γ) i (mod )} Thus a combinatorial proof that G n ( 1) = g n may be had by (1) identifying a distinguished subset Γ n of Γ n (with Γ n = if g n =0 and, more generally, Γ n = g n,withγ n beingasubsetofγ (0) n or Γ (1) n, depending on whether g n is positive or negative), and () constructing an involution γ γ of Γ n Γ n for which s(γ) ands(γ ) have opposite parity (In what follows, we call the parity of s(γ) the s-parity of γ, andthemapγ γ an s-parity changing involution of Γ n Γ n) In addition to conveying a visceral understanding of why G n ( 1) taes its particular value, such an exercise furnishes a combinatorial proof of the congruence G n g n (mod ) Shattuc [9] has, for example, given such a combinatorial proof of the congruence ( ) n / 1 S(n, ) (mod ) n for Stirling numbers of the second ind, answering a uestion posed by Stanley [10, p 46, Exercise 17b], the electronic journal of combinatorics 1 (005), #N10

The polynomials F n (, t) andl n (, t), or special cases thereof, have appeared previously in several guises In a paper of Carlitz [1], F n (, 1) arises as the generating function for the statistic a 1 +a + +(n 1)a n 1 on the set of binary words a 1 a a n 1 with no consecutive ones In the same paper, L n (, 1) occurs (though not explicitly in the simple form entailed by (14)) as the generating function for the statistic a 1 +a + + na n on the set of binary words a 1 a a n with no consecutive ones, and with a 1 = a n =1 forbidden as well Cigler [6] has shown that F n (, t) arises as the bivariate generating function for a pair of statistics on the set of lattice paths from (0, 0) to (n, 0) involving only horizontal moves, and northeast moves, followed immediately by southeast moves Finally, Carlitz [] has studied the -Fibonacci polynomial Φ n (a, ) =a n 1 F n 1 (, a ) from a strictly algebraic point of view See also the related paper of Cigler [3] In below, we treat the -Fibonacci polynomials F n (, t) and evaluate F n (1, 1) and F n ( 1,t) In 3 we treat the -Lucas polynomials L n (, t) and evaluate L n (1, 1) and L n ( 1,t) While the combinatorial proofs presented below could of course be reformulated in terms of the aforementioned statistics on binary words or lattice paths, our approach, based on statistics on domino arrangements, yields the most transparent constructions of the relevant parity changing involutions Linear Domino Arrangements A well nown problem of elementary combinatorics ass for the number of ways to place indistinguishable non-overlapping dominos on the numbers 1,,, n, arranged in a row, where a domino is a rectangular piece capable of covering two numbers It is useful to place suares (pieces covering a single number) on each number not covered by a domino The original problem then becomes one of determining the cardinality of R n,,thesetof coverings of the row of numbers 1,,,n by dominos and n suares Since each such covering corresponds uniuely to a word in the alphabet {d, s} comprising d s and n s s, it follows that ( ) n R n, =, 0 n/, (1) for all n P (In what follows we will simply identify coverings with such words) If we set R 0,0 = { }, the empty covering, then (1) holds for n = 0 as well With it follows that R n := 066 n/ R n = 066 n/ where F 0 = F 1 =1andF n = F n 1 + F n for n R n,, n N, () ( ) n = F n, (3) the electronic journal of combinatorics 1 (005), #N10 3

Given c R n,letν(c):= the number of dominos in the covering c, letσ(c) :=the sum of the numbers covered by the left halves of each of those dominos, and let F n (, t) := σ(c) t ν(c) (4) c R n Categorizing covers of 1,,,naccording as n is covered by a suare or a domino yields the recurrence relation F n (, t) =F n 1 (, t)+ n 1 tf n (, t), n, (5) with F 0 (, t) =F 1 (, t) = 1 The following theorem gives an explicit formula for F n (, t) Theorem 1 For all n N, F n (, t) = 066 n/ Proof It clearly suffices to show that ( ) n σ(c) = c R n, ( ) n t (6) Each c R n, corresponds uniuely to a seuence (d 0,d 1,,d ), where d 0 is the number of suares following the th domino (counting from left to right) in the covering c, d is the number of suares preceding the first domino, and, for 0 <i<, d i is the number of suares between dominos i and i + 1 Here, σ(c) =(d +1)+(d + d 1 +3)+ + (d + d 1 + + d 1 +( 1)) = +0d 0 +1d 1 +d + + d Hence, by (1) c R n, σ(c) = d 0 +d 1 + +d =n d i N 0d 0+1d 1 + +d = ( n Corollary 11 The ordinary generating function of the seuence (F n (, t)) n>0 is given by F n (, t)x n = t x (1 x)(1 x) (1 x) (7) n>0 >0 Proof The result follows from (6), summation interchange, and (15) As noted earlier, F n (1, 1) = F n Hence (7) generalizes the well nown result, F n x n 1 = 1 x x (8) n>0 We now evaluate F n (1, 1) and F n ( 1,t) Substituting (, t) =(1, 1) into (5) and solving the resulting recurrence yields ), the electronic journal of combinatorics 1 (005), #N10 4

Theorem For all n N, 0, if n or 5 (mod 6); F n (1, 1) = 1, if n 0 or 1 (mod 6); 1, if n 3 or 4 (mod 6) (9) A slight variation on the strategy outlined in 1 above yields a combinatorial proof of (9) Let R n consist of those c = x 1 x in R n satisfying the conditions x i 1 x i = ds, 1 i n/3 If n 0or1(mod6),thenR n is a singleton whose sole element has even ν-parity If n 3or4(mod6),thenR n is a singleton whose sole element has odd ν-parity If n (mod 3), then R n is a doubleton containing two members of opposite ν-parity, which we pair The foregoing observations establish (9) for 0 n since R n = R n for such n Thus, it remains only to construct a ν-parity changing involution of R n R n for n 3 Such an involution is furnished by the pairings and (ds) sdv (ds) sssv, (ds) ddu (ds) ssdu where 0 < n/3, (ds) 0 denotes the empty word, and u and v are (possibly empty) words in the alphabet {d, s} Note that the above argument also furnishes a combinatorial proof of the well nown fact that F n is even if and only if n (mod3) Remar Neither (9), nor its corollary (31) below, is new Indeed, (9) is a special case of the well nown formula ( ) ( 1) ( ) n 066 n/ = { ( 1) n/3 n(n 1)/6, if n 0, 1 (mod 3); 0, if n (mod 3) See, eg, Ehad and Zeilberger [7], Kupershmidt [8], and Cigler [4] Our interest here, and in Theorem 3 below, has been to furnish new proofs of (9) and (31) based on parity changing involutions Theorem 3 For all m N, and where F 1 (, t) :=0 F m ( 1,t)=F m (1,t ) tf m 1 (1,t ) (10) F m+1 ( 1,t)=F m (1,t ), (11) the electronic journal of combinatorics 1 (005), #N10 5

Proof Taing the even and odd parts of both sides of (7) and replacing x with x 1/ yields and m>0 F m ( 1,t)x m =(1 tx) >0 m>0 F m+1 ( 1,t)x m = >0 t x (1 x) +1, t x (1 x) +1 from which (10) and (11) follow from (7) For a combinatorial proof of (10) and (11), we first assign to each domino arrangement c R n the weight w c := ( 1) σ(c) t v(c),wheret is an indeterminate Let R n consist of those c = x 1 x x r in R n satisfying the conditions x i 1 = x i,1 i r/ Suppose c = x 1 x x r R n R n,withi 0 being the smallest value of i for which x i 1 x i Exchanging the positions of x i0 1 and x i0 within c produces a σ-parity changing involution of R n R n which preserves v(c) Then F m+1 ( 1,t)= w c = w c = ws = F m(1,t ) c R m+1 s R m c R m+1 and F m ( 1,t)= c R m w c = = s R m w s t c R m w c = c R m σ(c) even w c + c R m σ(c) odd w c s R m 1 w s = F m(1,t ) tf m 1 (1,t ), since members of R m+1 end in a single s, while members of R m or in a single d, depending on whether σ(c) is even or odd end in a double letter When t = 1 in Theorem 3, we get for m N, F m ( 1, 1) = F m and F m+1 ( 1, 1) = F m (1) The arguments given above then specialize when t = 1 to furnish combinatorial proofs of the congruences F m F m (mod ) and F m+1 F m (mod ) 3 Circular Domino Arrangements If n P and 0 n/, letc n, be the set of coverings by dominos and n suares of the numbers 1,,, n arranged clocwise around a circle: the electronic journal of combinatorics 1 (005), #N10 6

n 1 By the initial half of a domino occurring in such a cover, we mean the half first encountered as the circle is traversed clocwise Classifying members of C n, according as (i) n is covered by the initial half of some domino, (ii) 1 is covered by the initial half of some domino, or (iii) 1 is covered by a suare, and applying (1) to count these three classes yields the well nown result ( ) ( ) ( n 1 n 1 n C n, = 1 + = n n Note that C,1 =, the relevant coverings being ), 0 n/ (31) (1) T 1 T»» and () 1»» T T In covering (1), the initial half of the domino covers 1, and in covering (), the initial half covers With C n := C n,, n P, (3) it follows that C n = 066 n/ 066 n/ ( ) n n = L n, (33) n where L 1 =1,L =3,andL n = L n 1 + L n for n, also a well nown result Given c C n,letν(c) := the number of dominos in the covering c, letσ(c) :=thesum of the numbers covered by the initial halves of each of those dominos, and let L n (, t) := c C n σ(c) t ν(c) (34) the electronic journal of combinatorics 1 (005), #N10 7

Theorem 31 For all n P, L n (, t) = 066 n/ n ( ) n t (35) (n ) Proof It suffices to show that n σ(c) = (n ) c C n, ( ) n (36) for 0 n/ Partitioning C n, into the categories (i), (ii), and (iii) employed above in deriving (31), and applying (6) yields ( ) ( ) ( ) n 1 n 1 σ(c) = +n + + + n 1 1 c C n, 1 = n ( ) n (n ) Corollary 311 The ordinary generating function of the seuence (L n (, t)) n>1 is given by L n (, t)x n = x 1 x + t x ( 1+ (1 x) ) (1 x)(1 x) (1 x) (37) n>1 >1 Proof This result follows from (35), using the identity ( ) ( ) ( ) n n n n 1 = + n, (38) (n ) 1 summation interchange, and (15) As noted earlier, L n (1, 1) = L n Hence (37) generalizes the well nown result L n x n = x +x 1 x x (39) n>1 The L n (, t) are related to the F n (, t) by the formula which reduces to the familiar L n (1,t)=F n (1,t)+tF n (1,t), n 1, (310) L n = F n 1 +F n, n 1, (311) when t =1 WenowevaluateL n (1, 1) and L n ( 1,t) Substituting t = 1 into (310) and applying (9) yields the electronic journal of combinatorics 1 (005), #N10 8

Theorem 3 For all n P, 1, if n 1 or 5 (mod 6); 1, if n or 4 (mod 6); L n (1, 1) =, if n 0 (mod 6);, if n 3 (mod 6) (31) For a combinatorial proof of (31), write C n = C n C n,wherec C n iff 1 is covered by a suare of c, or 1 and by a single domino of c, andc C n iff n and 1 are covered by a single domino of c Associate to each c C n awordu c = v 1 v in the alphabet {d, s}, where { s, if the i-th piece of c is a suare; v i := d, if the i-th piece of c is a domino, and one determines the i-th piece of c by starting at 1 and proceeding clocwise if c C n, and counterclocwise if c C n Note that although distinct elements of C n may be associated with the same word, each c C n is associated with a uniue word, and each c C n is associated with a uniue word Let { } C n := c C n : u c =(sd) n/3 or (sd) n/3 s { c } C n : u c =(ds) n/3 or (ds) n/3 d It is straightforward to chec that Cn =1ifn 1 or (mod 3) and Cn =ifn 0 (mod 3) In the former case, the sole element of Cn has even ν-parity if n 1or5 (mod 6), and odd ν-parity if n or 4 (mod 6) In the latter case, both elements of Cn have even ν-parity if n 0(mod6),andoddν-parity if n 3(mod6) To complete the proof it suffices to identify a ν-parity changing involution of C n Cn, and of C n Cn, whenever these sets are nonempty In the former case, this occurs when n, and such an involution is furnished by the pairing (sd) du (sd) ssu, where 0 (n )/3, andu is a (possibly empty) word in {d, s} In the latter case, this occurs when n 4, and such an involution is furnished by the pairing d(sd) j dv d(sd) j ssv, where 0 j (n 4)/3, andv is a (possibly empty) word in {d, s} Theorem 33 For all m P, L m ( 1,t)=L m (1,t ) (313) the electronic journal of combinatorics 1 (005), #N10 9

and L m 1 ( 1,t)=F m 1 (1,t ) tf m (1,t ), (314) where F 1 (, t) :=0 Proof Taing the even and odd parts of both sides of (37) and replacing x with x 1/ yields m>1 L m ( 1,t)x m = x 1 x +( x) >1 t x (1 x) +1 and m>1 L m 1 ( 1,t)x m =(1 tx) >0 t x +1 (1 x) +1, from which (313) and (314) follow from (37) and (7) The following observation leads to a combinatorial proof of (313) and (314) for n : If c C n, let c be the result of reflecting the arrangement of dominos and suares constituting c in the diameter through the point on the relevant circle We illustrate pairs c and c for n =8andn =9below c = 3 1 8 7 6 5 4 fi fifi L L L aaa fi fi fi aaa c = 3 1 8 7 6 5 4 L L L aaa!!! L LL!!! c = 3 1 8 7 6 5 4 9 L LL hhh ((( " " b b c = 3 1 8 7 6 5 4 9 L LL hhh ((( fi fifi e e Note that c and c have opposite σ-parity in both cases More generally, it may be verified that if n is even, then c and c have opposite σ-parity iff ν(c) is odd, and if n is odd, then c and c have opposite σ-parity iff is covered by a domino in c In what follows we use the same encoding of covers as words in {d, s} that we employed in the combinatorial proof of Theorem 3 We also assign to each c C n the weight w c =( 1) σ(c) t v(c) the electronic journal of combinatorics 1 (005), #N10 10

Suppose that n =m LetC m := 066m C m, and let Cm consist of those c C m for which u c = v 1 v satisfies v i 1 = v i for all i We extend the σ-parity changing, v-preserving involution of C m C m defined in our initial observation to C m Cm Since Cm = C m for m, we may restrict attention to the case m 3 Let c C m Cm, with u c = v 1 v,andleti 0 be the largest i for which v i 1 v i, whence i 0 Interchanging the (i 0 1) th and (i 0 ) th pieces of c furnishes such an involution Then L m ( 1,t)= w c = w c = L m (1,t ) c C m c C m Suppose now that n =m 1, where m Let C m 1 consist of those c C m 1 in which is covered by a suare There is an obvious v-preserving bijection b : C m 1 R m for which σ(c) σ(b(c)) (mod ) for all c C m 1 In view of the involution of C m 1 C m 1 defined in our initial observation, we have L m 1 ( 1,t)= w c = w c = w c c C m 1 c R m by (10) c C m 1 = F m 1 (1,t ) tf m (1,t ), When t = 1 in Theorem 33, we get for m P, L m ( 1, 1) = L m and L m 1 ( 1, 1) = F m 3 (315) The arguments given above then specialize when t = 1 to furnish combinatorial proofs of the congruences L m L m (mod ) and L m 1 F m 3 (mod ) 4 Some Concluding Remars Formulas such as (10), (11), (313), and (314) show that the = 1 casediffersin many respects from the general -case The referee points out to us that the reason for this behavior may lie in the fact that the Fibonacci and Lucas polynomials can be written as a sum of -binomial coefficients, which are nown to reduce to the ordinary binomial coefficients when = 1 Similar reductions also occur in some cases when is a root of unity, and analogues to formulas such as (10) and (313) might be expected in these cases For example, when = ρ = 1+i 3, a third root of unity, we have the following formulas: F 3m (ρ, t) =F m (1,t 3 ) tf m 1 (1,t 3 ), (41) F 3m+1 (ρ, t) =F m (1,t 3 )+ρt F m 1 (1,t 3 ), (4) F 3m+ (ρ, t) =(1+ρt)F m (1,t 3 ), (43) L 3m (ρ, t) =L m (1,t 3 ), (44) L 3m+1 (ρ, t) =F m (1,t 3 )+ρt(1 ρt)f m 1 (1,t 3 ), (45) the electronic journal of combinatorics 1 (005), #N10 11

and L 3m+ (ρ, t) =(1 t)f m (1,t 3 ) t F m 1 (1,t 3 ) (46) The combinatorial arguments given when = 1 can be extended to identities such as these For example, to prove (41) (43), instead of pairing members of R n with the same number of dominos and opposite σ-parity, we partition R n into tripletons whose members each contain the same number of dominos but have different σ-values mod 3 Assign to each member c of R n the weight w c := ρ σ(c) t v(c),whereρ = 1+i 3 and t is an indeterminate Then for a tripleton {c 1,c,c 3 } as described, we have w c1 + w c + w c3 =0 since 1 + ρ + ρ =0 Let R n consist of those c = x 1 x x r in R n satisfying x 3i = x 3i 1 = x 3i, 1 i r/3 (47) Suppose that c = x 1 x x r R n R n,withi 0 being the smallest value of i for which (47) fails to hold Group the three members of R n R n gotten by circularly permuting x 3i0, x 3i0 1, andx 3i0 within c = x 1 x x r, leaving the rest of c undisturbed Note that these three members of R n contain the same number of dominos but have different σ-values mod 3 If n =3m, then F 3m (ρ, t) = w c = w c = ws 3 +(ρ + ρ )t ws 3 = F m(1,t 3 ) c R 3m s R m s R m 1 c R 3m +(ρ + ρ )tf m 1 (1,t 3 )=F m (1,t 3 ) tf m 1 (1,t 3 ), which proves (41), since members of R 3m may end in a triple letter, in ds, orin sd Formulas (4) and (43) follow similarly, since members of R 3m+1 end in s or dd, while members of R 3m+ end in ss or d We conclude by remaring that Cigler has studied the generalized -Fibonacci polynomials [5] ( ) F n (j, s, t, ) := j( ) n (j 1)( +1) t s n (+1)j+1 (48) 06j6n j+1 Note that a close variant of (48), F n (j) (, t, s) :=F n+j 1(j, s, t, ), n N, (49) furnishes a natural generalization of F n (, t), reducing to the latter when j =ands =1 Analogues of (10) and (11) can be obtained for F n (j) (, t, 1) when j 3; for example, we have for m N, F m( 1,t,1) (3) = F m (3) (1, t, 1), (410) F m+1( 1,t,1) (3) = F m (3) (1, t, 1) tf m 1(1, (3) t, 1), (411) F m( 1,t,1) (4) = F m (4), 1) tf m (1,t (4), 1), (41) the electronic journal of combinatorics 1 (005), #N10 1

and F m+1( 1,t,1) (4) = F m (4) (1,t, 1) (413) It would be interesting to have combinatorial proofs of (410) (413) Acnowledgment The authors would lie to than the anonymous referee for a prompt, thorough reading of this paper and for many insightful suggestions which improved it We are indebted to the referee for formulas (41) (43), formulas (410) (413), and for Theorems 3 and 33 in their given form We also would lie to than the referee for calling to our attention the papers of Cigler [4, 5], Ehad and Zeilberger [7], and Kupershmidt [8] References [1] L Carlitz, Fibonacci notes 3: -Fibonacci numbers, Fib Quart 1 (1974), 317 3 [] L Carlitz, Fibonacci notes 4: -Fibonacci polynomials, Fib Quart 13 (1975), 97 10 [3] J Cigler, -Fibonacci polynomials, Fib Quart 41 (003), 31 40 [4] J Cigler, A new class of -Fibonacci polynomials, Elect J Combin 10 (003), #R19 [5] J Cigler, Some algebraic aspects of Morse code seuences, Discrete Math and Theoretical Computer Science 6 (003), 55 68 [6] J Cigler, -Fibonacci polynomials and the Rogers-Ramanujan identities, Ann Combin 8 (004), 69 85 [7] S Ehad and D Zeilberger, The number of solutions of X = 0 in triangular matrices over GF (), Elect J Combin 3 (1996), #R [8] B Kupershmidt, -Analogues of classical 6-periodicity, J Nonlinear Math Physics 10 (003), 318 339 [9] M Shattuc, Bijective proofs of parity theorems for partition statistics, J Integer Se, 8 (005), Art 515 [10] R Stanley, Enumerative Combinatorics, Vol I, Wadsworth and Broos/Cole, 1986 [11] C Wagner, Generalized Stirling and Lah numbers, Discrete Math 160 (1996), 199 18 [1] H Wilf, generatingfunctionology, Academic Press, 1990 the electronic journal of combinatorics 1 (005), #N10 13