Catalan triangle numbers and binomial coefficients

Transcription

1 Contemporary Mathematics Volume 73, Catalan triangle numbers and binomial coefficients Kyu-Hwan Lee and Se-jin Oh Abstract The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac Moody algebras We prove that any binomial coefficient can be written as weighted sums along rows of the Catalan triangle The coefficients in the sums form a triangular array, which we call the alternating Jacobsthal triangle We study various subsequences of the entries of the alternating Jacobsthal triangle and show that they arise in a variety of combinatorial constructions The generating functions of these sequences enable us to define their -analogue of q-deformation We show that this deformation also gives rise to interesting combinatorial sequences The starting point of this wor is certain identities in the study of Khovanov Lauda Rouquier algebras and fully commutative elements of a Coeter group Introduction It is widely accepted that Catalan numbers are the most frequently occurring combinatorial numbers after the binomial coefficients As binomial coefficients can be defined inductively from the Pascal s triangle, so can Catalan numbers from a triangular array of numbers whose entry in the n th row and th column is denoted by C(n, for0 n Set the first entry C(0, 0, and then each subsequent entry is the sum of the entry above it and the entry to the left All entries outside of the range 0 n are considered to be 0 Then we obtain the array shown in ( nown as Catalan triangle introduced by LW Shapiro [8] in 976 Notice that Catalan numbers C n appear on the hypotenuse of the triangle, ie C n C(n, n for n ( Mathematics Subject Classification Primary 05A0; Secondary 05E0 The first author was partially supported by a grant from the Simons Foundation (#38706 The second author was supported by NRF Grant # 206RCB c 208 American Mathematical Society This is a free offprint provided to the author by the publisher Copyright restrictions may apply

2 66 K-H LEE AND S-J OH The first goal of this paper is to write each binomial coefficient as weighted sums along rows of the Catalan triangle In the first case, we tae the sums along the n th and n + st rows of the Catalan triangle, respectively, and obtain 2-power weighted sums to epress a binomial coefficient More precisely, we prove: Theorem For integers n and 0 n +, we have the identities n + + (2 C(n, s2 s C(n +,s2 ma( s,0 and (3 n/2 n + C( n/2,2 (n +/2 ma( n/2,0 0 It is quite intriguing that the two most important families of combinatorial numbers are related in this way By replacing 2-powers with -powers in the identities, we define Catalan triangle polynomials and mae a conjecture on staced ( directed n + animals studied in [] (see Section 23 It is also interesting that (n +/2 appearing in (3 is eactly the number of fully commutative, involutive elements of the Coeter group of type A n (See (42 in [9] Actually, a clue to the identities (2 and (3 was found in the study of the homogeneous representations of Khovanov Lauda Rouquier algebras and the fully commutative elements of type D n in the paper [2] of the first-named author and G Feinberg, where they proved the following: Theorem 2 [2] For n, we have n n +3 (4 C n C(n, 2 n 2, 2 where C n is the n th Catalan number 0 We note that n+3 2 C n is the number of the fully commutative elements of type D n (See [9] The identity (4 is obtained by decomposing the set of fully commutative elements of type D n into pacets Liewise, we epect interesting combinatorial interpretations and representation-theoretic applications of the identities (2 and (3 In particular, C(n, appear as weight multiplicities of finite-dimensional simple Lie algebras and affine Kac Moody algebras of types A and C [6, 0, ] To generalize Theorem, we use other rows of the Catalan triangle and there appears a natural sequence of numbers A(m, t, defined by A(m, 0, to yield the following result: A(m, t A(m,t A(m,t, Theorem 3 For any n> m t, we have m n + + (5 C(n + m, s2 m s + A(m, tc(n + m, m + t t This is a free offprint provided to the author by the publisher Copyright restrictions may apply

3 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 67 In particular, when m, we have n + + (6 A(, tc(n +, t t0 The sequence consisting of A(m, t is listed as A in the On-line Encyclopedia of Integer Sequences (OEIS However, the identity (6 does not seem to have been nown The identity (6 clearly suggests that the triangle consisting of the numbers A(m, t be considered as a transition triangle from the Catalan triangle to the Pascal triangle We call it the alternating Jacobsthal triangle The triangle has (sums of subsequences of the entries with interesting combinatorial interpretations In particular, diagonal sums are related to the Fibonacci numbers and horizontal sums are related to the Jacobsthal numbers The second goal of this paper is to study a -analogue of q-deformation of the Fibonacci and Jacobsthal numbers through a -analogue of the alternating Jacobsthal triangle This deformation is obtained by putting the parameters q and into the generating functions of these numbers Our constructions give rise to different polynomials than the Fibonacci and Jacobsthal polynomials which can be found in the literature (eg [4, 7] For eample, the -analogue J,m (q ofq-deformation of the Jacobsthal numbers is given by the generating function ( q ( q 2 2 ( (q + J,m (q m m When q and, we recover the usual generating function ( + ( 2 of the Jacobsthal numbers Interestingly enough, sequences given by special values of this deformation have various combinatorial interpretations For eample, the sequence (J 2,m ( m (,, 4, 6, 6, 28, 64, 20, is listed as A00779 in OEIS and has the interpretation as the numbers of equal dual pairs of some integrals studied in [3] (See Table on p365 in [3] Similarly, many subsequences of a -analogue of the alternating Jacobsthal triangle are found to have combinatorial meanings See the triangle (52, for eample An outline of this paper is as follows In the net section, we prove Theorem to obtain Catalan triangle epansions of binomial coefficients as 2-power weighted sums We also introduce Catalan triangle polynomials and study some of their special values In Section 3, we prove Theorem 3 and investigate the alternating Jacobsthal triangle to obtain generating functions and meaningful subsequences The following section is concerned with q-deformation of the Fibonacci and Jacobsthal numbers The last section is devoted to the study of a -analogue of the q-deformation of the Fibonacci and Jacobsthal numbers using the -analogue of the alternating Jacobsthal triangle 2 Catalan epansion of binomial coefficients In this section, we prove epressions of binomial coefficients as 2-power weighted sums along rows of the Catalan triangle Catalan trapezoids are introduced for the This is a free offprint provided to the author by the publisher Copyright restrictions may apply

4 68 K-H LEE AND S-J OH proofs In the last subsection, Catalan triangle polynomials are defined and some of their special values will be considered 2 Catalan triangle We begin with a formal definition of the Catalan triangle numbers Definition 2 For n 0and0 n,wedefinethe(n, -Catalan triangle number C(n, recursively by if n 0; C(n, + C(n, if 0 <<n; (2 C(n, C(n, 0 if 0; C(n, n if n, and define the n th Catalan number C n by C n C(n, n for n 0 The closed form formula for the Catalan triangle numbers is well nown: for n 0and0 n, (n +!(n + C(n,!(n +! In particular, we have C n 2n, n + n and it can be easily verified ( n + (22 C(n, Theorem 22 [2] For n, we have (23 n + n n +3 C n C(n, 2 n As mentioned in the introduction, n+3 2 C n is the number of the fully commutative elements of type D n ([9] and the identity (23 is obtained by decomposing the set of fully commutative elements of type D n into pacets Theorem 23 For n Z 0, we have n/2 n + (24 C( n/2,s2 ma( n/2 s,0 (n +/2 n + Proof Set Q n : for convenience We will consider n even (n +/2 and n odd separately First, assume n 2 for some Z 0 Thenwehave 2 + Q 2 + By (23 in Theorem 22, we have (25 C(, s2 2 s +3 C ( This is a free offprint provided to the author by the publisher Copyright restrictions may apply

5 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 69 On the other hand, 2 C(, s2 2 s C(, s2 2 s +2C(, Note that C(, C Multiplying (25 by 4, we have C(, s2 s +5C(, Hence C(, s2 s C + ( 2 Net, assume n 2 +forsome Z 0 Thenwehave 2 +2 Q 2+ + Again from (23, we obtain (26 C( +,s2 s +4 C Q On the other hand, C( +,s2 s C( +,s2 s +2C( +, Note that C( +,C + Multiplying (26 by 2, we have + C( +,s2 s +4C( +, C( +,s2 ma( s,0 +2C( +, Since C( +, + ourassertionistrueinthiscaseaswell Q + 2+, Corollary 24 For n Z 0, the dual form of (24 holds; ie, n/2 n + C( n/2,s2 n/2 s (n +/2 In particular, we have the following identity by replacing n with 2n : n 2n n C(n,s2 n s (n + s!(n s n 2 2 n s n n s!n! s This is a free offprint provided to the author by the publisher Copyright restrictions may apply

6 70 K-H LEE AND S-J OH 2 + Proof The assertion for n 2 follows from the fact that ( and for n 2 + follows from the fact that From Theorem 23, we have, for n 2 ( Z, 2 C(, s2 ma( s,0 C(, s2 s + 2 +( 2 Since 2 2, we obtain a new identity: +( 2 (27 C(, s2 s More generally, we have ( the following identity which is an interesting epression n + + of a binomial coefficient as a 2-power weighted sum of the Catalan triangle along the n th row Theorem 25 For 0 n +, we have min(n, n + + (28 C(n, s2 s Proof We will use an induction on n+ The cases when n+,n,n are already proved Indeed, the cases n+ and n follow from Corollary 24, and the case n is nothing but (27 Assume that we have the identity (28 for any n and such that 2 n + m Consider the case n + m + By assumption, we have + n + +2 (29 C(n, s2 + s + Since C(n, + n n + +2, the identity (29 can be written as n ( n n n + +2 Now, simplifying the left-hand side n + +2 n n n we obtain the desired identity n + + ( n n + +2 C(n, s2 + s C(n, s2 s n ( n + +, This is a free offprint provided to the author by the publisher Copyright restrictions may apply

7 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 7 Eample 26 ( 7 3 ( C(3,s2 3 s , 3 C(4,s2 3 s Catalan trapezoid As a generalization of Catalan triangle, we define a Catalan trapezoid by considering a trapezoidal array of numbers with m complete columns (m Let the entry in the n th row and th column of the array be denoted by C m (n, for0 m + n Set the entries of the first row to be C m (0, 0 C m (0, C m (0,m, and then each subsequent entry is the sum of the entry above it and the entry to the left as in the case of Catalan triangle All entries outside of the range 0 n + m are considered to be 0 Then, in particular, we obtain C m (n, 0 for all n 0 For eample, when m 3,weobtain ( Alternatively, the numbers C m (n, can be defined in the following way Definition 27 For an integer m, set C (n, C(n, for0 n and C 2 (n, C(n +,for0 n +, and define inductively (2 ( n ++ if 0 <m, ( C m (n +, n ++ n + m ++ m if m n + m, m 0 if n + m< Using the numbers C m (n,, we prove the following theorem Theorem 28 For any triple of integers (m,, n such that m n+m, we have (22 min(n, n + + m m C(n, s2 s C(n + m, s2 m s + C(n ++s, s This is a free offprint provided to the author by the publisher Copyright restrictions may apply

8 72 K-H LEE AND S-J OH Proof By Theorem 25, the second case in (2 can be re-written as follows: C m (n +, min(n, m C(n, s2 s C(n + m, s2 m s if m n + m On the other hand, for m n + m, weuse(22toget m n ++ n + m ++ m C(n ++s, s m Thus we obtain min(n, m C(n, s2 s C(n + m, s2 m s m C(n ++s, s for m n + m This completes the proof By specializing ( (22 ( at m, we obtain different epressions of a binomial n + + n ++ coefficient as a 2-power weighted sum of the Catalan triangle along the n + st row (cf (28 Corollary 29 We have the following identities: For, n ++ C(n, s2 s C(n +,s2 s + C(n +, (23 C(n +,s2 ma( s,0 Note that, combining Theorem 23 and Corollary 29, we have proven Theorem Remar 20 The identities in Corollary 29 can be interpreted combinatorially, and one can prove them bijectively As it reveals combinatorics ( behind n ++ the identities, we setch a bijective proof of the first identity C(n, s2 s Bijective proof n ++ We interpret as the number of the lattice paths from (0, 0 to (n ++, n + withup-steps(, and down-steps (, For such a path λ, let P 0,P,,P s be the intersections of λ with the -ais, decomposing λ into s +partsd 0,D,,D s Then the last part D s of λ from P s to the end point (n ++, n + is a Dyc path, ie it stays at or above the -ais We remove from each of D i, i 0,,s, the first step and the last step and from D s the first step, and flip them, if necessary, to obtain Dyc paths, and concatenate them, putting a step (,, between them The resulting path is a Dyc path from (0, 0 to (n + s, n + s This gives a 2 s -to- map from the This proof was communicated to us by Jang Soo Kim We than him for allowing us to use his proof This is a free offprint provided to the author by the publisher Copyright restrictions may apply

9 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 73 set of the lattice paths from (0, 0 to (n ++, n + tothesetofthedyc paths from (0, 0 to (n + s, n + s Since C(n, s is equal to the number of the Dyc paths from (0, 0 to (n + s, n + s, we have proven n ++ C(n, s2 s C(n, s2 s 23 Catalan triangle polynomials The identities in the previous subsections naturally give rise to the following definition Definition 2 For 0 n, wedefinethe(n, th Catalan triangle polynomial F n, ( by (24 F n, ( C(n, s s C(n, s s + C(n, Note that the degree of F n, is Evaluations of F n, ( at the first three nonnegative integers are as follows: F n, (0 C(n, F n, ( C(n ( +,C 2 (n, n + + F n, (2 C n (n +, Clearly, ( n n + n, C(n, C(n, + C(n,, F n, (d F n, (d+f n, (d for any d Z Let us recall the description of C(n, in terms of binomial coefficients: n + n + + (25 C(n, 2 (2 Theorem 22 For any d Z, we have which recovers (25 when d 2 C(n, d F n, (d (d F n, (d Proof Since F n, (d C(n, sd s + C(n,, we have F n, (d C(n, d C(n, s d s Hence the equation (2 and the fact that C(n, 0 C(n, 0 yield F n, (d F n,(d C(n, (26 F n, (d d By multiplying (26 by d, our assertion follows s This is a free offprint provided to the author by the publisher Copyright restrictions may apply

10 74 K-H LEE AND S-J OH With the consideration of Corollary 29, we define a natural variation of F n, ( Definition 23 For 0 n, we define the modified (n, th triangle polynomial in the following way: (27 F n, ( C(n +,s ma( s,0 Note that the degree of F n, ( is Catalan Evaluations of F n, ( at the first three nonnegative integers are as follows: F n, (0 C(n +, + C(n +, F n, ( C(n ( +2, F n + + n, (2 Similarly, F n, (d F n, (d+ F n, (d for any d Z We have the same result in Theorem 22 for F n, (d also: Theorem 24 For any d Z, we have C(n, d F n, (d (d F n, (d which recovers (25 when d 2 Let σ n be the number of n-celled staced directed animals in a square lattice See [] for definitions The sequence (σ n n 0 is listed as A05974 in OEIS We conjecture σ n F n,n (3 for n 0 We epect that one can find a direct, combinatorial proof of this conjecture A list of the numbers σ n is below, and the conjecture is easily verified for the numbers in the list: σ 0,σ 3,σ 2,σ 3 44,σ 4 84, σ 5 789, σ , σ 7 500, σ , σ , σ , σ For eample, when n 7, we have the sequence (C(8,, 0 7 equal to (, 8, 35, 0, 275, 572, 00, 430, and compute F 7,7 ( Alternating Jacobsthal triangle In the previous section, the binomial coefficient n++ is written as sums along the n th and the n + st row of the Catalan triangle, respectively In this section, we consider other rows of the Catalan triangle as well and obtain a more general result In particular, the n + th row will produce a canonical sequence of numbers, which form the alternating Jacobsthal triangle We study some subsequences of the triangle and their generating functions in the subsections This is a free offprint provided to the author by the publisher Copyright restrictions may apply

11 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 75 Define A(m, t Z recursively for m t 0by (3 A(m, 0, A(m, t A(m,t A(m,t Here we set A(m, t 0whent>m Then, by induction on m, one can see that A(m, t and A(m, m t Using the numbers A(m, t, we prove the following theorem which is a generalization of the identities in Corollary 29: Theorem 3 For any n> m t, we have m n + + (32 C(n + m, s2 m s + A(m, tc(n + m, m + t Proof We will use an induction on m Z Form, we already proved the identity in Corollary 29 Assume that we have the identity (32 for some m Z By specializing (22 at m +,wehave m n + + C(n + m +,s2 m s + C(n ++s, s m t m C(n + m +,s2 m s + + C(n + m +, m By the induction hypothesis applied to (22, we have m C(n ++s, s C(n ++s, s A(m, tc(n + m, m + t t Then our assertion follows from the fact that C(n + m, m + t C(n + m +, m + t C(n + m +, m + t (33 We obtain the triangle consisting of A(m, t (m t 0: The triangle in (33 will be called the alternating Jacobsthal triangle The0 th column is colored in blue (online to indicate the fact that some formulas do not tae entries from this column This is a free offprint provided to the author by the publisher Copyright restrictions may apply

12 76 K-H LEE AND S-J OH Eample 32 For m 3,wehave 3 n + + C(n +3,s2 3 s +C(n +3, 2 C(n +3, +C(n +3, By specializing (32 at m, the th row of alternating Jacobsthal ( triangle n + + and n + th row of Catalan triangle yield the binomial coefficient : Corollary 33 For any n>, we have n + + (34 A(, t C(n +, t t0 Eample 34 8 A(3, 0C(7, 0 + A(3, C(7, + A(3, 2C(7, 2 + A(3, 3C(7, A(3, 0C(8, 0 + A(3, C(8, + A(3, 2C(8, 2 + A(3, 3C(8, A(4, 0C(8, 0+A(4, C(8, +A(4, 2C(8, 2+A(4, 3C(8, 3+A(4, 3C(8, Generating function The numbers A(m, t can be encoded into a generating function in a standard way Indeed, from (3, we obtain (35 A(m, t m t ( m A(, t A(m,t A(m 2,t +( m t A(t,t Lemma 35 We have ( ( + t (36 A(m, t m t Proof When t 0,wehave Inductively, when t>0, we have ( ( + t ( + mt ( ( + t ( +2 A(m, 0 m m0 A(m, t m t+ mt Then we obtain (36 from (35 32 Subsequences The alternating Jacobsthal triangle has various subsequences with interesting combinatorial interpretations First, we write ( ( + t a m+,t 2m b m+,t 2m+ m 0 m 0 This is a free offprint provided to the author by the publisher Copyright restrictions may apply

13 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 77 to define the subsequences {a m,t } and {b m,t } of {A(m, t} Thenwehave a m,t A(t +2m 2,t and b m,t A(t +2m,t Clearly, a m,t,b m,t 0 Using (35, we obtain m a m,t a,t + b,t and b m,t b,t + a,t It is easy to see that a n,2 n and b n,2 n Thenwehave n n n(n + n(n a n,3 a,2 + b,2 + n Similarly, b n,3 n b,2 + n a,2 We compute more and obtain a n,4 n(n n(n + 2 n(n + (4n, b n,4 6 n(n + n(n + (4n +5, 6 a n,5 n(n + (2n2 +2n, b n,5 n(n +2 (n Note also that ( + ( t a m+,t 2m + b m+,t 2m+ m 0 m 0 Net, we define B(m, t A(m, m t for m t to obtain the triangle ( Lemma 36 For m t, we have m B(m, t B(, t Proof We use induction on m Whenm t, wehaveb(m, t A(m, 0 Assume that the identity is true for some m t Sincewehave B(m +,ta(m +,m+ t A(m, m t A(m, m + t B(m, t B(m, t, t This is a free offprint provided to the author by the publisher Copyright restrictions may apply

14 78 K-H LEE AND S-J OH we obtain B(m +,t+ B(, t B(m, t B(m, t + B(, t t m t B(, t B(m, t + B(, t t by the induction hypothesis Using Lemma 36, one can derive the following formulas: B(n, 0 A(n, n andb(n, A(n, n 2 n, n(n 5 B(n, 2 A(n, n 2 4 +, 2 B(n, 3 A(n, n 3 8 n(n2 9n We consider the columns of the triangle (37 and let c m,t ( t B(m+t+,t for each t for convenience Then the sequences (c m,t m for the first several t s appear in the OEIS Specifically, we have: (c m,2 (2, 4, 7,, 6, 22, corresponds to A00024, (c m,3 (2, 6, 3, 24, 40, 62, corresponds to A003600, (c m,4 (3, 9, 22, 46, 86, 48, corresponds to A22378, (c m,5 (3, 2, 34, 80, 66, 34, corresponds to A257890, (c m,6 (4, 6, 50, 30, 296, 60, corresponds to A Diagonal sums As we will see in this subsection, the sums along lines of slope in the alternating Jacobsthal triangle are closely related to Fibonacci numbers We begin with fiing a notation For s 0, define B s A(m, t t+m 2s, t>0 Using the generating function (36, we have 2t 2 F ( : ( ( + t B s s t Then we obtain ( 2 2 t 2 F ( t and the formula (38 F ( ( ( + 2 It is nown that the function F ( is the generating function of the sequence of the alternating sums of the Fibonacci numbers; precisely, we get s+ (39 B s ( Fib(+( s Fib(s (s 0, where (Fib(s s 0 is the Fibonacci sequence (See A9282 in OEIS From the construction, the following is obvious: B s+ B s + B s +(s and B 0,B 0 t This is a free offprint provided to the author by the publisher Copyright restrictions may apply

15 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 79 4 q-deformation In this section, we study a q-deformation of the Fibonacci and Jacobsthal numbers by putting the parameter q into the identities and generating functions we obtained in the previous section We also obtain a family of generating functions of certain sequences by epanding the q-deformation of the generating function of the numbers A(m, t intermsofq 4 q-fibonacci numbers For s 0, define B s (q A(m, tq m t Z[q] t+m 2s, t>0 Then we obtain 2t 2 F (, q : ( q( + q t t B s (q s ( q( + q 2 Note that B s+ (q qb s (q+b s (q+q s (s and B 0 (q,b (q 0 Motivated by (39, we define a q-analogue of Fibonacci number by B s (q :( s B s (q+( s+ q s A(m, t q m t +( s+ q s t+m 2s, t>0 In particular, we have B (q q, B2 (q, B3 (q q 3 + q, B4 (q 2q 2 +, B5 (q q 5 +2q 3 +2q, B 6 (q 3q 4 +4q 2 +, B7 (q q 7 +3q 5 +6q 3 +3q, B8 (q 4q 6 +9q 4 +7q 2 + These polynomials can be readily read off from the alternating Jacobsthal triangle (33 We observe that B 2s (q Z 0 [q 2 ]and B 2s+ (q Z 0 [q 2 ]q and that B s (q is wealy unimodal Note that we have F (, q q ( q( + q 2 q 2 q ( q( + q 2 B s (q s (q s B s (q( s Thus the generating function CF(, q of B s (q isgivenby CF(, q: B s (q s 2 (4 + q ( + q( q 2 Remar 4 The well-nown Fibonacci polynomial F s (q can be considered as a different q-fibonacci number whose generating function is given by 2t 2 ( q t q 2 F s (q s t This is a free offprint provided to the author by the publisher Copyright restrictions may apply

16 80 K-H LEE AND S-J OH Recall that the polynomial F s (q can be read off from the Pascal triangle When q 2,thenumberF s (2 is nothing but the s th Pell number On the other hand, it does not appear that the sequence ( B s (2 s (2,, 0, 9, 52, 65, 278, 429, 520, has been studied in the literature 42 q-jacobsthal numbers Recall that the Jacobsthal numbers J m are defined recursively by J m J m +2J m 2 with J andj 2 Then the Jacobsthal sequence (J m isgivenby (,, 3, 5,, 2, 43, 85, 7, Consider the function (42 Define Q(, q : H m (q : t t ( q( + q t A(m, tq m t for m t For eample, we can read off H 5 (q q 4 2q 3 +4q 2 3q + from the alternating Jacobsthal triangle (33 Using (36, we obtain Q(, q A(m, tq m t m t mt m t A(m, tq m t m H m (q m m A standard computation also yields (43 Q(, q ( q(+(q By taing q 0orq, the equation (43 becomes hand, by taing q, the equation (43 becomes ( + ( 2, On the other which is the generating function of the Jacobsthal numbers J m That is, we have H m (0 H m ( for all m, H m ( is the m th Jacobsthal number J m for each m Since J m H m ( m t A(m, t, we see that an alternating sum of the entries along a row of the triangle (33 is equal to a Jacobsthal number Moreover, we have a natural q-deformation J m (q of the Jacobsthal number J m, which is defined by J m (q :H m ( q A(m, t q m t For eample, we have J 3 (q q 2 + q +, J 4 (q 2q 2 +2q +, J 5 (q q 4 +2q 3 +4q 2 +3q + t This is a free offprint provided to the author by the publisher Copyright restrictions may apply

17 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 8 Note that J m (q is wealy unimodal We also obtain Q(, q ( + q( (q + J m (q m m The following identity is well-nown ([4, 5]: (m /2 m r J m 2 r r Hence we have J m (m /2 r0 ( m r r r0 2 r A(m, t H m ( J m ( Remar 42 In the literature, one can find different Jacobsthal polynomials See [7], for eample 43 A family of generating functions Now let us epand Q(, q with respect to q to define the functions L l (: t Q(, q ( q( + q t L l (q l t Lemma 43 For l 0, we have L l+ ( L l(+ l+2 Proof Clearly, we have L 0 ( n Weseethat n t + L l+ (q l+ Q(, q l0 On the other hand, we obtain + l0 q This completes the proof l0 { L l(q l+ + (ql+ ( q( + (q } ( q( + (q + Q(, q ( q( + (q Using Lemma 43, we can compute first several L l (: L 0 ( n, n L ( 2 ( ( 2 n n+2, L 2 ( n 4 ( ( + 2 ( 3 q q This is a free offprint provided to the author by the publisher Copyright restrictions may apply

18 82 K-H LEE AND S-J OH L 3 ( 4 ( + 2 ( ( ( 4 One can chec that L 2 ( is the generating function of the sequence A00024 in OEIS and that L 3 ( is the generating function of the sequence A Note that the lowest degree of L l ( in the power series epansion is larger than or equal to l + More precisely, the lowest degree of L l ( isl ++δ(l (mod 2 5 -analogue of q-deformation In this section, we consider -analogues of the q-deformations we introduced in the previous section This construction, in particular, leads to a -analogue of the alternating Jacobsthal triangle for each Z \{0} Specializations of this construction at some values of and q produce interesting combinatorial sequences Define A (m, t by A (m, 0 m/2 and A (m, t A (m,t A (m,t Then we have ( 2 ( + t mt A (m, t m t in the same way as we obtained (36 As in Section 42, we also define H,m (q A (m, tq m t We obtain the generating function Q (, q ofh,m (q by t Q (, q : ( q 2 2 ( + q t ( + q ( q 2 2 ( + (q H,m (q m t In particular, when q,wehave Q (, t m t ( + ( 2 ( + t 2 m ( 2m + 2m m Note that H,m ( (m /2 Moreover, the triangle given by the numbers A (m, t can be considered as a -analogue of the alternating Jacobsthal triangle (33 See the triangles (5 and (52 Thus we obtain infinitely many triangles as varies in Z \{0} Similarly, we define a -analogue of the q-jacobsthal number by J,m (q :H,m ( q, and the number J,m ( m t A (m, t canbeconsideredasthe-analogue of the m th Jacobsthal number This is a free offprint provided to the author by the publisher Copyright restrictions may apply

19 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 83 For eample, if we tae 2, the polynomial H 2,m (q can be read off from the following triangle consisting of A 2 (m, t: ( We have J 2,m ( m t A 2(m, t, and the sequence (J 2,m ( m (,, 4, 6, 6, 28, 64, 20, appears as A00779 in OEIS As mentioned in the introduction, this sequence has the interpretation as the numbers of equal dual pairs of some integrals studied in [3] (See Table on p365 in [3] Define B (m, t A (m, m t Then we obtain the following sequences from (5 which appear in OEIS: (B 2 (m, 2 m 3 (2, 3, 5, 8, 2, 7, 23, 30, A002856,A52948, ( B 2 (m, 3 m 5 (3, 8, 6, 28, 45, 68, A We also consider diagonal sums and find the positive diagonals ( A 2 (m, t corresponds to m+t2s, t>0 s (, 3, 8, 2, 55, 44, 377, (Fib(2s, where Fib(s is the Fibonacci number; the negative diagonals A 2 (m, corresponds to m+2s+,t>0 s (0,, 5, 8, 57, 69, A25809, whose s th entry is the number of Dyc paths of length 2(s+ and height 3 This is a free offprint provided to the author by the publisher Copyright restrictions may apply

20 84 K-H LEE AND S-J OH (52 Similarly, when, we obtain the following triangle consisting of A (m, t: We find some meaningful subsequences of this triangle and list them below ( A (m, 4 m 4 (, 3, 5, 7, 0, 4, 8, 22,( ( n 2 /2 n 3 A0848, (B (m, 2 m 5 (2, 5, 9, 4, 20, 27, A22342, ( B ( (m, 3 m 6 (2, 7, 6, 30, 50, 77, A00558, m+t2s, t 2 A (m, t s 2(,, 4, 9, 25, 64, 69, 44,(Fib(n 2 n, ( t2 A (m, t m 2 (, 2, 3, 6, 3, 26, 5, 02, A00790 Define B,s (q andf (, q byb,s (q : A (m, tq m t Z[q] and F (, q : t+m 2s, t>0 2t 2 ( q 2 2 ( + q t +q ( q 2 2 ( + q 2 t B,s (q s s Let us consider the following to define B,s (q: +q ( q 2 2 ( + q 2 +q q 2 2 ( + q( q + 2 ( q 2 2 ( + q 2 Define a -analogue CF (, q of the function CF(, q by CF (, q: ( q(q + 2 ( q 2 2 ( q 2 B,s (q s B,s (q( s The polynomial B,s (q can be considered as a -analogue of the q-fibonacci number B s (q Finally, we define L,l+ ( by t Q (, q ( q 2 2 ( + q t L,l (q l t l0 Then, using a similar argument as in the proof of Lemma 43, one can show that L,l+ ( L,l(+ (l+/2 l+2 This is a free offprint provided to the author by the publisher Copyright restrictions may apply

21 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 85 Remar 5 We can consider the Jacobsthal Lucas numbers and the Jacobsthal Lucas polynomials starting with the generating function +4 ( 2 ( t, and study their (-analogue of q-deformation Acnowledgments We would lie to than Jang Soo Kim for helpful comments The second-named author would lie to than the faculty and staff of the Department of Mathematics at the University of Oregon, where part of this paper was completed In particular, he is thanful to Benjamin Young for helpful discussions on topics related to this paper References [] M Bousquet-Mélou and A Rechnitzer, Lattice animals and heaps of dimers, Discrete Math 258 (2002, no -3, MR [2] G Feinberg and K-H Lee, Fully commutative elements of type D and homogeneous representations of KLR-algebras, JComb6 (205, no 4, MR [3] J Heading, The equality of multiple integrals in the series development of a reflection coefficient for waves governed by a general linear differential equation, JPhysA4 (98, no, MR [4] V E Hoggatt Jr and M Bicnell-Johnson, Convolution arrays for Jacobsthal and Fibonacci polynomials, Fibonacci Quart 6 (978, no 5, MR55833 [5] A F Horadam, Jacobsthal representation numbers, Fibonacci Quart 34 (996, no, MR37475 [6] J S Kim, K-H Lee and S-j Oh, Weight multiplicities and Young tableau through affine crystals, arxiv: [7] T Koshy, Fibonacci and Lucas numbers with applications, Pure and Applied Mathematics (New Yor, Wiley-Interscience, New Yor, 200 MR [8] L W Shapiro, A Catalan triangle, Discrete Math 4 (976, no, MR [9] J R Stembridge, The enumeration of fully commutative elements of Coeter groups, J Algebraic Combin 7 (998, no 3, MR6606 [0] S Tsuchioa, Catalan numbers and level 2 weight structures of A ( p, New trends in combinatorial representation theory, RIMS Kôyûrou Bessatsu, B, Res Inst Math Sci (RIMS, Kyoto, 2009, pp MR [] S Tsuchioa and M Watanabe, Pattern avoidance seen in multiplicities of maimal weights of affine Lie algebra representations, Proc Amer Math Soc 46 (208, no, 5 28 MR37237 Department of Mathematics, University of Connecticut, Storrs, Connecticut address: hlee@mathuconnedu Department of Mathematics, Ewha Womans University, Seoul, , Korea address: sejin092@gmailcom South This is a free offprint provided to the author by the publisher Copyright restrictions may apply