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1 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS arxiv: v2 [mathco] 7 Oct 207 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 Coxeter 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,) for 0 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 0 () Date: October 4, 208 This wor was partially supported by a grant from the Simons Foundation (#38706) This wor was supported by NRF Grant # 206RCB20335

2 2 K-H LEE, 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 express a binomial coefficient More precisely, we prove: Theorem For integers n and 0 n+, we have the identities ( ) n++ (2) C(n,s)2 s C(n+,s)2 max( s,0) and (3) ( ) n/2 n+ C( n/2,)2 max( n/2,0) (n+)/2 0 It is quite intriguing that the two most important families of combinatorial numbers are related in this way By replacing 2-powers with x-powers in the identities, we define Catalan triangle polynomials and mae a conjecture ( on staced ) directed animals studied in [] (see n+ Section 23) It is also interesting that appearing in (3) is exactly the number (n+)/2 of fully commutative, involutive elements of the Coxeter 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 (4) where C n is the n th Catalan number n+3 n C n C(n,)2 n 2, 2 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 expect 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 appear a natural sequence of numbers A(m, t), defined by to yield the following result: A(m,0), A(m,t) A(m,t ) A(m,t),

3 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 3 Theorem 3 For any n > m t, we have ( ) m n++ (5) C(n+m,s)2 m s + In particular, when m, we have ( ) n+ + (6) m A(m,t)C(n+m, m+t) t A(,t)C(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 example, the -analogue J,m (q) of q-deformation of the Jacobsthal numbers is given by the generating function x( qx) ( q 2 x 2 )( (q +)x) J,m (q)x m m x When q and, we recover the usual generating function of the (+x)( 2x) Jacobsthal numbers Interestingly enough, sequences given by special values of this deformation have various combinatorial interpretations For example, 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 example An outline of this paper is as follows In the next section, we prove Theorem to obtain Catalan triangle expansions 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 about q-deformation of the Fibonacci and Jacobsthal numbers The last section is devoted to the

4 4 K-H LEE, S-J OH study of a -analogue of the q-deformation of the Fibonacci and Jacobsthal numbers using the -analogue of the alternating Jacobsthal triangle 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 2 Catalan expansion of binomial coefficients In this section, we prove expressions of binomial coefficients as 2-power weighted sums along rows of the Catalan triangle Catalan trapezoids are introduced for the 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 0 and 0 n, we define the (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 0 and 0 n, C(n,) (n+)!(n +)!(n+)! In particular, we have Theorem 22 [2] For n, we have (22) C n ( ) 2n n+ n n+3 n 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 (22) 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+ (23) C( n/2,s)2 max( n/2 s,0) (n+)/2

5 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 5 ( ) n+ Proof Set Q n : for convenience Assume n 2 for some Z (n+)/2 0 Then we have ( ) 2 + Q 2 + By (22) in Theorem 22, we have C(,s)2 2 s +3 2 C +3 ( ) 2 (24) 2 +2 On the other hand, 2 C(,s)2 2 s C(,s)2 2 s +2C(, ) Note that C(, ) C Multiplying (24) by 4, we have C(,s)2 s C(, ) + Hence C(,s)2 s ( ) 2 5C 2+ + Assume n 2 + for some Z 0 Then we have ( ) 2 +2 Q 2+ + By replacing the in (24) with +, we have (25) ( ) 2 C(+,s)2 s +4 2 C ( ) 2 ( ) 2+ Q + 2 ( ) On the other hand, C(+,s)2 s C(+,s)2 s +2C( +,) Note that C(+,) C + Multiplying (25) by 2, we have Since + C(+,s)2 s +4C( +,) C(+,s)2 max( s,0) +2C( +,) ( ) C( +,) + ( ) ( ) 2+2 Q + 2+,

6 6 K-H LEE, S-J OH our assertion is true in this case as well Corollary 24 For n Z 0, the dual form of (23) holds; ie, ( ) n/2 n+ C( n/2,s)2 n/2 s (n+)/2 In particular, we have the following identity by replacing n with 2n : ( ) n 2n n C(n,s)2 n s (n+s )!(n s) n ( ) 2 2 n s n n s!n! s ( ) 2 + Proof The assertion for n 2 follows from the fact that ( ) ( n 2 + follows from the fact that 2 + From Theorem 23, we have, for n 2 ( Z ), ( ) 2 C(,s)2 max( s,0) Since (26) ( ) 2 + ( ) 2 ( ) and for ) ( ) 2 C(,s)2 s + + ( ) 2, we obtain a new identity: ( ) 2 C(,s)2 s More generally, ( we have) the following identity which is an interesting expression of a binomial coefficient as a 2-power weighted sum of the Catalan triangle along the n n+ + th row Theorem 25 For 0 n+, we have ( ) n+ + (27) min(n,) C(n,s)2 s Proof We will use an induction on n+ The cases when n+,n,n are already proved by Theorem 23, Corollary 24 and (26) Assume that we have the identity (27) for + < n: ( ) n++2 + (28) C(n,s)2 + s +

7 Since C(n, +) n n+ +2 ( n CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 7 ( ) n++2, the identity (28) can be written as + ) n n++2 Now, simplifying the left-hand side ( ) n++2 n ( ) n++2 + n we obtain the desired identity ( ) n++ ( ) n n+ +2 C(n,s)2 + s C(n,s)2 s ( ) n ( n++ ), Example 26 ( ) 7 3 ( ) C(3,s)2 3 s , 3 C(4,s)2 3 s Catalan trapezoid As a generalization of Catalan triangle, we define a Catalan trapezoid by considering a trapezoidal array of numberswith m complete columns (m ) Let the entry in the n th row and th column of the array bedenoted by C m (n,) for 0 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 For example, when m 3, we obtain (29) Alternatively, the numbers C m (n,) can be defined in the following way

8 8 K-H LEE, S-J OH Definition 27 For an integer m, set C (n,) C(n,) for 0 n and C 2 (n,) C(n+,) for 0 n+, and define inductively ( ) n++ if 0 < m, ( ) ( ) (20) 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 ( ) n++ m m (2) C(n,s)2 s C(n+m,s)2 m s + C(n++s, s) Proof By Theorem 25, the equation (20) can be re-written as follows: C(n,s)2 s if 0 < m, C m (n+,) m C(n,s)2 s C(n+m,s)2 m s if m n+m, 0 if n+m < On the other hand, for m n+m, we have ( ) ( ) n++ n+m++ m m Thus we obtain m C(n,s)2 s C(n+m,s)2 m s for m n+m This completes the proof m m C(n++s, s) C(n++s, s) ( By specializing ) ( (2) at ) m, we obtain different expressions of a binomial coefficient n++ n++ as a 2-power weighted sum of the Catalan triangle along the n+ st row (cf (27)) Corollary 29 We have the following identities: For, ( ) n++ C(n,s)2 s C(n+,s)2 s +C(n+,) (22) C(n+,s)2 max( s,0)

9 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 9 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 the identities, we setch a ( ) n++ bijective proof of the first identity C(n,s)2 s Bijective proof We interpret ( n + + ) as the number of the lattice paths from (0,0) to (n+ +,n+ )withup-steps(,) anddown-steps(, ) Forsuchapathλ, let P 0,P,,P s be the intersections of λ with the x-axis, decomposing λ into s+ parts D 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 x-axis 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 Theresultingpath is adyc path from (0,0) to (n+ s,n +s) This gives a 2 s -to- map from the set of the lattice paths from (0,0) to (n++,n+ ) to the set of the Dyc 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, s)2 s C(n,s)2 s 23 Catalan triangle polynomials The identities in the previous subsections naturally give rise to the following definition Definition 2 For 0 n, we define the (n,) th Catalan triangle polynomial F n, (x) by (23) F n, (x) C(n,s)x s C(n,s)x s +C(n,) Note that the degree of F n, is Evaluations of F n, (x) 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 This proof was communicated to us by Jang Soo Kim We than him for allowing us to use his proof

10 0 K-H LEE, S-J OH Let us recall the description of C(n, ) in terms of binomial coefficients: ( ) ( ) n+ n++ (24) C(n,) 2 (2 ) Theorem 22 For any d Z, we have which recovers (24) when d 2 C(n,) df n, (d) (d )F n, (d) Proof Since F n, (d) C(n,s)d 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,) (25) F n, (d) d By multiplying (25) by d, our assertion follows s With the consideration of Corollary 29, we define a natural variation of F n, (x) Definition 23 For 0 n, we define the modified (n,) th Catalan triangle polynomial in the following way: (26) F n, (x) C(n+,s)x max( s,0) Note that the degree of F n, (x) is Evaluations of F n, (x) 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 which recovers (24) when d 2 C(n,) d F n, (d) (d ) F n, (d) 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

11 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS We expect 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 theses 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 example, when n 7, we have the sequence (C(8,), 0 7) equal to and compute (, 8, 35, 0, 275, 572, 00, 430), F 7,7 (3) Alternating Jacobsthal triangle In the previous section, the binomial coefficient ( n++) is written as sums along the n th andthen+ st row of the Catalan triangle, respectively Inthis section, weconsider 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 (3) Define A(m,t) Z recursively for m t 0 by A(m,0), A(m,t) A(m,t ) A(m,t) Here we set A(m,t) 0 when t > m Then, by induction on m, one can see that m 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,s)2 m s + m A(m,t)C(n+m, m+t) Proof We will use an induction on m Z For m, we already proved the identity in Corollary 29 Assume that we have the identity (32) for some m Z By specializing (2) at m+, we have ( ) n++ m m t C(n+m+,s)2 m s + m C(n++s, s) m C(n+m+,s)2 m s + +C(n+m+, m) C(n++s, s)

12 2 K-H LEE, S-J OH By the induction hypothesis applied to (2), we have m m C(n++s, s) A(m,t)C(n+m, m+t) Then our assertion follows from the fact that (33) t C(n+m, m+t) C(n+m+, m+t) C(n+m+, m+t ) We obtain the triangle consisting of A(m,t) (m t 0): The triangle in (33) will be called the alternating Jacobsthal triangle The 0 th column is colored in blue to indicate the fact that some formulas do not tae entries from this column Example 32 For m 3, we have ( ) n+ + 3 C(n+3,s)2 3 s +C(n+3, 2) C(n+3, )+C(n+3,) By specializing (32) at m, the th row of alternating ( Jacobsthal ) triangle and n+ th n++ row of Catalan triangle yield the binomial coefficient : Corollary 33 For any n >, we have ( ) n++ (34) A(,t)C(n+,t) Example 34 ( ) 8 A(3,0)C(7,0) +A(3,)C(7,) +A(3,2)C(7,2) +A(3,3)C(7,3) ( ) 9 A(3,0)C(8,0) +A(3,)C(8,) +A(3,2)C(8,2) +A(3,3)C(8,3) 3 t ( ) 9 A(4,0)C(8,0) +A(4,)C(8,) +A(4,2)C(8,2) +A(4,3)C(8,3) +A(4,3)C(8,3) 4

13 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 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) Lemma 35 We have (36) m t Proof When t 0, we have t > 0, we have ( x)(+x) t (+x) Then we obtain (36) from (35) ( ) m A(,t ) A(m,t ) A(m 2,t ) +( ) m t A(t,t ) ( x)(+x) t A(m,t)x m t mt x +x+x2 + m0 A(m,0)xm Inductively, when ( x)(+x) t ( x+x2 ) mt A(m,t )x m t+ 32 Subsequences The alternating Jacobsthal triangle has various subsequences with interesting combinatorial interpretations First, we write ( x)(+x) t a m+,t x 2m b m+,t x 2m+ m 0 m 0 to define the subsequences {a m,t } and {b m,t } of {A(m,t)} Then we have 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 m a m,t a,t + b,t and b m,t m b,t + It is easy to see that a n,2 n and b n,2 n Then we have n n a n,3 a,2 + b,2 n(n+) + n(n ) 2 2 Similarly, b n,3 n b,2 + We compute more and obtain n a n,4 n(n+)(4n ) 6 a,2 n(n+) 2 + n(n+) 2 m a,t n 2 n(n+), b n,4 n(n+)(4n+5), 6

14 4 K-H LEE, S-J OH a n,5 n(n+)(2n2 +2n ), b n,5 n(n+)2 (n+2) 6 3 Note also that (+x)( x) t a m+,t x 2m + m+,t x m 0 m 0b 2m+ Next, we define to obtain the triangle (37) Lemma 36 For m t, we have B(m,t) A(m,m t) for m t m B(m,t) B(,t ) Proof We use induction on m When m t, we have B(m,t) A(m,0) Assume that the identity is true for some m t Since we have we obtain m t t B(m,t) A(m,m t) A(m,m t ) A(m,m t) B(m,t) B(m,t ), B(,t )+B(m,t) by the induction hypothesis m t m 2 t B(,t )+B(m,t) B(m,t ) B(,t )+B(m,t) Using Lemma 36, one can derive the following formulas: B(n,0) A(n,n) and B(n,) A(n,n ) 2 n, B(n,2) A(n,n 2) 4+ n(n 5), 2 B(n,3) A(n,n 3) 8 n(n2 9n+32) 6

15 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 5 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 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 fixing a notation For s 0, define B s A(m,t) t+m 2s, t>0 Using the generating function (36), we have x 2t 2 F(x) : ( x)(+x) t B s x s Then we obtain and the formula (38) t ( x)x 2 F(x) F(x) ( x 2 t +x ) t ( x)(+x x 2 ) x 2 +x x 2 It is nown that the function F(x) is the generating function of the sequence of thealternating 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 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 expanding the q-deformation of the generating function of the numbers A(m,t) in terms of q

16 6 K-H LEE, S-J OH 4 q-fibonacci numbers For s 0, define B s (q) A(m,t)q m t Z[q] Then we obtain Note that F(x,q) : t+m 2s, t>0 t x 2t 2 ( qx)(+qx) t B s (q)x s ( qx)(+qx x 2 ) 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 In particular, we have t+m 2s, t>0 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(x,q) qx ( qx)(+qx x 2 ) qx x 2 qx ( qx)(+qx x 2 ) B s (q)x s (qx) s B s (q)( x) s Thus the generating function CF(x,q) of B s (q) is given by CF(x,q): B s (q)x s x 2 (4) +qx (+qx)( qx x 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 x 2t 2 ( qx) t qx x 2 F s (q)x s t Recall that the polynomial F s (q) can be read off from the Pascal triangle When q 2, the number F 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

17 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 7 42 q-jacobsthal numbers Recall that the Jacobsthal numbers J m are defined recursively by J m J m +2J m 2 with J and J 2 Then the Jacobsthal sequence (J m ) is given by (,,3,5,,2,43,85,7,) (42) Consider the function Define For example, we can read off Q(x,q) : H m (q) : t x t ( qx)(+qx) t m A(m,t)q m t for m t H 5 (q) q 4 2q 3 +4q 2 3q + from the alternating Jacobsthal triangle (33) Using (36), we obtain Q(x,q) A(m,t)q m t x m t mt m t A standard computation also yields (43) Q(x, q) m A(m,t)q m t x m H m (q)x m m x ( qx)(+(q )x) By taing q 0 or q, the equation (43) becomes q, the equation (43) becomes x (+x)( 2x), x On the other hand, by taing x 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 m J m (q) : H m ( q) A(m,t) q m t For example, 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

18 8 K-H LEE, S-J OH Note that J m (q) is wealy unimodal We also obtain Q(x, q) x (+qx)( (q +)x) 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 J m (q)x m m A(m,t) H m ( ) J m () Remar 42 In the literature, one can find different Jacobsthal polynomials See [7], for example 43 A family of generating functions Now let us expand Q(x,q) with respect to q to define the functions L l (x): x t Q(x,q) ( qx)(+qx) t L l (x)q l Lemma 43 For l 0, we have Proof Clearly, we have L 0 (x) t t l0 L l+ (x) x x L l(x)+ xl+2 x n x n x We see that x x x + L l+ (x)q l+ Q(x,q) l0 On the other hand, we obtain x x + { x x L l(x)q l+ + x } x (qx)l+ l0 x x qx x This completes the proof x ( qx)(+(q )x) x ( qx)(+(q )x) + x x x ( qx)(+(q )x) Q(x,q) Using Lemma 43, we can compute first several L l (x): L 0 (x) x n x x, n qx qx

19 CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 9 L (x) x2 ( x) 2 + x2 x x3 ( x) 2 nx n+2, L 2 (x) n x 4 ( x) 3 + x3 x x3 ( x+x 2 ) ( x) 3 L 3 (x) x4 ( x+x 2 ) ( x) 4 + x4 x x5 (2 2x+x 2 ) ( x) 4 One can chec that L 2 (x) is the generating function of the sequence A00024 in OEIS and that L 3 (x) is the generating function of the sequence A Note that the lowest degree of L l (x) in the power series expansion is larger than or equal to l + More precisely, the lowest degree of L l (x) is l++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 Then we have A (m,0) m/2 and A (m,t) A (m,t ) A (m,t) ( x 2 )(+x) t A (m,t)x m t in the same way as we obtained (36) As in Section 42, we also define m H,m (q) A (m,t)q m t t We obtain the generating function Q (x,q) of H,m (q) by x t Q (x,q) : ( q 2 x 2 )(+qx) t x(+qx) ( q 2 x 2 )(+(q )x) H,m (q)x m In particular, when q, we have x t x(+x) Q (x,) ( x 2 )(+x) t x 2 m ( x 2m +x 2m) t 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), mt m

20 20 K-H LEE, S-J OH and the number J,m () m t A (m,t) can be considered as the -analogue of the m th Jacobsthal number For example, if we tae 2, the polynomial H 2,m (q) can be read off from the following triangle consisting of A 2 (m,t): (5) 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 corresponds m+t2s, t>0 A 2 (m,t) s (,3,8,2,55,44,377,) (Fib(2s)), where Fib(s) is the Fibonacci number; the negative diagonals corresponds m+2s+, t>0 A 2 (m,) s (0,,5,8,57,69,) A25809,

21 (52) CATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS 2 whose s th entry is the number of Dyc paths of length 2(s+) and height 3 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 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) and F (x,q) by B,s (q) : A (m,t)q m t Z[q] and F (x,q) : t t+m 2s, t>0 x 2t 2 ( q 2 x 2 )(+qx) t B,s (q)x s s Let us consider the following to define B,s (q): +qx ( q 2 x 2 )(+qx x 2 ) +qx ( q 2 x 2 )(+qx x 2 ) +qx q 2 x 2 (+qx)( qx+x 2 ) ( q 2 x 2 )(+qx x 2 ) Define a -analogue CF (x,q) of the function CF(x,q) by CF (x,q): ( qx)(qx+x 2 ) ( q 2 x 2 )( qx x 2 ) B,s (q)x s B,s (q)( x) 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+ (x) by x t Q (x,q) ( q 2 x 2 )(+qx) t L,l (x)q l t l0

22 22 K-H LEE, S-J OH Then, using a similar argument as in the proof of Lemma 43, one can show that L,l+ (x) x x L,l(x)+ (l+)/2 x l+2 x Remar 5 We can consider the Jacobsthal Lucas numbers and the Jacobsthal Lucas polynomials starting with the generating function +4x ( x 2 )( x) t, and study their (-analogue of) q-deformation References [] M Bousquet-Mélou and A Rechnitzer, Lattice animals and heaps of dimers, Discrete Math 258 (2002), [2] G Feinberg and K-H Lee, Fully commutative elements of type D and homogeneous representations of KLR-algebras, J Comb 6 (205), no 4, [3] J Heading, Theorem relating to the development of a reflection coefficient in terms of a small parameter, J Phys A: Math Gen 4 (98), [4] V E Hoggatt Jr and M Bicnell-Johnson, Convolution arrays for Jacobsthal and Fibonacci polynomials, Fibonacci Quart 6 (978), no 5, [5] A F Horadam, Jacobsthal Representation Numbers, Fibonacci Quart 34 (996), no, [6] J S Kim, K-H Lee and S-j Oh, Weight multiplicities and Young tableaux through affine crystals, arxiv: [7] T Koshy, Fibonacci and Lucas numbers with applications, Pure and Applied Mathematics, Wiley- Interscience, New Yor, 200 [8] L W Shapiro, A Catalan triangle, Discrete Math 4 (976), [9] J R Stembridge, The enumeration of fully commutative elements of Coxeter groups, J Algebraic Combin 7 (998), no 3, [0] S Tsuchioa, Catalan numbers and level 2 weight structures of A () p, New trends in combinatorial representation theory, 45 54, RIMS Kôyûrou Bessatsu, B, Res Inst Math Sci (RIMS), Kyoto, 2009 [] S Tsuchioa and M Watanabe, Pattern avoidances seen in multiplicities of maximal weights of affine Lie algebra representations, preprint, arxiv: Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA address: hlee@mathuconnedu Department of Mathematics, Ewha Womans University, Seoul, , South Korea address: sejin092@gmailcom