Some new binomial sums related to the Catalan triangle

In this paper, we derive many new identities on the classical Catalan triangle C = (Cn,k)n>k>0, where Cn,k = k+1 n+1 ( 2n−k n ) are the well-known ballot numbers. The first three types are based on the determinant and the fourth is relied on the permanent of a square matrix. It not only produces many known and new identities involving Catalan numbers, but also provides a new viewpoint on combinatorial triangles.


Introduction
In 1976, by a nice interpretation in terms of pairs of paths on a lattice Z 2 , Shapiro [44] first introduced the Catalan triangle B = (B n,k ) n k 0 with B n,k = k+1 n+1 2n+2 n−k and obtained where is the nth Catalan number.Table 1.1 illustrates this triangle for small n and k up to 6.Note that the entries in the first column of the Catalan triangle B are indeed the Catalan numbers B n,0 = C n+1 , which is the reason why B is called the Catalan triangle.Since then, much attentions have been paid to the Catalan triangle and its generalizations.In 1979, Eplett [20] deduced the alternating sum in the nth row of B, namely, In 1981, Rogers [42] proved that a generalization of Eplett's identity holds in any renewal array.In 2008, Gutiérrez et al. [27] established three summation identities and proposed as one of the open problems to evaluate the moments Ω m = n k=0 (k + 1) m B 2 n,k .Using the WZ-theory (see [40,52]), Miana and Romero computed Ω m for 1 m 7. Later, based on the symmetric functions and inverse series relations with combinatorial computations, Chen and Chu [13] resolved this problem in general.By using the Newton interpolation formula, Guo and Zeng [26] generalized the recent identities on the Catalan triangle B obtained by Miana and Romero [37] as well as those of Chen and Chu [13].
Some alternating sum identities on the Catalan triangle B were established by Zhang and Pang [53], who showed that the Catalan triangle B can be factorized as the product of the Fibonacci matrix and a lower triangular matrix, which makes them build close connections among C n , B n,k and the Fibonacci numbers.Motivated by a matrix identity related to the Catalan triangle B [46], Chen et al. [14]  the electronic journal of combinatorics 21(1) (2014), #P1.33 Aigner [3], in another direction, came up with the admissible matrix, a kind of generalized Catalan triangle, and discussed its basic properties.The numbers in the first column of the admissible matrix are called Catalan-like numbers, which are investigated in [5] from combinatorial views.The admissible matrix A = (A n,k ) n k 0 associated to the Catalan triangle B is defined by A n,k = 2k+1 2n+1 2n+1 n−k , which is considered by Miana and Romero [38] by evaluating the moments Φ m = n k=0 (2k + 1) m A 2 n,k .Table 1.2 illustrates this triangle for small n and k up to 6.
The interlaced combination of the two triangles A and B forms the third triangle C = (C n,k ) n k 0 , defined by the ballot numbers The triangle C is also called the "Catalan triangle" in the literature, despite it has the most-standing form C = (C n,n−k ) n k 0 first discovered in 1961 by Forder [24], see for examples [1,6,9,23,30,38,47].Three relations, C n,0 = C n , C n+1,1 = C n+1 and n k=0 C n,k = C n+1 bring the Catalan numbers and the ballot numbers in correlation [4,29,43].Many properties of the Catalan numbers can be generalized easily to the ballot numbers, which have been studied intensively by Gessel [25].The combinatorial interpretations of the ballot numbers can be found in [5,8,10,11,14,18,19,21,22,28,31,35,39,41,44,50,51].It was shown by Ma [33] that the Catalan triangle C can be generated by context-free grammars in three variables.
The Catalan triangles B and C often arise as examples of the infinite matrix associated to generating trees [7,12,34,36].In the theory of Riordan arrays [45,46,49], much interest has been taken in the three triangles A, B and C, see [2,14,15,16,17,32,34,48,51].In fact, A, B and C are Riordan arrays A = (C(t), tC(t) 2 ), B = (C(t) 2 , tC(t) 2 ), and C = (C(t), tC(t)), the electronic journal of combinatorics 21(1) (2014), #P1.33 where is the generating function for the Catalan numbers C n .Recently, Sun and Ma [51] studied the sums of minors of second order of M = (M n,k (x, y)) n k 0 , a class of infinite lower triangles related to weighted partial Motzkin paths, and obtained the following theorem.Recall that a partial Motzkin path is a lattice path from (0, 0) to (n, k) in the XOYplane that does not go below the X-axis and consists of up steps u = (1, 1), down steps d = (1, −1) and horizontal steps h = (1, 0).A weighted partial Motzkin path (not the same as stated in [14]) is a partial Motzkin path with the weight assignment that all up steps and down steps are weighted by 1, the horizontal steps are endowed with a weight x if they are lying on X-axis, and endowed with a weight y if they are not lying on X-axis.The weight w(P ) of a path P is the product of the weight of all its steps.The weight of a set of paths is the sum of the total weights of all the paths.Denote by M n,k (x, y) the weight sum of the set M n,k (x, y) of all weighted partial Motzkin paths ending at (n, k).
Table 1.4 illustrates few values of M n,k (x, y) for small n and k up to 4 [51].The triangle M can reduce to A, B and C when the parameters (x, y) are specalized, namely, In this paper, we derive many new identities on the Catalan triangle C. The first three types are special cases derived from (2) which are presented in Section 2 and 3 respectively.Section 4 is devoted to the fourth type based on the permanent of a square matrix, and gives a general result on the triangle M in the x = y case.It not only produces many known and new identities involving Catalan numbers, but also provides a new viewpoint on combinatorial triangles.

The first two operations on the Catalan triangle
Let X = (X n,k ) n k 0 and Y = (Y n,k ) n k 0 be the infinite lower triangles defined on the Catalan triangle C respectively by Table 2.1 and 2.2 illustrate these two triangles X and Y for small n and k up to 5, together with the row sums.It indicates that the two operations contact the row sums of X and Y with the first two columns of C.More generally, we obtain the first result which is a consequence of Theorem 1.1.
Taking m = n in ( 5) and ( 6), we have which yield the following results.
It should be pointed out that both (8) and ( 9) are still correct for any integer −1 if one notices that they hold trivially for any integer > n and both sides of them can be transferred into polynomials in .Specially, after shifting n to n − 1, the case = −1 in (8) and ( 9) generates the following corollary.
Corollary 4. For any integer n 1, there hold 3 The third operation on the Catalan triangle ) n k 0 be the infinite lower triangle defined on the Catalan triangle C by .
Note that a weighted partial Motzkin path with no horizontal steps is just a partial Dyck path.Then the relation C n,k = M 2n−k,k (0, 0) signifies that C n,k counts the set C n,k of partial Dyck paths of length 2n − k from (0, 0) to (2n − k, k) [35].Such partial Dyck paths have exactly n up steps and n − k down steps.For any step, we say that it is at level i if the y-coordinate of its end point is i.
For k = 0, a partial Dyck path is an (ordinary) Dyck path.For any Dyck path P of length 2n + 2m + 2, its (2n + 1)-th step L (along the path) must end at odd level, say 2k + 1 for some k 0, then P can be uniquely partitioned into P = P 1 LP 2 , where Hence, the cases p = 0, 1 and 2, i.e., m = n − 1, n and n + 1 in (11) produce the following corollary.

The fourth operation on the Catalan triangle
Let W = (W n,k ) n k 0 be the infinite lower triangle defined on the Catalan triangle by where per(A) denotes the permanent of a square matrix A. This, in general, motivates us to consider the permanent operation on the triangle M = (M n,k (x, y)) n k 0 .Recall that M n,k (x, y) is the weight sum of the set M n,k (x, y) of all weighted partial Motzkin paths ending at (n, k).For any step of a partial weighted Motzkin path P , we say that it is at level i if the y-coordinate of its end point is i.For 1 i k, an up step u of P at level i is R-visible if it is the rightmost up step at level i and there are no other steps at the same level to its right.
Theorem 7.For any integers m, n, r with m n 0, there holds where the electronic journal of combinatorics 21(1) (2014), #P1.33 Proof.We just give the proof of the part when r 0, the other part can be done similarly and is left to interested readers.Define n,m,k to be the subset of A n,m,k such that for any (P, It is easily to see that the weights of the sets For 0 i < r, the weight of the set i k=0 C (r,i) n,m,k is M n+i,0 (y, y)M m+r−i−1,0 (y, y).This claim can be verified by the following argument.For any (P, y, y) such that the last (i + 1)-th step of P Q 1 is at level k.Summing k for 0 k i, all P Q 1 ∈ M n+i,0 (y, y) contribute the total weight M n+i,0 (y, y) and all Q 2 ∈ M m+r−i−1,0 (y, y) contribute the total weight M m+r−i−1,0 (y, y).Hence, w( n,m,k , P Q is exactly an element of M m+n+r,1 (y, y).Note that in this case, the first R-visible up step of P is still the one of P Q and it is at most the (n + r)-th step of P Q.
For any (P, where Q 1 ∈ M j,k (y, y) for some j r, then P Q 1 u * Q 2 forms an element of M m+n+r,1 (y, y).Note that in this case, the last R-visible up step u * of Q is still the one of P Q 1 u * Q 2 .Moreover, the u * step is at least the (n + r + 1)-th step of Conversely, for any path in M m+n+r,1 (y, y), it can be partitioned uniquely into P Q, where P ∈ M n+r,k (y, y) for some k 0. If the unique R-visible up step u * of P Q is lying in P , then k 1 and (P, Q) ∈ B (r) n,m,k−1 ; If the u * step is lying in Q, P Q can be repartitioned into P 1 P 2 u * Q 1 with P 1 ∈ M n,j (y, y) for some j 0, then (P n,m,j .Clearly, the above procedure is invertible.Hence, ϕ is indeed a bijection as desired and ( 12) is proved. where Proof.To prove (13), replacing n, m, r respectively by 2n, 2m, 2p − 1 and setting (y, y) = (0, 0) in ( 12), together with the relation C n,k = M 2n−k,k (0, 0) and ( 7), we have , as desired.
The case p = 0 in ( 13) and ( 14), after some routine computation, gives Corollary 9.For any integers m n 1, there hold Specially, the m = n case produces The cases p = 1 in ( 13) and p = −1 in ( 14), replacing n and m in ( 14) by n + 1 and m + 1, after some routine computation, yield Subtracting ( 16) from (15), after some routine simplification, one gets The case p = 0 in (17), after some routine computation, generates

Corollary 6 .
For any integer n 0, according to the (2n + 1)-th step u or d, we have (a) The number of Dyck paths of length 4n is bisected; (b) The parity of the number of Dyck paths of length 4n + 2 is C 2 n ; (c) The parity of the number of Dyck paths of length 4n + 4 is 2C n C n+1 .

Table 1 .
1.The first values of B n,k .

Table 1 .
derived many nice matrix identities on weighted partial Motzkin paths.2. The first values of A n,k .

Table 1
.3 illustrates this triangle for small n and k up to 7.

Table 1 .
3. The first values of C n,k .

Table 2 .
1.The first values of X n,k .

Table 2 .
2. The first values of Y n,k .

Table 3 .
1 illustrates the triangle Z for small n and k up to 6, together with the row sums and the alternating sums of rows.It signifies that the sums and the alternating sums of rows of Z are in direct contact with the first column of C. Generally, we have the second result which is another consequence of Theorem 1.1.

Table 3 .
1.The first values of Z n,k .Theorem 5.For any integers m, n 0, let p = m − n + 1.Then there hold min{m,n} k=0

Table 4 .
1 illustrates the triangle W for small n and k up to 8, together with the row sums.

Table 4 .
1.The first values of W n,k .