On Total Positivity of Catalan-stieltjes Matrices

Recently Chen-Liang-Wang (Linear Algebra Appl. 471 (2015) 383–393) present some sufficient conditions for the total positivity of Catalan-Stieltjes matrices. Our aim is to provide a combinatorial interpretation of their sufficient conditions. More precisely, for any Catalan-Stieltjes matrix A we construct a digraph with a weight, which is positive under their sufficient conditions, such that every minor of A is equal to the sum of the weights of families of nonintersecting paths of the digraph. We have also an analogous result for the minors of a Hankel matrix associated to the first column of a Catalan-Stieltjes matrix.


Introduction
The study of totally positive matrices appears in various areas such as orthogonal polynomials, combinatorics, algebraic geometry, stochastic processes, game theory, differential equations, representation theory, Brownian motion, electrical networks, and chemistry; see [13,5,6,19,9].In this paper we shall consider the total positivity of some special lower triangular matrices.Recall that an infinite real matrix M is said to be totally positive (TP) if every minor of M is nonnegative.Definition 1.1.An infinite lower triangular matrix of real numbers A γ,σ,τ := (a n,k ) n,k 0 is called a Catalan-Stieltjes matrix if there are three sequences of positive numbers γ := (r k ) k 0 , σ := (s k ) k 0 and τ := (t k ) k 1 such that a n,0 = s 0 a n−1,0 + t 1 a n−1,1 ; a n,k = r k−1 a n−1,k−1 + s k a n−1,k + t k+1 a n−1,k+1 (k 1, n 1), (1.1) where a 0,0 = 1 and a n,k = 0 unless n k 0.
the electronic journal of combinatorics 23(4) (2016), #P4.33 Although a matrix defined by (1.1) is called a Catalan matrix in [2, p. 291], we prefer to call it a Catalan-Stieltjes matrix because, when r k = 1, Stieltjes [23] first introduced such a matrix in his step-by-step method to expand a continued J-fraction into a power series; see [24, Section 53] and [10].Indeed, the matrix (1.1) implies the following continued fraction expansion of the ordinary generating function of the first column of A γ,σ,τ : where λ k+1 = r k t k+1 for k 0. The sequence (a n,0 ) is usually called a moment sequence in the theory of orthogonal polynomials (see [26]).Conversely, starting from (1.2) with nonnegative s k and λ k+1 (k 0), and any factorisation r k t k+1 of λ k+1 such that r k , t k+1 0 (k 0), we can recover the moment sequence (a n,0 ) using matrix (1.1).
Recently Chen-Liang-Wang [7] proved some sufficient conditions for the total positivity of Catalan-Stieltjes matrices.At the end of their paper they asked for a combinatorial interpretation of their results.The aim of this paper is to present such a combinatorial interpretation using a classical lemma of Lindström [16].As in [5,12,11], our strategy is to first interpret the matrix A as a path matrix of some planar network within two boundary vertex sets, and then apply Lindström's lemma [16] to write every minor of A as a sum of positive weights of families of nonintersecting paths.We first recall some basic definitions of this methodology.Let G = (V, E) be an infinite acyclic digraph, where V is the vertex set and E the edge set.If S := (A i ) i 0 and T := (B i ) i 0 are two sequences of vertices in G, we say that the triple (G, S, T ) is a network.We assume that there is a weight function w : E → R and define the associated path matrix M = (m i,j ) i,j 0 by where the sum is over all the paths γ from A i to B j and the weight of a path is the product of its edge weights.By convention we define m i,j = 1 if A i = B j .
Let I = (i 1 , . . ., i n ) and J = (j 1 , . . ., j n ) be two positive increasing integer sequences.The I, J minor of a matrix M is defined by det M I,J , where M I,J is the submatrix of M corresponding to row set I and column set J. Let A I = {A i : i ∈ I}, B J = {B j : j ∈ J} be two n-sets of vertices of G, which need not be disjoint.For any permutation σ ∈ S n denote by N (G; A I , B σ(J) ) the set of n-tuples (p 1 , . . ., p n ) where p i is a path from A i k to B j σ(k) such that any two paths in the tuple are vertex-disjoint.The weight of P is defined by w(P ) = n i=1 w(p i ).The following result is due to Lindström's [16].See also [12,2].
The basic idea is to find a planar graph G along with two sequences of vertices S and T so that N (G; A I , B σ(J) ) is empty except when σ is identity.As there is no general method for constructing a simple planar network in order to prove that a given matrix is totally positive, to motivate our approach, we will start with the Motzkin path description of the matrix coefficients a n,k in (1.1).Consider the digraph M = (V, E), where V = Z × N and A path in M is called a Motzkin path.An example of two nonintersecting Motzkin paths in M is depicted in Figure 1.1.It is well-known and easy to verify (see [10]) that the coefficient a n,k is equal to the sum of weights of Motzkin paths from A n := (n, 0) to B k := (0, k), where the arrows are weighted as follows: w((i + 1, j) → (i, j + 1)) = r i , w((i + 1, j) → (i, j)) = s j , w((i + 1, j + 1) → (i, j)) = t j+1 , for i 1, j 0. It follows that the Catalan-Stieltjes matrix A is a path matrix of the Motzkin network (M, (A i ), (B i )).Unfortunately, the signed expression (1.3) does not manifest any positivity for det A I,J in general, except for the special case t k = 0; see Proposition 3.1.Actually it is easy to see that M is not a planar graph.
In the next section we will suitably modify the Motzkin network along with its weight in order to make a planar network and recover the total positivity conditions in [7].We also show how to use our path model to carry these positivity conditions over to the Hankel matrix associated to the first column of A γ,σ,τ .In Section 3 we sepcialize our general results to some well-known combinatorial matrices as well as coefficientwise total positivity of their polynomial analogous.We conclude this paper with two open problems in Section 4 and an appendix about the computation of some Catalan-Stieltjes matrices in Section 5.

Minors of Catalan-Stieltjes and Hankel matrices
We consider the graph G = (V, E) where Comparing with the Motzkin graph M we see that each crossing point (i + 1 2 , j + 1 2 ) of two edges is transformed to a vertex in G as shown below: Figure 2.2: Five types of arrows where (a, b) ∈ Z × N.

Catalan-Stieltjes network
Definition 2.1.Let A i := (i, 0) and B i := (0, i) for i 0. The Catalan-Stieltjes network is defined to be the triple (G, Lemma 2.2.For any of the following four weight functions on the edges of G: • w((i, j) the electronic journal of combinatorics 23(4) (2016), #P4.33 Proof.Let w i,j be the right-hand side of (2.1).It suffices to prove that w i,j satisfy the recurrence (1.1).Among the four weight functions, we just prove the first one because the other cases can be verified in the same manner.Firstly, by definition w 0,0 = 1.We can classify the paths from A n+1 to B k according to their intersecting points with the line x = 1 as follows: • All the paths from the electronic journal of combinatorics 23(4) (2016), #P4.33 It is clear that the sum of the weights of the paths from which is the recurrence (1.1).
Theorem 2.3.Let I = (i 1 , . . ., i n ) and J = (j 1 , . . ., j n ) be two positive increasing integer sequences.For any of the four weight functions in Lemma 2.2, we have then the Catalan-Stieltjes matrix A defined by (1.1) is totally positive.
We will give several examples of Catalan-Stieltjes networks in Section 3.For the reader's convenience, in Figure 2.4 we present a rotated version of the Catalan-Stieltjes network in Figure 2.3.

Hankel network
) that the H-TP condition on α is equivalent to sat that α is a Stieltjes moment sequence, i.e., there is an integral representation of the form where µ is a non-negative measure µ on [0, +∞).
A sequence α = (a n ) n 0 is said to be generated by a Catalan-Stieltjes matrix (1.1) if it coincides with its first column, namely, a n = a n,0 for all n ∈ N. In [15] Liang-Mu-Wang gave some sufficient conditions on the H-TP of a sequence generated by a Catalan-Stieltjes matrix.Actually, for such a sequence α, we can derive from Lemma 2.2 a lattice path interpretation for each minor of the associated Hankel matrix H(α), which implies sufficient conditions on the H-TP of α.Recall that a sequence α = (a n ) n 0 is strongly log-convex if a n a m+1 a m a n+1 for all m n 0. Clearly the H-TP of α implies that it is strongly log-convex.
Theorem 2.7.Let α = (a n ) n 0 be a sequence generated by a Catalan-Stieltjes matrix (1.1) and H = (a i+j ) the associated Hankel matrix.Then, for any two positive increasing integer sequences I = (i 1 , . . ., i n ) and J = (j 1 , . . ., j n ) we have where w is any of the four weight functions in Lemma 2.2.
Proof.By Lemma 2.2, the coefficient a i+j,0 is the sum of weights of paths from A i to C j in Hankel network H; see Figure 2.3.In other words, the matrix H is the path matrix of H from (A i ) to (C j ).It is clear that the only possible permutation in Lindström's lemma is identity, so each minor of H reduces to the sum of positive weights of nonintersecting paths families from (A i ) i∈I to (C j ) j∈J in H.

Applications to some combinatorial matrices
It is known [28,7] that instead of the total order of real numbers we can consider the partial order of the commutative ring R[x] of polynomials with real coefficients as follows: a polynomial in R[x] is coefficientwise nonnegative if it has nonnegative coefficients and p(x) q(x) if p(x) − q(x) is coefficientwise nonnegative.Thus we can generalize the previous notions to coefficentwise log-convexity and coefficentwise-Hankel total positivity.For example, a sequence in R[x] is called coefficientwise-Hankel totally positive if the associated Hankel matrix is coefficientwise totally positive.Clearly the totally positive results in the previous sections can be restated in terms of coefficientwise totally positive sequence.In what follows we consider some special cases of Catalan-Stieltjes network and Hankel network in connection with some classical combinatorial sequences.One souce of such examples can be found in Viennot's Lecture Note [26] because almost all the moment sequences of classical orthogonal polynomials have interesting combinatorial interpretations.

Stirling network
If t k = 0 for all k 1, the recurrence (1.1) reduces to In particular A is coefficientwise totally positive if r k and s k are polynomials in x with nonnegative coefficients for all k 0.
Remark 3.2.Mongelli [18, Theorem 5] gave the above combinatorial interpretation in the special case r k = 1 and s k = k(z + 1).Generalizing the positivity part of Mongelli's result Zhu [29] proved the above total positivity result in the special case where r k and s k are quadratic polynomials of k.
Since these three sequences satisfy all the conditions in Corollary 2.4, the matrix A γ,σ,τ is TP and the sequence (N n (x)) is coefficientwise-H-TP.When x = 1 the matrix reduces to the Catalan triangle of Aigner [1]: . The corresponding Narayana network of type A is depicted in Figure 3.2.As the weight of arrows (i, j) → (i − 1, j) the electronic journal of combinatorics 23(4) (2016), #P4.33 is s j − t j − r j = 0 for all i 1 and j 0, so there is no such arrows in Figure 3.2.For example, if we choose I = {2, 3} and J = {0, 1}, then: and the three pairs of nonintersecting paths from {A 2 , A 3 } to {B 0 , B 1 } are drawn as red, green and blue pairs of paths in Figure 3.2.

Catalan-Shapiro network
Shapiro [20] proved that the ballot numbers B n,k = k n 2n n+k (n, k 1) satisfy the recurrence Thus the sequence (B n,1 ) n 1 ) of Catalan numbers is the moment sequence generated by the Catalan-Stieltjes matrix B n+1,k+1 n,k 0 : Clearly all the four conditions of Corollary 2.4 are satisfied, so the matrix B is TP and the sequence (B n,1 ) n 1 ) is H-TP.Note that the total positivity of B was first proved in [25].The corresponding Catalan-Shapiro network is depicted in Figure 3.3, where the edge (i, j) → (i − 1, j) has weight 1 if j = 0, and 0 otherwise.

Bell network
The Bell polynomials are defined by It is known [26,10] that B n (x)'s are generated by the Catalan-Stieltjes matrix (a n,k ): where a n,0 = B n (x) for n 0. Since recurrence (3.3) satisfies just the second condition (ii) of Corollary 2.8, the sequence (B n (x)) is coefficientwise-H-TP.When x = 1 it reduces to the Bell triangle [3] . It satisfies the first and the third conditions in Corollary 2.4.The corresponding Bell network is depicted in Figure 3.4.

Restricted hexagonal network
A hex tree is an ordred tree of which each vertex has updegree 0, 1, or 2, and an edge from a vertex of updegree 1 is either left, median, or right.The so-called restricted hexagonal number (see [15]) h n is also the number of hex trees with n edges (see [17, A002212]).
. It is easy to see that all the four conditons of Corollary 2.4 are satisfied.Thus the matrix H is TP and the sequence (h n ) is H-TP.The corresponding restricted hexagonal network is depicted in Figure 3.5.In a recent paper [14] Kim and Stanley studied a related polynomial sequence p n (x) where p n (x) = n j=0 1 j+1 2j j n j x n−j for n 0. In particular they proved that this sequence is the moment sequence associated to the Catalan-Stieltjes matrix (p n,k (x)): p n+1,0 (x) = (x + 1)p n,0 (x) + p n,1 (x), p n+1,k+1 (x) = p n,k (x) + (x + 2)p n,k+1 + p n,k+2 (k 1).
where p n,0 = p n (x) for n 0. Clearly all the conditions of (2.8) are satisfied, so the sequence (p n (x)) is coefficientwise-H-TP.

Eulerian polynomials
The Eulerian polynomials (A n (x)) can be defined by the electronic journal of combinatorics 23(4) (2016), #P4.33 where a n,0 = A n (x) for n 0. By the condition (i) of Corollary 2.8, the sequence (A n (x)) is coefficientwise-H-TP.

Rising factorials
The rising factorial µ n := (x) n is defined by They can be generated by the Catalan-Stieltjes matrix (see [26]) where a n,0 = µ n .Since recurrence (3.4) satisfies only the second point of Corollary 2.8, the sequence (µ n ) is coefficientwise-H-TP in R[x].

Schröder polynomials
The Schröder polynomials r n (x) (see [4,28]) are defined by r n (x) = N n (x + 1), i.e., and generated by the Catalan-Stieltjes matrix (see Appendix): Since the recurrence satisfies only the condition (i) of Corollary 2.8, so the sequence (r n (x)) is coefficientwise-H-TP.

Central Delannoy polynomials
The central Delannoy numbers [21,28] are defined by and generated by the Catalan-Stieltjes matrix (see Appendix): Since the recurrence satisfies only the condition (i) of Corollary 2.8, so the sequence (D n (x)) is coefficientwise-H-TP.

Narayana polynomials of type B
The Narayana polynomials of type B [28] are defined by and generated by the Catalan-Stieltjes matrix (see Appendix) Since the recurrence satisfies only the condition (i) of Corollary 2.8, so the sequence (W n (x)) is Hankel totally positive if x 1.

Two open problems
Sokal [22] and Wang-Zhu [27] actually independently proved that the polynomial sequence (W n (x)) n 0 is coefficientwise-Hankel totally positive.Can one find a planar network proof of this result ?A toy example of Lindström-Gessel-Viennot's methodology is a lattice path model for the total positivity of the Pascal matrix P := n k n,k 0 .As the Hadamard product of two totally positive matrices is not totally positive in general (see [8]), we speculate that the Hadamard product P • P = n is totally positive.

Appendix
For the reader's convenience, we indicate a quick path to the last four Catalan-Stieltjes matrices of Section 4. Our derivation of the Catalan-Stieltjes matrices relies on the correspondance between (1.1) and (1.2).Introduce two formal power series F and V given by We derive then from (5.3) the continued fraction expansion of G(z, x) with r k = 1 (k 0), s 0 = x, s k = x + 1 and t k = x for k 1.

. 1 ) 1 . 3 . 1 .
the electronic journal of combinatorics 23(4) (2016), #P4.33Since the Stirling numbers of second kind S(n, k) satisfy (3.1) with r k = 1 and t k = k we call the corresponding graph Stirling network.On the other hand, as there is no down step in the Motzkin paths, the corresponding network (see Figure1.1) reduces to Figure3.Proposition Let I = (i 1 , . . ., i n ) and J = (j 1 , . . ., j n ) be two positive increasing integer sequences.The minors of the matrix A = (a i,j ) satisfying (3.1) has the following combinatorial interpretation det A I,J = P ∈N (G;A I ,B J )
.2)Proof.By Lemma 2.2 and Lindström's lemma, we can write det A I,J as a double sum as (1.3) except that the nonintersection condition forces the path p k to go from A i k to B j k for all k = 1, ..., n, namely σ ∈ S n must be identity.From Theorem 2.3 we derive immediately the main results of Chen-Liang-Wang [7, Theorems 2.10 and 2.11, Corollary 2.12].Corollary 2.4.If the three sequences of nonnegative numbers (r k ) k 0 , (s k ) k 0 and (t k ) k 1 satisfy one of the following conditions: real numbers is Hankel-totally positive if the associated Hankel matrix H := H(α) = (a i+j ) i,j 0 is totally positive.For brevity, we use H-TP to denote Hankel-totally positive or Hankel-totally positivity in what follows.It is known (see[19,