Some Positivities in Stirling Arrays with Higher Level

In this paper, we prove that the sequences of the Stirling numbers of the first and second kind with higher level are both Pólya frequency and log-concave. Then, we show that some polynomials related to the above Stirling numbers with higher level are q-log-convex or strongly q-log-convex. Furthermore, we establish that the linear transformation related to the Stirling numbers of the first kind with level 2 preserves the log-convexity.


Introduction
A set partition of a set [n] := {1, 2, . . ., n} is a collection of non-empty disjoint subsets, called blocks, whose union is [n].The number of set partitions of [n] into k non-empty blocks is given by the Stirling numbers of the second kind n k .If n, k ≥ 0, then let Π (n,k) denote the set of all partitions of [n] having exactly k non-empty blocks.Throughout, we will assume that the blocks are arranged left-to-right in ascending order of minimal elements.Given a partition π in Π n , let min(π) denote the set of the minimal elements in each block of π.The combinatorial, arithmetical, and analytical properties of the Stirling numbers are well known, see for example [1,2].The classical generalizations of Stirling numbers come from a combinatorial setting, for example, by imposing restrictions on the size of the blocks (cf.[3]), and from an analytical approach (cf.[4]).
Recently, Komatsu et al. [5][6][7] have started a combinatorial study on the Stirling numbers with higher level (level s).Note that the first appearance of this sequence was in 1918 by Tweedie [8].Given a positive integer s, let n k s denote the number of ordered s-tuples (n,k) , such that min(π 1 ) = min(π 2 ) = • • • = min(π s ).This sequence is called Stirling numbers of the second kind with higher level.Alternatively, the Stirling numbers of the second kind with higher level can be defined as the connecting coefficients in the following expression: This sequence can be determined by the recurrence relation with the initial conditions n n s = 1, n 0 s = 0 (n ≥ 1), and n k s = 0 for n < k.Notice that for s = 1, we recover the classical Stirling numbers of the second kind.Domaratzki [9] called this generalization the generalized factorial numbers because for s = 2, the sequence n k 2 is related to the central factorial numbers T (n, k) by the equality n k 2 = T (2n, 2k), where Given a positive integer s, let n k s denote the connection coefficients in the polynomial identity The numbers n k s are called Stirling numbers of the first kind with higher level and have recently been studied [7].Let S n denote the set of permutations of [n].We will assume that permutations are expressed in standard cycle form, i.e., minimal elements first within each cycle, with cycles arranged left-to-right in ascending order of minimal elements.If n, k ≥ 0, then let S n,k denote the set of permutations of S n having exactly k cycles.Given a permutation σ in S n , let min(σ) denote the set of the minimal elements in each cycle of σ.Given a positive integer s, the sequence n k s counts the number of ordered s-tuples This sequence can be determined by the recurrence relation with the initial conditions n n s = 1, n 0 s = 0 n s = 0 for n ≥ 1.The Stirling numbers of the first kind with higher level satisfy the following recurrence relation [6]: Moreover, the orthogonality relationships (cf.[5,6]) between the Stirling numbers with higher level of the both kinds are max{n,m} and min{n,m} where δ n,m is the Kronecker delta.
In this paper, we prove that some sequences related to the Stirling numbers with higher level of the both kinds are total positivity, Pólya frequency, log-concave or unimodal.We also characterize some linear transformations preserving the double log-convexity.

Total Positivity of the Stirling Triangle of the Second Kind With Higher Level
Let A = (a n,k ) n,k≥0 be an infinite matrix.It is called totally positive of order r (TP r , for short) if its minors of all orders ≤ r are nonnegative.It is called TP if its minors of all orders are nonnegative.For example, the Pascal array is a TP matrix [10, p.137].The totally positive matrices play an important role in the theory of total positivity (cf.[10][11][12]).
Theorem 2.1.The matrix n k s n,k≥0 is TP.
Proof.From the orthogonality relationships between the Stirling numbers with higher level of the both kinds, we obtain the equality If T n,k = n k s and f (k) = k s , then the above equality can be written in matrix form as T n = Q n × J n , where Note that J n is totally positive.Thus, by induction on n, we derive that the nonnegative matrices T n and Q n are TP using the classical Cauchy-Binet formula, which expresses a determinant of any submatrix of T n as a sum of products of determinants of submatrix of Q n and J n ; see Karlin [10] for details.This completes the proof.

Pólya Frequency Sequences
Let (a n ) n≥0 be an infinite sequence of nonnegative numbers.It is called a Pólya frequency sequence of order r (a PF r sequence, for short) if its Toeplitz matrix , is TP r .It is called PF if its Toeplitz matrix is TP.A finite sequence a 0 , a 1 , . . ., a n is said to be PF r (PF, resp.), if the corresponding infinite sequence a 0 , a 1 , . . ., a n , 0, 0, . . . is PF r (PF, resp.).We will use the following result given by Aissen, Schoenberg, and Whitney [13], see also [10, p. 399].

Theorem 2.2.
A finite sequence a 0 , a 1 , . . ., a n is PF if and only if its generating function n i=0 a i x i has only real zeros.The generating function of the Stirling numbers of the first kind with higher level is given in (3).Therefore, by the Theorem 2.2, we obtain the following result.

Theorem 2.3. Each row sequence of n
k s n,k≥0 is a PF sequence.
The Bell polynomials with higher level of degree n are defined by Notice that for s = 1, we recover the classical Bell polynomials ( [1]).We now introduce the polynomials Notice that from (2), we obtain the recursion where Proof.It is well known that the Stirling numbers of the second kind are also defined as where (x where Using the Harper method [14, Lemma 1] and Lemma 2.4, we can obtain the following similar result.

Theorem 2.5. Each row sequence of
Proof.We know that the column generating function of n k s is given by Then, it follows from Lemma 2.6 that the column sequence n k s n≥k is PF since the sequence (i sn ) n≥0 is also PF.

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Let (r i ) i≥0 and (s j ) j≥0 be the sequences of the real zeros of polynomials f of degree n and g of degree n − 1 in nonincreasing order, respectively.The polynomial g interlaces f , denoted by Lemma 2.8.( [16]) Let F, f, g 1 , . . ., g k be real polynomials satisfying the following conditions. (1) All zeros of f and g j are real and g j f for each j.
(3) F and g 1 , . . ., g k have leading coefficients of the same sign.Suppose that b j (u) ≤ 0 whenever f (u) = 0. Then all the zeros of F are real and f F .Theorem 2.9.For all n ≥ 1, the Bell polynomials with higher level B n,s (x) have the interlacing property, that is B n,s (x) B n+1,s (x).
Proof.From Lemma 2.4 and ( 6), we take n,s (x) for j = 1, . . ., s.It is clear that these are real polynomials.Moreover, by the Rolle's theorem all zeros of B n,s (x) and B (j) n (x) are real and B (j) n (x) B n,s (x).Therefore, the condition (2) in Lemma 2.8 holds for all 1 ≤ n.The condition (3) is clear.Finally, all the zeros of Bell polynomials with higher level are real and negative.Hence, b j (u) ≤ 0 whenever f (u) = 0 for all j, which completes the proof.

Log-Concavity and Unimodality of Sequences
i for all i > 0, which is equivalent to (for relevant results one can see [15,17,18]) Clearly, the sequence (a n ) is log-concave if and only if its is PF 2 , i.e., its Toeplitz matrix (a i−j ) i,j≥0 is TP 2 (cf.[19]).A finite sequence of real numbers (a k ) 0≤k≤m is said to be unimodal if there exists an index 0 ≤ m * ≤ m, called the mode of the sequence, such that a k increases up to k = m * and decreases from then on, that is, It is easy to see that if a sequence is log-concave, then it is unimodal [15].From Theorems 2.3 and 2.5, we obtain the following results.is log-concave and, therefore, unimodal.
Proof.To prove that n0−ak k0+bk s k≥0 is log-concave, it suffices to show where n = n 0 − ak and k = k 0 + bk.By the Corollaries 2.11 and 2.12, we have n k On the other hand, by Theorem 2.1 Thus, it follows from ( 10) and (11) that This completes the proof.
It is well known that the Stirling numbers of the first kind have as mode the integer closest to the Harmonic number H n = n j=1 1 j .This surprising result was demonstrated by J. Hammersley [21].Therefore, to find the mode of the Stirling numbers of the first kind with higher level, we need the following lemma.
for each n ≥ 2.
Proof.Let P (x) = n k=0 n k s x k .By (3), we have Making use of Lemma 2.14, we have On the other hand, we have The result follows immediately.
By setting s = 1 in the above theorem, we obtain the J. Hammersley result about the mode of the Stirling numbers of the first kind.In fact, 15  3 2 is the maximal, see Fig. 2.

Strong q-log-convexity of row generating functions
For two polynomials with real coefficients f (q) and g(q), denote f (q) ≥ q g(q) if the difference f (q)−g(q) has only nonnegative coefficients.For a polynomial sequence (f n (q)) n≥0 , it is called q-log-convex (resp.q-log-concave) if for all n ≥ 1 This notion was introduced by Liu and Wang [22] (first suggested by Stanley).A polynomial sequence (f n (q)) n≥0 is called strongly q-log-convex (resp.strongly q-log-concave) if for m ≥ n ≥ 1, see Chen et al. [23].Clearly, their strong q-log-convexity of polynomial sequences implies the q-log-convexity.However, the converse does not hold (cf.[23]).It is easy to see that if the sequence (f n (q)) n≥0 is q-logconvex, then for each fixed nonnegative number q, the sequence (f n (q)) n≥0 is log-convex.The q-log-concavity and q-log-convexity of polynomials have been extensively studied, see [22][23][24][25] for instance.
In the next two theorems, we discuss the q-log-convexity of row generating functions of the Stirling numbers of the both kinds with higher level.Theorem 2.17.Let C n (q) = n k=0 n k s q k be the row generating functions of n k s .Then the polynomials sequence (C n (q)) n is q-log-convex.
Remark 2.18.For n ≥ 1, the sequence n Theorem 2.20.Let T n (q) = n k=0 n k s q k be the row generating functions of n k s .Then the polynomials sequence (T n (q)) n is strongly q-log-convex.Proof.To proof that (T n (q)) n is strongly q-log-convex, it suffices to show T n−1 (q)T m+1 (q) − T n (q)T m (q) ≥ q 0 for m ≥ n.By the recurrence relation (7) and Lemma 2.4, we have It follows that Noting that n−1 (q)T m (q) ≥ q 0. Therefore, by ( 14), T n−1 (q)T m+1 (q) − T n (q)T m (q) ≥ q 0. This completes the proof.

Linear Transformations Preserving Log-Convexity
Let us consider the following two linear transformations of sequences: and where (a(n, k)) 0≤k≤n is a triangular array of positive real numbers.The linear transformation (15) preserves the log-convexity (resp.the log-concavity) of sequences if the log-convexity (resp.the log-concavity) of (x n ) implies that of (t n ).We also say that the corresponding triangle (a(n, k)) 0≤k≤n preserves the log-convexity (resp.the log-concavity).The linear transformation (16) preserves the double log-convexity (resp.the double log-concavity) of sequences if the log-convexity (resp.the log-concavity) of (x n ) and (y n ) implies that of (z n ).We also say that the corresponding triangle (a(n, k)) 0≤k≤n preserves the double log-convexity (resp.the double log-concavity).Now, define the reciprocal triangle (a (n, k)) 0≤k≤n of (a(n, k)) 0≤k≤n by

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The triangle (a(n, k)) 0≤k≤n preserves the double log-convexity if the both triangles (a(n, k)) 0≤k≤n and (a (n, k)) 0≤k≤n preserve the log-convexity.Menon [27] proved that the log-concavity is preserved under the ordinary convolution.However, the ordinary convolution of two log-convex sequences need not be log-convex.Even the sequence of partial sums of a log-convex sequence is not log-convex in general.On the other hand, Davenport and Pólya [28] showed that the log-convexity is preserved under the binomial convolution.This convolution preserves also the log-concavity, see Walkup [29].It is also established that the q-binomial convolution preserves the log-concavity, see [30].In [31][32][33][34], the authors established the preserving log-convexity and logconcavity properties for the bi s nomial coefficients and the p, q-binomial coefficients, respectively.Liu and Wang [22] obtained a sufficient condition on a triangular array which ensures the linear transformation ( 15) is log-convexity preserving.Given a triangular array (a(n, k)) 0≤k≤n define a k (n, t) by where 1 ≤ n, 0 ≤ t ≤ 2n, and 0 ≤ k ≤ t/2 .The sufficient condition of Liu and Wang is stated as follows.
Theorem 2.21.[22,Theorem 4.8] Assume that the polynomials form a q-log-convex sequence.(A): For any given n and t, if there exists an integer k = k (n, t) such that a k (n, t) ≥ 0 for k ≤ k and a k (n, t) ≤ 0 for k > k .Then the linear transformation with respect to the triangular array (a(n, k)) 0≤k≤n preserves the log-convexity.
As applications of the above theorem, Liu and Wang [22] and Chen et al. [23] proved that the linear transformations preserve the log-convexity, respectively.To generalize these works, we propose the following proposition.Proposition 2.22.Suppose that the triangle (a(n, k)) 0≤k≤n satisfies Theorem 2. 21 and (a (n, k)) 0≤k≤n satisfies condition (A).Then, the linear transformation with respect to the triangular array (a (n, k)) 0≤k≤n preserves the logconvexity and, therefore, the triangular array (a(n, k)) 0≤k≤n preserves the double log-convexity.
Proof.Let A n (q) = n k=0 a (n, k)q k .Clearly, it suffices to show that the polynomials A n (q) form a q-log-convex sequence.We have which has nonnegative coefficients by the q-log-convexity of (A n (q)) n≥0 , as desired.
Therefore, the following examples are immediate consequences of Proposition 2.22.
Example 2.23.The following transformations preserve the double log-convexity. (1) We can generalize the above theorem, so that we will give a sufficient condition on a triangular array which ensures the linear transformation (16) preserves the double log-convexity.Theorem 2.24.Assume that the polynomials form a q-log-convex sequence.For any given n and t, suppose that the following two conditions hold: Then, the triangular array {a(n, k)} 0≤k≤n preserves the double log-convexity.
Liu and Wang conjectured that the sufficient condition of Theorem 2.21 holds for the polynomial of square binomial coefficients n k=0 n k 2 q k .Chen et al. in [24] showed this conjecture, and they established the following result.Proof.Let (x k ) k≥0 be a log-convex sequence.We need to show that the sequence (y n ) n≥0 is log-convex.We proceed by induction on n.It is easy to verify that y 0 y 2 ≥ y 2  1 .Now assume that n ≥ 3 and y 0 , y 1 , . . ., y n−1 is logconvex.Recall from relation (5) that Hence, Let z j = j k=0 j k 2 x k+1 for 0 ≤ j ≤ n−1.Then the sequence z 0 , z 1 , . . ., z n−1 is log-convex by the induction hypothesis, so is the sequence y 0 , y 1 , . . ., y n−1 , y n by Proposition 2.26.This completes the proof.

Corollary 2 . 11 .
Each row sequences of n k s n,k≥0 and n k s n,k≥0 are log-concave and, therefore, unimodal.From Theorem 2.1, we have also the following result.

Lemma 2 . 14 .,Theorem 2 . 15 .
([19]) Let (c k ) 0≤k≤n be a sequence of positive real numbers such that the polynomial n k=0 c k x k has only real roots, that is, (c k ) 0≤k≤n is a PF sequence.Then every mode k * of the sequence {c k } 0≤k≤n satisfies where x and x denote the floor and ceiling of x, respectively.Every mode k * of the sequence { n k s } 1≤k≤n satisfies n−1 j=0

Proposition 2 . 25 .Proposition 2 . 26 .
[24, Theorem 1.3]The square binomial transformation z n = n k=0 n k 2 x k preserves the log-convexity.By Theorem 2.24, we can extend the above result to the double logconvexity preserving as follows.The square binomial convolution z n = n k=0 n k 2 x k y n−k preserves the double log-convexity.Using Proposition 2.26, we obtain the following interesting result.Theorem 2.27.The Stirling transformation y n = n k=0 n k 2 x k of the first kind with level 2 preserves the log-convexity.