1 Jun 2006 on the Log-Convexity of Combinatorial Sequences

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is implied by the Pólya frequency property. An infinite sequence a0, a1, a2,... is called a Pólya frequency (PF, for short) sequence if all minors of the infinite Toeplitz matrix

(ai−j)i,j≥0 are nonnegative, where ak = 0 if k < 0. A ﬁnite sequence a0, a1,...,an−1, an

is PF if the inﬁnite sequence a0, a1,...,an−1, an, 0, 0,... is PF. Clearly, a PF sequence is log-concave. PF sequences are much better behaved and have been studied deeply in the theory of total positivity (Karlin [33]). For example, the fundamental representation

theorem of Schoenberg and Edrei states that a sequence a0 =1, a1, a2,... of real numbers is PF if and only if its generating function has the form

(1 + αjz) a zn = j≥1 eγz n (1 β z) n≥0 Qj≥1 j X − Q in some open disk centered at the origin, where α , β ,γ 0 and (α +β ) < + (see j j ≥ j≥1 j j ∞ Karlin [33, p. 412] for instance). In particular, a finite sequenceP of nonnegative numbers is PF if and only if its generating function has only real zeros ([33, p. 399]). So it is often more convenient to show that a sequence is PF even if our original interest is only in the log-concavity. Indeed, many log-concave sequences arising in combinatorics are actually PF sequences. Brenti [11, 13, 14] has successfully applied total positivity techniques and results to study the log-concavity problems. The object of this paper is to give a survey for the log-convexity of certain well-known combinatorial sequences. We develop techniques to investigate the log-convexity problem of sequences. The paper is organized as follows. In 2 we review some basic results about § the log-convexity of sequences. In particular, we survey miscellaneous methods to the log- convexity of the Bell numbers. In 3 we discuss the log-convexity of sequences satisfying a § three-term recurrence. As consequences, some famous combinatorial sequences, including the central binomial coefficients, the Catalan numbers, the Motzkin numbers, the Fine numbers, the central Delannoy numbers, the little and large Schröder numbers are log- convex. And also, each of bisections of combinatorial sequences satisfying three-term linear recurrences, including the Fibonacci, Lucas and Pell numbers, must be log-convex or log-concave respectively. In 4 we introduce the concept of the q-log-convexity of sequences § of polynomials in q and show the q-log-convexity of the Bell polynomials, the Eulerian polynomials and the q-Schröder numbers. We also present certain linear transformations preserving the log-convexity of sequences and establish the link with the q-log-convexity.

2 2 Basic techniques and Bell numbers

In this section we review some known results about the log-convexity of sequences and in particular, survey miscellaneous methods to the log-convexity of the Bell numbers. We ﬁrst consider operators on sequences that can preserve the log-convexity. The product of two log-convex sequences is obviously log-convex. It is somewhat surprising that the log-convexity can also be preserved by sums.

Proposition 2.1. If both x and y are log-convex, then so is the sequence x +y . { n} { n} { n n} Proof. It follows immediately that

(x + y )(x + y ) (√x x + √y y )2 (x + y )2 n−1 n−1 n+1 n+1 ≥ n−1 n+1 n−1 n+1 ≥ n n from the well-known Cauchy’s inequality and the log-convexity of x and y . { n} { n} Given two sequences x and y , deﬁne their ordinary convolution z′ { n}n≥0 { n}n≥0 { n}n≥0 and binomial convolution z′′ by { n}n≥0 n n n z′ = x y and z′′ = x y n k n−k n k k n−k Xk=0 Xk=0 respectively. It is known that the log-concavity of sequences can be preserved by both ordinary and binomial convolutions (see Wang and Yeh [60] for instance). 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 P´olya [17] showed that the log-convexity is preserved under the binomial convolution. Proposition 2.1 can provide an interpretation of this result.

Davenport-P´olya Theorem ([17]). If both x and y are log-convex, then so is { n} { n} their binomial convolution n n z = x y , n =0, 1, 2,.... n k k n−k Xk=0 Proof. It is easy to verify that z z =(x y )(x y +2x y + x y ) (x y + x y )2 = z2. 0 2 0 0 0 2 1 1 2 0 ≥ 0 1 1 0 1 We proceed by induction on n. Note that

n−1 n 1 n−1 n 1 z = − x y + − x y . n k k n−k k k+1 n−k−1 Xk=0 Xk=0 Two sums in the right hand side are the binomial convolutions of x with { k}0≤k≤n−1 y and x with y respectively. Hence both are log-convex by { k}1≤k≤n { k}1≤k≤n { k}0≤k≤n−1 the induction hypothesis. Thus the sequence z is log-convex by Proposition 2.1. { n} 3 Example 2.2. A permutation π of the n-element set [n] = 1, 2,...,n is alternating if { } π(1) > π(2) < π(3) > π(4) < π(n). The number E of alternating permutations of [n] ··· n is known as an Euler number and the ﬁrst few Euler numbers are

E = 1, 1, 1, 2, 5, 16, 61, 272,... . (Sloane’s A000111) { n}n≥0 { }

n n The sequence has the exponential generating function k=0 Enx /n! = tan x + sec x and satisfies the recurrence n P n 2E = E E n+1 k k n−k Xk=0 (see Comtet [16, p. 258] and Stanley [47, p. 149] for instance). Let zn = En/2. Then z = n n z z . Clearly, z , z , z , z is log-convex. Thus the sequence z is n+1 k=0 k k n−k 0 1 2 3 { n} log-convex by induction and Davenport-Pólya Theorem, and so is the sequence E . P { n} The following corollary is an immediate consequence of Davenport-Pólya Theorem.

n n Corollary 2.3. The binomial transformation zn = k=0 k xk can preserve the log- convexity. P It is possible to study the log-convexity problems using the theory of total positivity. The following proposition is a special case of Brenti [11, Theorem 2.2.5]. For completeness, we give a proof of it. We need some notation and terminology. An inﬁnite matrix M of

nonnegative numbers is said to be totally positive of order 2 (or a TP2 matrix, for short) if all minors of order 2 of M are nonnegative. Let a be a sequence of nonnegative { n}n≥0 number and let a0 a1 a2

a1 a2 a3  (ai+j)= . a a a .  2 3 4       ···  be the associated Hankel matrix. Clearly, the sequence a is log-convex if and only { n}n≥0 if a a a a for 1 m n, or equivalently, all minors of order 2 of the matrix m n ≤ m−1 n+1 ≤ ≤ (ai+j) are nonnegative. Given an inﬁnite lower triangular matrix A = (an,k)n,k≥0, let n k An(u)= k=0 an,ku denote the n-th row generating function of A. PropositionP 2.4. Let A,B,C be three inﬁnite lower triangular matrices satisfying the following conditions.

(i) Both B and C are TP2.

(ii) A (u)= B (u)C (u) for all i, j N. i+j i j ∈

4 If the sequence x is log-convex, then so is the sequence z deﬁned by { n}n≥0 { n} n

zn = an,kxk, n =0, 1, 2,.... (2.1) Xk=0 Proof. Let Z = (zi+j)i,j≥0 and X = (xi+j)i,j≥0 be the associated Hankel matrices of the sequences z and x respectively. Then (2.1) is equivalent to the identity Z = BXCt { n} { k} by the condition (ii). Now all minors of order 2 of the matrices B,CT are nonnegative since B,C are TP , and those of the matrix X are nonnegative since x is log-convex. So 2 { n}n≥0 all minors of order 2 of the product matrix Z are nonnegative by the well-known Cauchy-

Binet formula (see [33, p. 1] for instance). Thus the sequence zn is log-convex. n { } Let P =(pnk)n,k≥0 be the Pascal matrix, where pnk = k is the binomial coefficients. It is known that the matrix P is TP2 (all minors of P are actually nonnegative, see Gessel and Viennot [29] for a combinatorial proof of this fact). Clearly, the row generating function n Pn(u)=(1+ u) . Thus we can obtain Corollary 2.3 again by taking A = B = C = P in Proposition 2.4. In the remaining part of this section we concentrate our attention on the Bell numbers. The log-convexity of the Bell numbers was first obtained by Engel [28] in 1994 and here we survey several different proofs. We first review some basic facts about the Stirling numbers of the second kind which are closely related to the Bell numbers. Denote [n] = 1, 2,...,n and let 1 k n. The Stirling number S(n, k) of the { } ≤ ≤ second kind is the number of partitions of the set [n] having exactly k blocks. The triangle S(n, k) of Stirling numbers of the second kind is { }1≤k≤n 1 1 1 1 3 1 (Sloane’s A008277) 1 7 6 1 1 15 25 10 1

··· with S(0, 0) = 1. It is well known that the Stirling numbers satisfy the recurrence

S(n +1,k)= kS(n, k)+ S(n, k 1), n,k 1, − ≥ with S(n, 0) = S(0,k) = 0 except S(0, 0) = 1. For each ﬁxed n, the sequence S(n, k) { }0≤k≤n is PF and therefore log-concave (Harper [32]). For each ﬁxed k, the sequence S(n, k) { }n≥k is convex ([16, p. 291]). Also, note that the generating function 1 S(n, k)xn−k = (1 x)(1 2x) (1 kx) Xn≥k − − ··· − 5 ([16, p. 207]). Hence the sequence S(n, k) is PF (and therefore log-concave) by the { }n≥k fundamental representation theorem of PF sequences.

The Bell number Bn is the total number of partitions of [n], i.e.,

Bn = S(n, k). (2.2) Xk=0 The ﬁrst few Bell numbers are

B = 1, 1, 2, 5, 15, 52, 203, 877,... . (Sloane’s A051130) { n}n≥0 { } The Bell numbers have the exponential generating function B n un = exp(eu 1) n! − n≥0 X ([16, p. 210]). In 1996, Bender and Canﬁeld [5] showed that the sequence B is { n}n≥0 log-convex and the sequence B /n! is log-concave simultaneously by establishing { n }n≥0 the following powerful result. (We refer the reader to Schirmacher [44] and Asai et al. [3] for generalizations of this result.)

Bender-Canﬁeld Theorem ([5]). Let X0 =1,X1,X2,... be a log-concave sequence and deﬁne the sequence P by { n}

n j Pnu /n! = exp Xju /j . n≥0 j≥1 ! X X Then P is log-convex and P /n! is log-concave. { n} { n }

Although there is not a nice closed formula for Bn, Dobinski formula gives the expression 1 ∞ kn B = n e k! Xk=0 ([16, p. 210]). It immediately follows that the sequence B is convex since kn is convex { n} in n for each k. Further, the log-convexity of B can also be followed from Dobinski { n} formula ([9]):

1 in−1jn−1 i2 + j2 1 in−1jn−1 B B = ij = B2. n−1 n+1 e2 i!j! 2 ≥ e2 i!j! n i,j≥0 i,j≥0 X X We can also prove the log-convexity of the Bell numbers by means of the recurrence

n n B = B n+1 k k Xk=0 6 ([16, p. 210]). Clearly, B0, B1, B2, B3 is log-convex. Thus the log-convexity of the full sequence B can be obtained by induction and Corollary 2.3. { n}n≥0 Another simple approach to see the log-convexity of the Bell numbers is to use (2.2) and the following.

n Proposition 2.5. The Stirling transformation of the second kind zn = k=0 S(n, k)xk can preserve the log-convexity. P Proof. Let x be a log-convex sequence. We need to show that the sequence z { k}k≥0 { n}n≥0 is log-convex. We proceed by induction on n. It is easy to verify that z2 z z . Now 1 ≤ 0 2 assume that n 3 and z , z ,...,z is log-convex. Recall that ≥ 0 1 n−1 n n 1 S(n, k)= − S(j 1,k 1) j 1 − − Xj=k − ([16, p. 209]). Hence

j n n n 1 n−1 n 1 z = − S(j 1,k 1)x = − S(j, k)x . n j 1 − − k j k+1 j=0 " # Xk=1 Xj=k − X Xk=0 Let y = j S(j, k)x for 0 j n 1. Then the sequence y ,y ,...,y is log- j k=0 k+1 ≤ ≤ − 0 1 n−1 convex byP the induction hypothesis, so is the sequence z0, z1,...,zn−1, zn by Corollary 2.3. This completes the proof. Taking x 1 in Proposition 2.5, we obtain the log-convexity of the Bell numbers. k ≡ The following are some more applications of Proposition 2.5.

Example 2.6. The Bell number Bn can be viewed as the number of ways of placing n labeled balls into n indistinguishable boxes. Let Sn be the number of ways of placing n labeled balls into n unlabeled (but 2-colored) boxes. Then the ﬁrst few are

S = 1, 2, 6, 22, 94, 454, 2430, 14214,... . (Sloane’s A001861) { n}n≥0 { } Clearly, S = n 2kS(n, k). Hence the sequence S is log-convex by Proposi- n k=1 { n}n≥0 tion 2.5. Note that S = n n B B . Hence the log-convexity of S can also be P n k=0 k k n−k { n} followed from Davenport-P´olya Theorem and the log-convexity of B . We will provide P { n} another proof of this result in Remark 4.5.

Example 2.7. Consider the ordered Bell number c(n), i.e., the number of ordered partitions of the set [n]. The ﬁrst few are

c(n) = 1, 1, 3, 13, 75, 541, 4683, 47293,... (Sloane’s A000670). { }n≥0 { }

7 Note that c(n)= n k!S(n, k) (Stanley [47, p. 146]). The sequence c(n) is there- k=0 { }n≥0 fore log-convex byP Proposition 2.5. We will provide another proof of this result in Re- mark 4.7.

Remark 2.8. Let c(n, k) be the signless Stirling number of the ﬁrst kind, i.e., the number of permutations of [n] which contain exactly k permutation cycles. Then n n 1 c(n, k)= − (n j)!c(j 1,k 1) j 1 − − − Xj=k − for 1 k n ([16, p. 215]). By the same method used in the proof of Proposition 2.5, we ≤ ≤ n can obtain that the linear transformation zn = k=0 c(n, k)xk preserves the log-convexity. We will provide another proof of the log-convexityP of the Bell numbers in Remark 4.5.

3 Sequences satisfying three-term recurrences

In this section we consider the log-convexity of certain famous combinatorial numbers, 2n including the central binomial coefficients b(n)= n , the Catalan numbers, the Motzkin numbers, the Fine numbers, the central Delannoy numbers, the little and large Schröder numbers. These numbers play an important role in enumerative combinatorics and count various combinatorial objects. We review some basic facts about these numbers from the viewpoint of the enumeration of lattice paths in the (x, y) plane. The central binomial coefficient b(n) counts the number of lattice paths from (0, 0) to 2n (n, n) with steps (0, 1) and (1, 0) in the first quadrant. It is clear that b(n) = n for n 0. The first few central binomial coefficients are ≥ b(n) = 1, 2, 6, 20, 70, 252, 924, 3432,... . (Sloane’s A000984) { }n≥0 { } The central binomial coefficients satisfy the recurrence (n + 1)b(n +1) = 2(2n + 1)b(n). The sequence b(n) is log-convex since b(n+1) = 2(2n+1) is increasing. { }n≥0 b(n) n+1 The Catalan number Cn counts the number of lattice paths, Dyck Paths, from (0, 0) to (2n, 0) with steps (1, 1), (1, 1) and never falling below the x-axis, or equally, the number − of lattice paths from (0, 0) to (n, n) with steps (0, 1) and (1, 0), never rising above the line y = x. The Catalan numbers have an explicit expression C = 1 2n for n 0 and the n n+1 n ≥ first few Catalan numbers are C = 1, 1, 2, 5, 14, 42, 132, 429,... . (Sloane’s A000108) { n}n≥0 { } The sequence C is one of the most ubiquitous and fascinating sequences of enumera- { n}n≥0 tive combinatorics. We refer the reader to Stanley [48, Exercise 6.19] for 66 combinatorial

8 interpretations of the Catalan numbers and to his homepage in MIT for an addendum.

The Catalan numbers satisfy the recurrence (n + 2)Cn+1 = 2(2n + 1)Cn. The sequence C is log-convex since Cn+1 = 2(2n+1) is increasing. { n}n≥0 Cn n+2 The Motzkin number Mn counts the number of lattice paths, Motzkin paths, starting from (0, 0) to (n, 0), with steps (1, 0), (1, 1) and (1, 1), and never falling below the x-axis, − or equally, the number of lattice paths from (0, 0) to (n, n) with steps (0, 2), (2, 0) and (1, 1), never rising above the line y = x. The ﬁrst few Motzkin numbers are

M = 1, 1, 2, 4, 9, 21, 51, 127,... . (Sloane’s A001006) { n}n≥0 { } The Motzkin numbers are closely related to the Catalan numbers. For example, it is n n well known that Cn+1 = k≥0 k Mk and Mn = k≥0 2k Ck. We refer the reader to Aigner [1], Donaghey andP Shapiro [21], and StanleyP [48, Exercise 6.37, 6.38, 6.46] for further information about the Motzkin numbers. The Motzkin numbers satisfy the three- term recurrence

(n + 3)Mn+1 = (2n + 3)Mn +3nMn−1 (3.1)

(see [51] for a bijection proof).

The Fine number fn is the number of Dyck paths from (0, 0) to (2n, 0) with no hills. (A hill in a Dyck path is a pair of consecutive steps giving a peak of height 1.) The ﬁrst few Fine numbers are

f = 1, 0, 1, 2, 6, 18, 57, 186,... . (Sloane’s A000957) { n}n≥0 { } It is known that the Catalan number C = 2f + f for n 1. See Deutsch and n n n−1 ≥ Shapiro [20] for a survey of the Fine numbers, including a partial list of the Fine numbers occurrences. The Fine numbers satisfy the three-term recurrence

2(n + 1)f = (7n 5)f + 2(2n 1)f (3.2) n − n−1 − n−2 (see [36] for a bijection proof). The central Delannoy number D(n) is the number of lattice paths, king walks, from (0, 0) to (n, n) with steps (1, 0), (0, 1) and (1, 1) in the ﬁrst quadrant (see [16, p. 81] and [4, 53] for the interesting backgrounds of the Delannoy numbers). Clearly, the number of king walks with n k diagonal steps is n+k b(k). Hence D(n)= n n+k b(k) ([53]). − n−k k=0 n−k The ﬁrst few central Delannoy numbers are P

D(n) = 1, 3, 13, 63, 321, 1683, 8989, 48639,... . (Sloane’s A001850) { }n≥0 { }

9 It is known that the central Delannoy numbers satisfy the three-term recurrence

nD(n) = 3(2n 1)D(n 1) (n 1)D(n 2) (3.3) − − − − − (see [37] for a bijection proof).

The (large) Schröder number rn is the number of king walks, Schröder paths, from (0, 0) to (n, n), and never rising above the line y = x. The large Schröder numbers bear the same relation to the Catalan numbers as the central Delannoy numbers do to the n n+k central binomial coefficients. Hence we have rn = k=0 n−k Ck ([27, 61]). The first few large Schröder numbers are P

r = 1, 2, 6, 22, 90, 394, 1806, 8558,... . (Sloane’s A006318) { n}n≥0 { } The Schröder paths consist of two classes: those with steps on the main diagonal and those without. These two classes are equinumerous, and the number of paths in either class is the little Schröder number (half the large Schröder number). The first few little Schröder numbers are

s = 1, 1, 3, 11, 45, 197, 903, 4279,... . (Sloane’s A001003) { n}n≥0 { } We refer the reader to Stanley [48, Exercise 6.39] for 19 combinatorial interpretations of the Schröder numbers and Stanley [49] for the fascinating historical story of the Schröder numbers. It is known that the Schröder numbers of two kinds satisfy the three-term recurrence (n + 2)z = 3(2n + 1)z (n 1)z (3.4) n+1 n − − n−1 (see Foata and Zeilberger [26] for a combinatorial proof and Sulanke [50] for another one). All these numbers presented previously satisfy a three-term recurrence. Aigner [1] first established algebraically the log-convexity of the Motzkin numbers and then Callan [15] gave a combinatorial proof. Recently, Doˇslićet al [22, 23] showed the log-convexity of the Motzkin numbers, the Fine numbers, the central Delannoy numbers, the large and little Schröder numbers using calculus.1 Motivated by these results, we investigate the log-convexity problem of combinatorial sequences satisfying a three-term recurrence by an algebraic approach. In what follows, we distinguish two cases according to the sign of coefficients in the recurrence relations. 1We appreciate Doˇslićfor acquainting us with his very recent work [24, 25], in which the techniques developed in [22, 23] are extended and more examples are presented.

10 3.1 The recurrence anzn+1 = bnzn + cnzn 1 − Let z be a sequence of positive numbers satisfying the recurrence { n}n≥0

anzn+1 = bnzn + cnzn−1 (3.5) for n 1, where a , b ,c are all positive. Consider the quadratic equation a λ2 b λ ≥ n n n n − n − cn = 0 associated with the recurrence (3.5). Clearly, the equation has a unique positive root 2 bn + bn +4ancn λn = . (3.6) p2an Deﬁne x = z /z for n 0. Then the sequence z is log-convex if and only if the n n+1 n ≥ { n}n≥0 sequence x is increasing. By (3.5), the sequence x satisﬁes the recurrence { n}n≥0 { n}n≥0 cn anxn = bn + (3.7) xn−1 for n 1. It follows that x x is equivalent to x λ , and is also equivalent ≥ n ≥ n−1 n−1 ≤ n to x λ . Thus the sequence z is log-convex if and only if the sequence x n ≥ n { n}n≥0 { n}n≥0 can be separated by the sequence λ : { n}n≥1 x λ x λ x λ x λ . (3.8) 0 ≤ 1 ≤ 1 ≤ 2 ≤···≤ n−1 ≤ n ≤ n ≤ n+1 ≤··· Theorem 3.1. Let z and λ be as above. Assume that z , z , z , z is log- { n}n≥0 { n}n≥1 0 1 2 3 convex and that the inequality

a λ λ b λ c 0 (3.9) n n−1 n+1 − n n−1 − n ≥ holds for n 2. Then the sequence z is log-convex. ≥ { n}n≥0 Proof. Let x = z /z for n 0. We prove the interlacing inequalities (3.8) by induc- n n+1 n ≥ tion. By the assumption that z , z , z , z is log-convex, we have x λ x λ x . 0 1 2 3 0 ≤ 1 ≤ 1 ≤ 2 ≤ 2 Now assume that λ x λ . Note that x λ is equivalent to x λ . On n−1 ≤ n−1 ≤ n n−1 ≤ n n ≥ n the other hand, x λ implies that n−1 ≥ n−1

bn cn 1 bn cn 1 xn = + + λn+1 an an xn−1 ≤ an an λn−1 ≤ by the inequality (3.9). Hence we have λ x λ . Thus (3.8) holds by induction. n ≤ n ≤ n+1 Corollary 3.2. The Fine sequence f is log-convex. { n}n≥2

11 Proof. Let z = f for n 0. Then z =1, z =2, z =6, z = 18 and n n+2 ≥ 0 1 2 3

2(n + 4)zn+1 = (7n + 16)zn + 2(2n + 5)zn−1

by (3.2). Solve the equation 2(n + 4)λ2 (7n + 16)λ 2(2n +5) = 0 to obtain − − 2(2n + 5) λ = . n (n + 4) It is easy to verify that

2(20n2 + 151n + 171) 2(n + 4)λ λ (7n + 6)λ 2(2n +5)= 0. n−1 n+1 − n−1 − (n + 3)(n + 5) ≥

Hence the sequence z , i.e., f , is log-convex by Theorem 3.1. { n}n≥0 { n}n≥2 Since the expression (3.6) of λn, sometimes it is inconvenient directly to check the inequality (3.9). However, the inequality may be veriﬁed by means of Maple. For example, the nth characteristic root of the Motzkin sequence satisfying the recurrence (3.1) is

2n +3+ √16n2 + 48n +9 λ = . n 2(n + 3) Using Maple it is easy to verify the inequality:

(n + 3)λ λ (2n + 3)λ 3n 0. n−1 n+1 − n−1 − ≥ Thus the log-convexity of the Motzkin numbers follows from Theorem 3.1.

Corollary 3.3. The Motzkin sequence M is log-convex. { n}n≥0 We can also give another criterion for the log-convexity of the sequence z . { n} Theorem 3.4. Let z and λ be as above. Suppose that there exists a sequence { n}n≥0 { n}n≥1 µ of positive numbers such that the following three conditions hold. { n}n≥1 (i) µ λ for all n 1. n ≤ n ≥ (ii) z µ z and z µ z . 1 ≤ 1 0 2 ≤ 2 1 (iii) a µ µ b µ + c for n 2. n n−1 n+1 ≥ n n−1 n ≥ Then the sequence z is log-convex. { n}n≥0 Proof. Let x = z /z for n 0. Then it suﬃces to show that the sequence x is n n+1 n ≥ { n} increasing. We prove this by showing the interlacing inequalities hold:

x µ x µ x µ x µ . (3.10) 0 ≤ 1 ≤ 1 ≤ 2 ≤···≤ n−1 ≤ n ≤ n ≤ n+1 ≤··· 12 The condition (ii) is equivalent to x µ and x µ . However, µ λ . Hence x λ , 0 ≤ 1 1 ≤ 2 1 ≤ 1 0 ≤ 1 and so x λ µ . Thus we have µ x µ . Now assume that µ x µ . 1 ≥ 1 ≥ 1 1 ≤ 1 ≤ 2 n−1 ≤ n−1 ≤ n Then x µ implies x µ since µ λ . On the other hand, x µ implies n−1 ≤ n n ≥ n n ≤ n n−1 ≥ n−1

bn cn 1 bn cn 1 xn = + + µn+1 an an xn−1 ≤ an an µn−1 ≤ by the condition (iii). Hence we have µ x µ . Thus (3.10) holds by induction. n ≤ n ≤ n+1 For convenience, we may choose µn in the theorem as an appropriate rational approx-

imation to λn. We present two examples to demonstrate this approach.

Example 3.5. The derangements number dn is the number of permutations of n elements with no ﬁxed points. The ﬁrst few are

d = 1, 0, 1, 2, 9, 44, 265, 1854,... . (Sloane’s A000166). { n}n≥0 { } The sequence d satisﬁes the recurrence d = n(d + d ) (Comtet [16, p. 182]). { n}n≥0 n+1 n n−1 Let z = d for n 0. Then z =(n + 2)(z + z ). We have n n+2 ≥ n+1 n n−1 (n +2)+ √n2 +8n + 12 (n +2)+(n + 3) 2n +5 λ = = . n 2 ≥ 2 2 Set µ = (2n + 5)/2. Then µ λ . Also, z /z = 2 < 7/2= µ and z /z = 9/2= µ . n n ≤ n 1 0 1 2 1 2 Furthermore, 2n +1 µ µ (n + 2)µ (n +2)= 0. n−1 n+1 − n−1 − 4 ≥ Thus the sequence z , i.e., d , is log-convex by Theorem 3.4. { n}n≥0 { n}n≥2

Remark 3.6. Generally, if an takes a constant value, both bn and cn are linear functions in n respectively, then we can show by means of Theorem 3.1 that the sequence z { n}n≥0 satisfying the recurrence (3.5) is asymptotically log-convex. In other words, there exists an index N such that z is log-convex. We leave the proof of this result to the { n}n≥N reader as an exercise.

Example 3.7. Let an be the number of directed animals of size n (Stanley [48, Exercise 6.46]). The ﬁrst few are

a = 1, 1, 2, 5, 13, 35, 96, 267,... . (Sloane’s A005773) { n}n≥0 { } The sequence satisﬁes the recurrence (n + 1)z = 2(n + 1)z + 3(n 1)z . We have n+1 n − n−1 4n 2 4n 1 6n λn =1+ − 1+ − = r n +1 ≥ 2n +1 2n +1

13 for n 1, where the inequality follows from x−1 x+1 x when x y > 1. Set ≥ y−1 y+1 ≥ y ≥ µ = 6n . Then µ λ , z /z =1 < 2= µ andqz /z =2 < 2.4= µ . Also, n 2n+1 n ≤ n 1 0 1 2 1 2 9(n 1) (n + 1)µ µ 2(n + 1)µ 3(n 1) = − 0. n−1 n+1 − n−1 − − (2n 1)(2n + 3) ≥ − Thus the sequence a is log-convex by Theorem 3.4. { n} Remark 3.8. The techniques developed in this subsection can also be used to study the log-concavity of the sequences z satisfying the recurrence (3.5). For example, it is { n}n≥0 clear that z is log-concave if and only if { n}n≥0 x λ x λ x λ x λ . 0 ≥ 1 ≥ 1 ≥ 2 ≥···≥ n−1 ≥ n ≥ n ≥ n+1 ≥···

Thus, if z0, z1, z2, z3 is log-concave and the inequality

a λ λ b λ c 0 n n−1 n+1 − n n−1 − n ≤ holds for n 2, then the sequence z is log-concave. ≥ { n}n≥0 Remark 3.9. When coeﬃcients an, bn,cn in the recurrence (3.5) take constant values respectively, the sequence z is neither log-convex nor log-concave since λ takes a { n}n≥0 n constant value. For example, the Fibonacci sequence

F = 0, 1, 1, 2, 3, 5, 8, 13,... (Sloane’s A000045) { n}n≥0 { } satisﬁes the recurrence Fn+1 = Fn + Fn−1. The sequence is neither log-convex nor log- concave. Actually, F F F 2 = ( 1)n. However, the bisection F with even n−1 n+1 − n − { 2n}n≥0 index is log-concave and the bisection F with odd index is log-convex since { 2n+1}n≥0 F F F 2 = ( 1)n−1. We will give a general result about such sequences in Corol- n−2 n+2 − n − lary 3.17.

3.2 The recurrence anzn+1 = bnzn cnzn 1 − − In this part we consider the log-convexity of the sequence z of positive numbers { n} satisfying the recurrence a z = b z c z (3.11) n n+1 n n − n n−1 for n 1, where a , b ,c are all positive. Let x = z /z for n 0. Then we need to ≥ n n n n n+1 n ≥ check whether the sequence x is increasing. { n}n≥0 By the recurrence (3.11), we have

bn cn 1 xn = . an − an xn−1

14 Hence b b c c 1 c 1 1 x x = n+1 n + n n+1 + n . (3.12) n+1 − n a − a a − a x a x − x n+1 n n n+1 n n n−1 n Observe that if an an+1 cn cn+1 xn + 0 (3.13) bn bn+1 an an+1 ≥

then x x implies x x from (3.12). Hence we can conclude that if x x n−1 ≤ n n ≤ n+1 0 ≤ 1 and the inequality (3.13) holds for n 1, then the sequence x is increasing, and ≥ { n}n≥0 the sequence z is therefore log-convex. { n} In what follows we consider the case that an, bn,cn are all linear functions in n. In this case, the inequality (3.13) is easily checked since two determinants take constant values respectively. More precisely, let

an = α1n + α0, bn = β1n + β0, cn = γ1n + γ0 and denote β β γ γ α α A = 0 1 , B = 0 1 , C = 0 1 . γ0 γ1 α0 α1 β0 β1

Then it is easy to see that two determinants in the inequality (3.13) are equal to C and

B respectively. Thus we have the following criterion.

Theorem 3.10. Let z be a sequence of positive numbers and satisfy the three-term { n}n≥0 recurrence (α n + α )z =(β n + β )z (γ n + γ )z 1 0 n+1 1 0 n − 1 0 n−1 for n 1, where α n+α , β n+β ,γ n+γ are positive for n 1. Assume that z , z , z is ≥ 1 0 1 0 1 0 ≥ 0 1 2 log-convex. Then the full sequence z is log-convex if one of the following conditions { n}n≥0 holds.

(i) B,C 0. ≥ (ii) B < 0,C > 0, AC B2 and z B + z C 0. ≥ 0 1 ≥ (iii) B > 0,C < 0, AC B2 and z B + z C 0. ≤ 0 1 ≥ Proof. Let x = z /z for n 0. Then x x since z z z2. Thus it suﬃces to show n n+1 n ≥ 0 ≤ 1 0 2 ≥ 1 that the inequality Cx + B 0 holds for n 0. If B 0 and C 0, then the inequality n ≥ ≥ ≥ ≥ is obvious. We next assume that BC < 0 and show that Cx + B 0 by induction on n. n ≥

15 We do it only for the case (ii) since the case (iii) is similar. Clearly, Cx + B 0 by the 0 ≥ consider z B + z C 0. Now assume that Cx + B 0 for n 1. Then 0 1 ≥ n−1 ≥ ≥ b c 1 Cx + B = C n n + B n a − a x n n n−1 C C b + c + B ≥ a n n B n C = (bnB + cnC)+ B. anB Note that b B + c C = a A since n n − n

an α1 α0 α1n + α0 α1 α0

anA + bnB + cnC = b β β = β n + β β β =0. n 1 0 1 0 1 0

cn γ1 γ0 γ1n + γ0 γ1 γ0

Hence AC Cx + B + B 0 n ≥ − B ≥ by the condition AC B2. This completes our proof. ≥ Remark 3.11. Similarly, assume that z0, z1, z2 is log-concave and one of the following conditions holds.

(i) B,C 0. ≤ (ii) B < 0,C > 0, AC B2 and z B + z C 0. ≤ 0 1 ≤ (iii) B > 0,C < 0, AC B2 and z B + z C 0. ≥ 0 1 ≤ Then the sequence z satisfying the recurrence (3.10) is log-concave. { n}n≥0 Corollary 3.12. The central Delannoy sequence D(n) is log-convex. { }n≥0 Proof. It is easy to obtain A =3, B = 1,C = 3. By the initial values D(0) = 1,D(1) = − 3,D(2) = 13, the log-convexity of D(n) follows immediately from Theorem 3.10 (ii).

Corollary 3.13. The little and large Schr¨oder numbers are log-convex respectively.

Proof. It suffices to show that the little Schröder numbers s is log-convex since the { n}n≥0 large Schröder numbers r =2s for n 1. n n ≥ We have A = 9, B = 3,C = 9 and s = s = 1,s = 3. Thus the log-convexity of − 0 1 2 s follows from Theorem 3.10 (ii). { n}n≥0

16 Example 3.14. Let hn be the number of the set of all tree-like polyhexes with n +1 hexagons (see Harary and Read [31]). It is known that hn counts the number of lattice paths from (0, 0) to (2n, 0) with steps (1, 1), (1, 1), (2, 0), never falling below the x-axis, − and with no peaks at odd level. The ﬁrst few are

h = 1, 1, 3, 10, 36, 137, 543, 2219,... (Sloane’s A002212) { n}n≥0 { } The sequence h satisﬁes the recurrence { n}n≥0 (n + 1)h = 3(2n 1)h 5(n 2)h n − n−1 − − n−2 with h = h = 1. Thus the sequence h is log-convex by Theorem 3.10 (ii). 0 1 { n}n≥0

Example 3.15. Let wn be the number of walks on cubic lattice with n steps, starting and ﬁnishing on the xy plane and never going below it (see Guy [30]). The ﬁrst few are

w = 1, 4, 17, 76, 354, 1704, 8421, 42508,... (Sloane’s A005572) { n}n≥0 { } It is known that the sequence w satisﬁes the recurrence { n}n≥0 (n + 2)w = 4(2n + 1)w 12(n 1)w n n−1 − − n−2 with w0 = 1 and w1 = 4, and is therefore log-convex by Theorem 3.10 (ii).

A special interesting case of Theorem 3.10 (i) and Remark 3.11 (i) is the following result.

Corollary 3.16. Suppose that the sequence z of positive numbers satisfies the re- { n}n≥0 currence az = bz cz for n 1, where a, b, c are positive constants. If z , z , z is n+1 n − n−1 ≥ 0 1 2 log-convex (resp. log-concave), then so is the full sequence z . { n}n≥0 Corollary 3.17. Suppose that the sequence z of positive numbers satisfies the re- { n}n≥0 currence az = bz + cz for n 1, where a, b, c are positive constants. If z , z , z is n+1 n n−1 ≥ 0 1 2 log-convex (resp. log-concave), then the bisection z is log-convex (resp. log-concave) { 2n} and the bisection z is log-concave (resp. log-convex). { 2n+1} Proof. By the recurrence az = bz + cz for n 1, we can obtain the recurrence n+1 n n−1 ≥ a2z =(b2 +2ac)z c2z n+2 n − n−2 for n 2. It is not difficult to verify that ≥ a2(z z z2)= b2(z z z2), a3(z z z2)= b2c(z2 z z ). 0 4 − 2 0 2 − 1 1 5 − 3 1 − 0 2 So the statement follows from Corollary 3.16.

17 Example 3.18. The Fibonacci numbers satisfy the recurrence Fn+1 = Fn + Fn−1. Hence both bisections of the Fibonacci sequence

F = 0, 1, 3, 8, 21, 55, 144, 377,... (Sloane’s A001906) { 2n}n≥0 { } and F = 1, 2, 5, 13, 34, 89, 233, 610,... (Sloane’s A001519) { 2n+1}n≥0 { } satisfy the same recurrence relation z =3z z . The former is log-concave and the n+1 n − n−1 latter is log-convex according to the initial values. The Lucas sequence

L = 1, 3, 4, 7, 11, 18, 29, 47,... (Sloane’s A000204) { n}n≥0 { } satisﬁes the same recurrence with the Fibonacci sequence and enjoys the similar property. The Pell sequence

P = 1, 2, 5, 12, 29, 70, 169, 408,... . (Sloane’s A000129) { n}n≥0 { } satisﬁes the recurrence P = 2P + P . The bisections P and P are log- n+1 n n−1 { 2n} { 2n+1} convex and log-concave respectively.

4 q-log-convexity

In this section we first introduce the concept of the q-log-convexity of polynomial sequences and then prove the q-log-convexity of certain well-known polynomial sequences, including the Bell polynomials, the Eulerian polynomials and the q-Schröder numbers. We also present certain linear transformations preserving the log-convexity of sequences and establish the link with the q-log-convexity. Let q be an indeterminate. Given two real polynomials f(q) and g(q), write f(q) ≤q g(q) if and only if g(q) f(q) has nonnegative coefficients as a polynomial in q. A sequence − of real polynomials P (q) is called q-log-convex if { n }n≥0 P 2(q) P (q)P (q) (4.1) n ≤q n−1 n+1 for all n 1. Clearly, if the sequence P (q) is q-log-convex, then for each fixed ≥ { n }n≥0 positive number q, the sequence P (q) is log-convex. The converse is not true in { n }n≥0 general. If the opposite inequality in (4.1) holds, then the sequence P (q) is called { n }n≥0 q-log-concave. The concept of the q-log-concavity was first suggested by Stanley and these

18 has been much interest in this subject. We refer the reader to Sagan [41, 42] for further information about the q-log-concavity. Perhaps the simplest example of q-log-convex polynomials is the q-factorial. It is well known that the factorial n! is log-convex. The standard q-analogue of an integer n is (n) = 1+ q + q2 + + qn−1 and the associated q-factorial is (n) ! = n (k) . It is q ··· q k=1 q easy to verify that the q-factorial (n)q! is q-log-convex by a direct calculation.Q We next provide more examples of q-log-convex sequences.

4.1 Bell polynomials and Eulerian polynomials

The Bell polynomial, or the exponential polynomial, is the generating function Bn(q)= n k k=0 S(n, k)q of the Stirling numbers of the second kind. It can be viewed as a q-analog of the Bell number and has many fascinating properties (see Roman [39, 4.1.3] for in- P § stance). Note that the Stirling numbers of the second kind satisfy the recurrence

S(n +1,k)= kS(n, k)+ S(n, k 1) − Hence the Bell polynomials satisfy the recurrence

′ Bn+1(q)= qBn(q)+ qBn(q).

It is well known that the Bell polynomials Bn(q) have only real zeros (see [59] for instance). In 2 we have shown that the linear transformation z = n S(n, k)x can § n k=0 k preserve the log-convexity of sequences. Therefore, for each positiveP number q, the sequence B (q) is log-convex. A further problem is whether the sequence B (q) { n }n≥0 { n }n≥0 is q-log-convex. Let π = a a a be a permutation of [n]. An element i [n 1] is called a descent 1 2 ··· n ∈ − of π if ai > ai+1. The Eulerian number A(n, k) is deﬁned as the number of permutations of [n] having k 1 descents. The Eulerian triangle A(n, k) is − { }1≤k≤n 1 1 1 1 4 1 (Sloane’s A008292) 1 11 11 1 1 26 66 26 1

··· with A(0, 0) = 1. The Eulerian numbers satisfy the recurrence

A(n, k)= kA(n 1,k)+(n k + 1)A(n 1,k 1). − − − − 19 n k Let An(q)= k=0 A(n, k)q be the Eulerian polynomial. Then P A (q)= nqA (q)+ q(1 q)A′ (q). n n−1 − n−1

It is well known that An(q) has only real zeros and A(n, k) is therefore log-concave in k for each ﬁxed n (see [59] for instance). By Frobenius formula

n A (q)= q k!S(n, k)(q 1)n−k n − Xk=1 and Proposition 2.5, the sequence A (q) is log-convex for each ﬁxed positive number { n }n≥0 q 1. We refer the reader to Comtet [16] for further information on the Eulerian numbers ≥ and the Eulerian polynomials. In what follows we show that both the Bell polynomials and the Eulerian polynomials are q-log-convex by establishing a general result. Let T (n, k) be an array of nonnegative numbers satisfying the recurrence { }n,k≥0 T (n, k)=(a n + a k + a )T (n 1,k)+(b n + b k + b )T (n 1,k 1) (4.2) 1 2 3 − 1 2 3 − − with T (n, k) = 0 unless 0 k n. It is natural to assume that a n + a k + a 0 for ≤ ≤ 1 2 3 ≥ 0 k < n and b n+b k+b 0 for 0

Theorem 4.1. Let T (n, k) be as above and the row generating function T (q) = { }n,k≥0 n n T (n, k)qk. Suppose that for 0

a b a b , a (b + b ) a b , a (b + b + b ) (a + a )b 2 1 − 1 2 2 1 2 − 1 2 2 1 2 3 − 1 3 2 are all nonnegative. Hence the polynomial Tn(q) in Theorem 4.1 has only real zeros for each n 0 by Wang and Yeh [59, Corollary 3]. ≥ Remark 4.3. If a2 = b2 = 0, then the condition (4.3) is trivially satisﬁed.

20 Proof of Theorem 4.1. Let T (q)T (q) T 2(q) = 2n A qt. We need to show that n−1 n+1 − n t=0 t A 0 for 0 t 2n. Note that the recurrence (4.2) is equivalent to t ≥ ≤ ≤ P

′ Tn(q) = (a1n + a3 + b1nq + b2q + b3q)Tn−1(q)+(a2 + b2q)qTn−1(q).

Hence

2n t ′ Atq = Tn−1(q)[(a1n + a1 + a3 + b1nq + b1q + b2q + b3q)Tn(q)+(a2 + b2q)qTn(q)] t=0 X T (q)[(a n + a + b nq + b q + b q)T (q)+(a + b q)qT ′ (q)] − n 1 3 1 2 3 n−1 2 2 n−1 = (a + b q)T (q)T (q)+(a + b q)q[T (q)T ′ (q) T ′ (q)T (q)]. 1 1 n−1 n 2 2 n−1 n − n−1 n t Thus At = k=0 ck(n, t), where P c = T (n, t k)[a T (n 1,k)+ b T (n 1,k 1) + a (t k)T (n 1,k) k − 1 − 1 − − 2 − − a kT (n 1,k)+ b (t k)T (n 1,k 1) b (k 1)T (n 1,k 1)] − 2 − 2 − − − − 2 − − − = T (n, t k)[(a + ta 2ka )T (n 1,k)+(b + b + tb 2kb )T (n 1,k 1)]. − 1 2 − 2 − 1 2 2 − 2 − − Clearly, c 0 if t is even and k = t/2. So, to prove that A 0, it suﬃces to prove that k ≥ t ≥ c + c 0 for 0 k < t k n. Let u = T (n 1,k) if 0 k n 1 and u = 0 k t−k ≥ ≤ − ≤ k − ≤ ≤ − k otherwise. Then the sequence u is log-concave. In what follows, we always assume { k}k≥0 that 0 k < t k n. By the recurrence (4.2), we have ≤ − ≤ c + c = [(a n + a (t k)+ a )u +(b n + b (t k)+ b )u ] k t−k 1 2 − 3 t−k 1 2 − 3 t−k−1 [(a + ta 2ka )u +(b + b + tb 2kb )u ] × 1 2 − 2 k 1 2 2 − 2 k−1 +[(a1n + a2k + a3)uk +(b1n + b2k + b3)uk−1] [(a ta +2ka )u +(b + b tb +2kb )u ] × 1 − 2 2 t−k 1 2 − 2 2 t−k−1 = xukut−k + yuk−1ut−k−1 + zukut−k−1 + wuk−1ut−k,

where

x = a (2a n + a t +2a )+ a2(t 2k)2, 1 1 2 3 2 − y = (b + b )(2b n + b t +2b )+ b2(t 2k)2, 1 2 1 2 3 2 − z = (a n + a k + a )(b + b ) + [b n + b (t k)+ b ]a 1 2 3 1 2 1 2 − 3 1 +(t 2k)[(a b a b )n + a b (t 2k)+(a b a b )], − 2 1 − 1 2 2 2 − 2 3 − 3 2 w = (b n + b k + b )a + [a n + a (t k)+ a ](b + b ) 1 2 3 1 1 2 − 3 1 2 (t 2k)[(a b a b )n a b (t 2k)+(a b a b )]. − − 2 1 − 1 2 − 2 2 − 2 3 − 3 2

21 By the assumption of the recurrence (4.2), the numbers a1, b1 +b2 and a1n+a2k+a3, b1n+ b (t k)+ b are all nonnegative for 0 k < t k n. On the other hand, by the 2 − 3 ≤ − ≤ assumption of the theorem, the number (a b a b )n + a b (t 2k)+(a b a b ) is 2 1 − 1 2 2 2 − 2 3 − 3 2 nonnegative since 0 < t 2k t k n. Hence z 0 for 0 k < t k n. Thus by − ≤ − ≤ ≥ ≤ − ≤ the log-convexity of u , we have { k} c + c xu u + yu u +(z + w)u u k t−k ≥ k t−k k−1 t−k−1 k−1 t−k = a1(2a1n + a2t +2a3)ut−k(uk + uk−1)

+(b1 + b2)(2b1n + b2t +2b3)uk−1(ut−k−1 + ut−k) +(t 2k)2(a2u u + b2u u +2a b u u ). − 2 k t−k 2 k−1 t−k−1 2 2 k−1 t−k Note that

2a n + a t +2a = [a n + a (t k)+ a ]+(a n + a k + a ), 1 2 3 1 2 − 3 1 2 3 2b n + b t +2b = [b n + b (t k)+ b ]+(b n + b k + b ). 1 2 3 1 2 − 3 1 2 3 Hence (2a n + a t +2a )u 0 and (2b n + b t +2b )u 0 for 0 k < t k n. 1 2 3 t−k ≥ 1 2 3 k−1 ≥ ≤ − ≤ Moreover, a2u u +b2u u +2a b u u 0 by the arithmetic-geometric mean 2 k t−k 2 k−1 t−k−1 2 2 k−1 t−k ≥ inequality and the log-concavity of u . Consequently we have c + c 0 for all { k} k t−k ≥ 0 t 2n and 0 k t/2 , the required result. This completes the proof of the ≤ ≤ ≤ ≤ ⌊ ⌋ theorem. Now the q-log-convexity of the Bell polynomials and the Eulerian polynomials follows immediately from Theorem 4.1.

Proposition 4.4. The Bell polynomials Bn(q) form a q-log-convex sequence.

Remark 4.5. An immediate consequence of Proposition 4.4 is the log-convexity of the Bell numbers. Also, note that 2-colored Bell number Sn = Bn(2) in Example 2.6. Hence the log-convexity of S can also be followed from Proposition 4.4. { n}n≥0

Proposition 4.6. The Eulerian polynomials An(q) form a q-log-convex sequence.

Remark 4.7. Note that the ordered Bell number c(n) = An(2)/2 in Example 2.7 by the Frobenius formula. Hence the log-convexity of c(n) can also be followed from { } Proposition 4.6.

4.2 Narayana polynomials and q-Schr¨oder numbers

Narayana number N(n, k) is deﬁned as the number of Dyck paths of length 2n with exactly k peaks (a peak of a path is a place at which the step (1, 1) is directly followed

22 by the step (1, 1)). The triangle N(n, k) of the Narayana numbers is − { }1≤k≤n 1 1 1 1 3 1 (Sloane’s A001263) 1 6 6 1 1 10 20 10 1

··· n with N(0, 0) = 1. By definition, k=0 N(n, k) = Cn, the Catalan numbers. Hence the Narayana numbers can be viewed asP a refinement of the Catalan numbers. The Narayana numbers have an explicit expression 1 n n N(n, k)= . n k k 1 − It is easily verified that N(n, k) is log-concave in k for each fixed n by a direct { }1≤k≤n calculation. See Stanley [48, Exercise 6.36] and Sulanke [52] for further information about the Narayana numbers. n k The Narayana polynomial Nn(q) = k=0 N(n, k)q is the generating function of the Narayana numbers. It is known that P

(n + 1)N (q)=(2n 1)(1 + q)N (q) (n 2)(1 q)2N (q) (4.4) n − n−1 − − − n−2

(see Sulanke [52] for a combinatorial proof). It is interesting that Nn(1) is exactly the Catalan number Cn and Nn(2) is exactly the large Schröder number rn ([52, 8]). Shaprio [45] asked whether the Narayana distribution is normal. A stronger result is that Nn(q) has only real zeros. Stanley noted that the result is implied by Brenti [11, Theorem 5.3.1] (see Bóna [7]) and Brändén [10] found a simple proof recently by ex- pressing the Narayana polynomial in terms of Jacobi polynomial. Very recently, we give a direct proof of the result in [35] by showing that N (q) forms a Sturm sequence, { n } i.e., zeros of Nn(q) are all real and interlace with those of Nn+1(q). Thus the sequence N(n, 0), N(n, 1),...,N(n, n) is PF for each n 1. ≥ The q-Schröder number rn(q), introduced by Bonin, Shapiro and Simion [8], is defined

as the q-analog of the large Schr¨oder number rn:

diag(P ) rn(q)= q P X where P takes over all Schr¨oder paths from (0, 0)to(n, n) and diag(P ) denotes the number n+k n−k of diagonal steps in the path P . Clearly, rn(1) = rn and rn(q)= k=0 n−k Ckq . The P 23 q-Schr¨oder number rn(q) is closely related to the Narayana polynomial Nn(q). Actually, r (q) = N (1 + q)(see [52] for instance). Now we prove that the sequence r (q) is n n { n }n≥0 q-log-convex.

Proposition 4.8. The q-Schr¨oder numbers rn(q) form a q-log-convex sequence.

Proof. Note that N (q)N (q) N 2(q) = q. Hence r (q)r (q) r2(q)=1+ q. Now 0 2 − 1 0 2 − 1 assume that r (q)r (q) r2 (q) has nonnegative coeﬃcients. Applying repeatedly the n−2 n − n−1 recurrence (4.4), we have

N (q)N (q) N 2(q) n−1 n+1 − n 2n +1 n 1 = N (q) (1 + q)N (q) − (1 q)2N (q) n−1 n +2 n − n +2 − n−1 2n 1 n 2 N (q) − (1 + q)N (q) − (1 q)2N (q) − n n +1 n−1 − n +1 − n−2 3 = N (q)[(1 + q)N (q) (1 q)2N (q)] (n + 1)(n + 2) n n−1 − − n−2 n 1 + − (1 q)2[N (q)N (q) N 2 (q)] n +2 − n−2 n − n−1 3 = N (q)[N (q) (1 + q)N (q)] (n 2)(n + 2) n n − n−1 − n 1 + − (1 q)2[N (q)N (q) N 2 (q)]. n +2 − n−2 n − n−1 Note that the coefficient of qk in N (q) (1 + q)N (q) is n − n−1 2 n 1 n 1 N(n, k) N(n 1,k) N(n 1,k 1) = − − 0. − − − − − n 1 k 2 k ≥ − − (It is easy to give a combinatorial interpretation of N(n, k) N(n 1,k)+N(n 1,k 1) ≥ − − − by the definition of Narayana numbers.) Hence r (q)r (q) r2 (q) has nonnegative n−1 n+1 − n coefficients. Thus the sequence r (q) is q-log-convex by induction. { n } We will provide another proof of Proposition 4.8 in Example 4.14. We can show that for each fixed nonnegative number q, the sequence N (q) is { n }n≥0 log-convex by means of Theorem 3.10 and the recurrence (4.4). We propose the following conjecture which obviously implies Proposition 4.8.

Conjecture 4.9. The Narayana polynomials Nn(q) form a q-log-convex sequence.

4.3 Linear transformations preserving log-convexity

In [60], Wang and Yeh established the link between linear transformations preserving the log-concavity and the q-log-concavity. A similar approach is eﬀective for the log- convexity. We demonstrate the general applicability of the method by an example.

24 Proposition 4.10. The linear transformation

n n + k z = x , n =0, 1, 2,... n n k k Xk=0 − preservers the log-convexity of sequences.

n n+k Example 4.11. Since the odd-indexed Fibonacci numbers F2n+1 = k=0 n−k , the log- convexity of the numbers F2n+1 follows immediately from PropositionP 4.10. n n+k Example 4.12. Since the large Schr¨oder numbers rn = k=0 n−k Ck, the log-convexity of the numbers rn is implied by the log-convexity of theP Catalan numbers Ck. n n+k Example 4.13. Since the central Delannoy numbers D(n) = k=0 n−k b(k), the log- convexity of the numbers D(n) is implied by the log-convexityP of the central binomial coeﬃcients b(k).

Proof of Proposition 4.10. Let x be a log-convex sequence. For n 1, denote { k}k≥0 ≥ ∆ = z z z2. We need to prove ∆ 0. Note that n n−1 n+1 − n n ≥ 2 n−1 n 1+ k n+1 n +1+ k n n + k ∆n = − xk xk xk (4.5) " n 1 k # " n +1 k # − " n k # Xk=0 − − Xk=0 − Xk=0 − is a quadratic form in n + 2 variables x0, x1,...,xn, xn+1. Let 0 t 2n. For 0 k ≤ ≤ ≤2n ≤ t/2 , denote by ak(n, t) the coeﬃcient of the term xkxt−k in ∆n. Then ∆n = t=0 St ⌊ ⌋ ⌊t/2⌋ where St = k=0 ak(n, t)xkxt−k. So it suﬃces to prove St 0. We do this in twoP steps. ⌊t/2⌋ ≥ First, we showP that A(n, t) = k=0 ak(n, t) is nonnegative. Second, we show that there exists an index r such that a (n, t) 0 for k r and a (n, t) 0 for k>r. But the kP ≥ ≤ k ≤ log-convexity of the sequence x implies that x x is decreasing for 0 k t/2 . It { k} k t−k ≤ ≤ ⌊ ⌋ follows that

⌊t/2⌋ ⌊t/2⌋ S = a (n, t)x x a (n, t)x x = A(n, t)x x 0, t k k t−k ≥ k r t−r r t−r ≥ Xk=0 Xk=0 the required result. Step 1. The nonnegativity of A(n, t). Observe that the generating function of A(n, t) is

2 2n n−1 n 1+ k n+1 n +1+ k n n + k A(n, t)qt = − qk qk qk . n 1 k n +1 k − n k t=0 " # " # " # X Xk=0 − − Xk=0 − Xk=0 − n n+k k Let fn(q)= k=0 n−k q , which is the n-th Morgan-Voyce polynomial (see [54] for further information). Then 2n A(n, t)qt = f (q)f (q) f 2(q). Thus the nonnegativity of P t=0 n−1 n+1 − n P 25 A(n, t) is equivalent to the q-log-convexity of the sequence f (q) . By the recurrence { n } relation of the binomial coeﬃcients, we can obtain n +1+ k n + k n + k 1 n 1+ k =2 + − − . n +1 k n k n k +1 − n 1 k − − − − − From this it follows that

f (q)=(2+ q)f (q) f (q), n+1 n − n−1 which is equivalent to f f 2+ q 1 f f n+1 n = − n n−1 . fn fn−1! 1 0 ! fn−1 fn−2! Now consider the determinants on the two sides of the equality. Then we have

f (q)f (q) f 2(q)= f (q)f (q) f 2 (q)= = f (q)f (q) f 2(q)= q n−1 n+1 − n n−2 n − n−1 ··· 0 2 − 1 2 by the induction and the initial conditions f0(q)=1, f1(q)=1+q and f2(q)=1+3q +q . Thus the sequence f (q) is q-log-convex and all A(n, t) are therefore nonnegative. { n }n≥0 Step 2. The existence of the index r. By (4.5), we have n 1+ k n +1+ t k n +1+ k n 1+ t k a (n, t) = − − + − − k n 1 k n +1 t + k n +1 k n 1 t + k − − − − − − n + k n + t k 2 − − n k n t + k − − (n 1+ k)!(n 1+ t k)! = − − − b (n, t) (2k)!(2t 2k)!(n +1 k)!(n +1 t + k)! k − − − when k

b (n, t) = (n + k)(n +1+ k)(n t + k)(n t + k + 1) k − − +(n k)(n k + 1)(n + t k)(n + t k + 1) − − − − 2(n + k)(n k + 1)(n + t k)(n t + k + 1). − − − −

Clearly, ak(n, t) has the same sign as that of bk(n, t) for each k. Using MATHEMATICA, we obtain that the derivative of b (n, t) with respect to k is 2(t 2k)[2(2n+1)2 t] 0. k − − − ≤ Thus bk(n, t) changes sign at most once (from nonnegative to nonpositive), and so does ak(n, t). But it is impossible that all ak(n, t) take negative values since their summation A(n, t) 0. Hence there exists an index r such that a (n, t) 0 for k r and a (n, t) 0 ≥ k ≥ ≤ k ≤ for k>r. This completes our proof.

26 From the proof of Proposition 4.10, we can see that the q-log-convexity plays a key role in showing that a linear transformation can preserve the log-convexity. More precisely, given a triangle a(n, k) of nonnegative real numbers, consider the sequence of { }0≤k≤n polynomials n k An(q)= a(n, k)q , n =0, 1, 2,... Xk=0 and the linear transformation n

zn = a(n, k)xk, n =0, 1, 2,.... Xk=0 For convenience, deﬁne a(n, k) = 0 unless 0 k n. Let 0 t 2n and denote ≤ ≤ ≤ ≤ a (n, t)= a(n 1,k)a(n +1, t k)+ a(n +1,k)a(n 1, t k) 2a(n, k)a(n, t k) k − − − − − − if 0 k

(C1). There exists an index r = r(n, t) such that a (n, t) 0 for k r and a (n, t) < 0 k ≥ ≤ k for k>r.

⌊t/2⌋ (C2). The summation A(n, t)= k=0 ak(n, t) is nonnegative.

Then the following results hold.P

n k (R1). The sequence of polynomials An(q)= k=0 a(n, k)q is q-log-convex.

n P (R2). The linear transformation zn = k=0 a(n, k)xk preserves the log-convexity.

(R3). If the sequence u is log-convexP and b(n, k) = a(n, k)u for 0 k n, then { k}k≥0 k ≤ ≤ the triangle b(n, k) also possesses the properties (C1) and (C2). { }0≤k≤n Actually, note that

2n ⌊t/2⌋ 2n A (q)A (q) A2 (q)= a (n, t) qt = A(n, t)qt n−1 n+1 − n  k  t=0 t=0 X Xk=0 X   and 2n ⌊t/2⌋ z z z2 = a (n, t)x x . n−1 n+1 − n  k k t−k t=0 X Xk=0   27 Hence (R1) is equivalent to (C2). Assume that x is log-convex. Then x x { k}k≥0 0 t ≥ x x x x . It follows that 1 t−1 ≥ 2 t−2 ≥··· 2n ⌊t/2⌋ 2n z z z2 a (n, t)x x = A(n, t)x x 0 n−1 n+1 − n ≥  k r t−r r t−r ≥ t=0 t=0 X Xk=0 X   by the conditions (C1) and (C2). Thus z is log-convex, and so (R2) holds. Finally, { n}n≥0 note that b (n, t)= a (n, t)u u . Hence B(n, t) A(n, t)u u , and so (R3) holds. k k k t−k ≥ r t−r n+k Example 4.14. In the proof of Proposition 4.10, the triangle a(n, k) = n−k is shown to possess the condition (C1) and (C2). Now consider the q-Schr¨oder numbers n n + k r (q)= C qn−k n n k k Xk=0 − and the q-central Delannoy numbers

n n + k D (q)= b(k)qn−k n n k Xk=0 −

(see Sagan [43]). Since both the Catalan numbers Ck and the central binomial coeﬃcients b(k) are log-convex, it immediately follows that both r (q) and D (q) are q-log-convex { n } { n } from the result (R3).

2n n n 2 It is known that the central binomial coeﬃcients n = k=0 k and the central n n 2 k Delannoy numbers D(n)= k=0 k 2 ([53]). So a problem naturallyP arises. P n 2 Conjecture 4.15. The triangle k possesses the properties (C1) and (C2). n o 5 Concluding remarks and open problems

In this paper we have explored the log-convexity of some combinatorial sequences by algebraic and analytic approaches. It is natural to look for combinatorial interpretations for the log-convexity of these sequences since their strong background in combinatorics. Callan [15] gave an injection proof for the log-convexity of the Motzkin numbers. It is possible to give combinatorial interpretations for the log-convexity of more combinatorial numbers. We feel that the lattice paths techniques of Wilf [62] and Gessel-Viennot [29] may be useful. As an example, we give an injection proof for the log-convexity of the Catalan numbers.

Recall that the Catalan number Cn is the number of lattice paths from (i, i) to (n + i, n + i) with steps (0, 1) and (1, 0), never rising above the line y = x (see Stanley [48,

28 Exercise 6.19 (h)] for instance). Let Cn(i) be the set of such paths. We next show that C2 C C by constructing an injection n ≤ n+1 n−1 φ : C (0) C (1) C (0) C (1). n × n → n+1 × n−1 Consider a path pair (p, q) C (0) C (1). Clearly, p and q must intersect. Let C be ∈ n × n their first intersect point. Then C splits p into parts p1 and p2, and splits q into parts q1 ′ and q2. Thus the concatenation p of p1 and q2 is a path in Cn+1(0), and the concatenation ′ ′ ′ q of q1 and p2 is a path in Cn−1(1). Define φ(p, q)=(p , q ). Then the image of φ consists of precisely (p′, q′) C (0) C (1) such that p′ and q′ intersect. It is easy to see that ∈ n+1 × n−1 if φ(p, q)=(p′, q′), then applying the same algorithm to (p′, q′) recovers (p, q). Thus φ is injective, as desired. It would be interesting to have a combinatorial interpretation for the log-convexity of combinatorial sequences satisfying a three-term recurrence. We refer the reader to Sagan [40] for combinatorial proofs for the log-concavity of combinatorial sequences satisfying a three-term recurrence. Another intriguing problem is to find a combinatorial interpretation for the log- convexity of the Bell numbers. We end this paper by proposing the following conjectures.

n Conjecture 5.1. The Eulerian transformation zn = k=0 A(n, k)xk preserves the log- convexity. P n Conjecture 5.2. The Narayana transformation zn = k=0 N(n, k)xk preserves the log- convexity. P

References

[1] M. Aigner, Motzkin numbers, European J. Combin. 19 (1998) 663–675.

[2] M. Aissen, I. J. Schoenberg and A. M. Whitney, On the generating functions of totally positive sequences I, J. Analyse Math. 2 (1952) 93–103.

[3] N. Asai, I. Kubo and H.-H. Kuo, Bell numbers, log-concavity, and log-convexity, Acta Appl. Math. 63 (2000) 79–87.

[4] C. Banderier and S. Schwer, Why Delannoy Numbers? J. Statist. Plann. Inference 135 (2005) 40–54.

29 [5] E. A. Bender and E. R. Canﬁeld, Log-concavity and related properties of the cycle index polynomials, J. Combin. Theory Ser. A 74 (1996) 57–70.

[6] F. R. Bernhart, Catalan, Motzkin, and Riordan numbers, Discrete Math. 204 (1999) 73–112.

[7] M. B´ona, Symmetry and Unimodality in Stack sortable permutations, J. Combin. Theory Ser. A 98 (2002) 201–209; correction: J. Combin. Theory Ser. A 99 (2002) 191–194.

[8] J. Bonin, L. Shapiro and R. Simion, Some q-analogues of the Schr¨oder numbers arising from combinatorial statistics on lattice paths, J. Statist. Plann. Inference 34 (1993) 35–55.

[9] J. M. Borwein and R. Theodorescu, Problems and Solutions: Solutions: Moments of the Poisson Distribution: 10738. Amer. Math. Monthly 107 (2000) p. 659.

[10] P. Br¨and´en, The generating function of two-stack sortable permutations by descents is real-rooted, arXiv: math.CO/0303149.

[11] F. Brenti, Unimodal, log-concave, and P´olya frequency sequences in combinatorics, Mem. Amer. Math. Soc. no. 413 (1989).

[12] F. Brenti, Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update, Contemp. Math. 178 (1994) 71–89.

[13] F. Brenti, Combinatorics and total positivity, J. Combin. Theory Ser. A 71 (1995) 175–218.

[14] F. Brenti, The applications of total positivity to combinatorics, and conversely, Math. Appl. 359 (1996) 451–473.

[15] D. Callan, Notes on Motzkin and Schr¨oder numbers, http://www.stat.wisc.edu/~callan/papersother/

[16] L. Comtet, Advanced combinatorics, D. Reidel Publishing Co., Dordrecht, 1974.

[17] H. Davenport and G. P´olya, On the product of two power series, Canadian J. Math. 1 (1949) 1–5.

[18] E. Deutsch, Dyck path enumeration, Discrete Math. 204 (1999) 167–202.

30 [19] E. Deutsch, A bijective proof of the equation linking the Schr¨oder numbers, large and small, Discrete Math. 241 (2001) 235–240.

[20] E. Deutsch and L. W. Shapiro, A survey of Fine numbers, Discrete Math. 241 (2001) 235–240.

[21] R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory Ser. A 23 (1977) 291–301.

[22] T. Doˇsli´cand D. Veljan, Calculus proofs of some combinatorial inequalities, Math. Ine. Appl. 2 (2003) 197–209.

[23] T. Doˇsli´c, D. Svrtan and D. Veljan, Enumerative aspects of secondary structures, Discrete Math. 285 (2004) 67–82.

[24] T. Doˇsli´c, Log-balanced combinatorial sequences, Int. J. Math. Math. Sci. 4 (2005) 507–522.

[25] T. Doˇsli´c, Logarithmic behavior of some combinatorial sequences, arXiv: math/CO. 0603379.

[26] D. Foata and D. Zeilberger, A classic proof of a recurrence for a very classical sequence, J. Combin. Theory Ser. A 80 (1997) 380–384.

[27] A. Ehrenfeucht, T. Harju, P. ten Pas and G. Rozenberg, Permutations, parenthesis words, and Schr¨oder numbers, Discrete Math. 190 (1998) 259–264.

[28] K. Engel, On the average rank of an element in a ﬁlter of the partition lattice, J. Combin. Theory Ser. A 65 (1994) 67–78.

[29] I. Gessel and G. Viennot, Binomial determinants, path, and hook length formulae, Adv. in Math. 58 (1985) 300–321.

[30] R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs. 3 (2000) #00.1.6

[31] F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinb. Math. Soc. (2) 17 (1970) 1–13.

[32] L. H. Harper, Stirling behavior is asymptotically normal, Ann. Math. Statist. 38 (1967) 410–414.

31 [33] S. Karlin, Total Positivity, Vol.I, Stanford University Press, 1968.

[34] D. C. Kurtz, A note on concavity properties of triangular arrays of numbers, J. Combin. Theory Ser. A 13(1972) 135–139.

[35] L. Liu and Y. Wang, A uniﬁed approach to polynomial sequences with only real zeros, arXiv: math. CO/0509207, to appear in Adv. in Appl. Math.

[36] P. Peart and W.-J. Woan, Bijective proofs of the Catalan and ﬁne recurrences, Congr. Numer. 137 (1999) 161–168.

[37] P. Peart and W.-J. Woan, A bijective proof of the Delannoy recurrence, Congr. Numer. 158 (2002) 29–33.

[38] J. Riordan, An introduction to combinatorial analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London 1958.

[39] S. Roman, The Umbral Calculus, New York: Academic Press, 1984.

[40] B. E. Sagan, Inductive and injective proofs of log concavity results, Discrete Math. 68 (1988) 281–292.

[41] B. E. Sagan, Log concave sequences of symmetric functions and analogs of the Jacobi- Trudi determinants, Trans. Amer. Math. Soc. 329 (1992) 795–811.

[42] B. E. Sagan, Inductive proofs of q-log concavity, Discrete Math. 99 (1992) 298–306.

[43] B. E. Sagan, Unimodality and the reﬂection principle, Ars Combin. 48 (1998) 65–72.

[44] E. Schirmacher, Log-concavity and the exponential formula, J. Combin. Theory Ser. A 85 (1999) 127–134.

[45] L. Shapiro, Some open questions about random walks, involutions, limiting distribu- tions, and generating functions, Adv. in Appl. Math. 27 (2001) 585–596.

[46] R. P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Ann. New York Acad. Sci. 576 (1989) 500–534.

[47] R. P. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cam- bridge, 1997.

[48] R. P. Stanley, Enumerative combinatorics, Vol. 2, Cambridge University Press, Cam- bridge, 1997.

32 [49] R. P. Stanley, Hipparchus, Plutarch, Schr¨oder, and Hough, Amer. Math. Monthly 104 (1997) 344–350.

[50] R. Sulanke, Bijective recurrences concerning Schr¨oder paths, Electron. J. Combin. 5 (1998), Research Paper 47, 11 pp.

[51] R. Sulanke, Bijective recurrences for Motzkin paths, Adv. in Appl. Math. 27 (2001) 627–640.

[52] R. Sulanke, The Narayana distribution, J. Statist. Plann. Inference 101 (2002) 311– 326.

[53] R. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003) Article 03.1.5, 19 pp.

[54] M. N. S. Swamy, Further properties of Morgan-Voyce polynomials, Fibonacci Quart. 6 (1968) 167–175.

[55] Y. Wang, A simple proof of a conjecture of Simion, J. Combin. Theory Ser. A 100 (2002) 399–402.

[56] Y. Wang, Proof of a conjecture of Ehrenborg and Steingr´imsson on excedance statis- tic, European J. Combin. 23 (2002) 355–365.

[57] Y. Wang, Linear transformations preserving log-concavity, Linear Algebra Appl. 359 (2003) 162–167.

[58] Y. Wang and Y. -N. Yeh, Proof of a conjecture on unimodality, European J. Combin. 26 (2005) 617–627.

[59] Y. Wang and Y. -N. Yeh, Polynomials with real zeros and P´olya frequency sequences, J. Combin. Theory Ser. A 109 (2005) 63–74.

[60] Y. Wang and Y.-N. Yeh, Log-concavity and LC-positivity, arXiv: math.CO/0504164, to appear in J. Combin. Theory Ser. A.

[61] J. West, Generating trees and the Catalan and Schr¨oder numbers, Discrete Math. 146 (1995) 247–262.

[62] H. S. Wilf, A uniﬁed setting for sequencing, ranking, and selection algorithms for combinatorial objects, Advances in Math. 24 (1977) 281–291.