Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn

Xun-Tuan Su; Fan-Bo Sun

doi:10.1515/math-2022-0555

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Refined ratio monotonicity of the coordinator polynomials of the root lattice of type B_n

Xun-Tuan Su und Fan-Bo Sun

Veröffentlicht/Copyright: 14. Februar 2023

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Abstract

Ratio monotonicity, a property stronger than both log-concavity and the spiral property, describes the behavior of the coefficients of many classical polynomials. It is known that the coordinator polynomials of the root lattice of type B n possess log-concavity. In this paper, we show that the coordinator polynomials of type B n have a refined version of ratio monotonicity except for the first and the last terms. To our knowledge, this refined version is novel.

Keywords: ratio monotonicity; coordinator polynomials; log-concavity; spiral property

MSC 2010: 05A15; 05A20; 11H06

1 Introduction

Unimodality and log-concavity have been extensively studied in many branches such as combinatorics, algebra, geometry and analysis. See the survey articles [1–3] for various techniques, problems, and results about unimodality and log-concavity. Let P ( x ) = ∑ i = 0 n a i x i be a polynomial of degree n with positive coefficients. It is said to be unimodal if there exists an index t such that

a 0 ≤ a 1 ≤ ⋯ ≤ a t − 1 ≤ a t ≥ a t + 1 ≥ ⋯ ≥ a n − 1 ≥ a n .

The polynomial P ( x ) is said to be spiral if

a n ≤ a 0 ≤ a n − 1 ≤ a 1 ≤ ⋯ ≤ a n 2 ,

where ⌊ x ⌋ is the floor function. The polynomial P ( x ) is said to be palindromic if a k = a n − k for all indices k = 0 , 1 , … , n . The polynomial P ( x ) is said to be log-concave if a k − 1 a k + 1 ≤ a k 2 for 1 ≤ k ≤ n − 1 , or equivalently,

a 0 a 1 ≤ a 1 a 2 ≤ ⋯ ≤ a n − 1 a n .

In the case that all the inequalities are strict in the definitions above, we say that P ( x ) is strictly unimodal, strictly spiral or strictly log-concave. Clearly, a palindromic and unimodal polynomial is in particular a spiral polynomial. Either log-concavity or the spiral property yields unimodality, while a log-concave polynomial is not necessarily spiral, and vice versa (see [4]). Many well-known combinatorial polynomials are unimodal or log-concave (see, e.g., [5–11]). For example, Wang and Zhao [12] proved that all coordinator polynomials of Weyl group lattices are log-concave.

Ratio monotonicity, introduced by Chen and Xia [13], is a property which implies both log-concavity and the spiral property. Precisely speaking, a polynomial of degree n with positive coefficients P ( x ) = ∑ i = 0 n a i x i is said to be ratio monotone if its coefficients satisfy

(1) a n a 0 ≤ a n − 1 a 1 ≤ ⋯ ≤ a n − k a k ≤ ⋯ ≤ a n − n − 1 2 a n − 1 2 ≤ 1

and

(2) a 0 a n − 1 ≤ a 1 a n − 2 ≤ ⋯ ≤ a k − 1 a n − k ≤ ⋯ ≤ a n 2 − 1 a n − n 2 ≤ 1 ,

where ⌊ x ⌋ is the floor function. In the case that all the inequalities are strict, we say that P ( x ) is strictly ratio monotone. It is known that the well-studied Boros-Moll polynomials, which are a special class of Jacobi polynomials, possess strict ratio monotonicity [4,13]. The notation of ratio monotonicity sometimes needs to be modified for different objects. For example, for the q -derangement numbers, the ratio monotonicity of a sequence { a i } i = 1 n of positive numbers is defined as follows:

a n a 2 ≤ a n − 1 a 3 ≤ ⋯ ≤ a n 2 + 2 a ⌈ n 2 ⌉ ≤ 1

and

a 1 a n ≤ a 2 a n − 1 ≤ ⋯ ≤ a n 2 a ⌈ n 2 ⌉ + 1 ≤ 1 ,

where ⌈ x ⌉ is the ceiling function. It is known that the q -derangement numbers D n ( q ) , which are the generating polynomials of the major index over all derangements on [ n ] , possess strict ratio monotonicity for n ≥ 6 except for the last term when n is even. See [14] for the details.

This paper is devoted to showing that the coordinator polynomials of the root lattice of type B n possess a refined version of strict ratio monotonicity, except for the first and the last terms. As a consequence, we obtain both the strict log-concavity and the spiral property for this family of coordinator polynomials. Note that the log-concavity of the coordinator polynomials of the root lattice of type B n was proved by Wang and Zhao [12], while the spiral property is new in the literature.

Let us first give a brief overview of the coordinator polynomials of the classical root lattices. Let ℒ denote a lattice, which is a discrete subgroup of a Euclidean vector space. The rank of ℒ is the dimension of the subspace spanned by ℒ . In what follows, assume that ℒ is a lattice of rank d generated as a monoid by a finite collection of vectors ℳ . That is, each vector in ℒ can be expressed as a nonnegative integer linear combination of the vectors in ℳ . The length of each vector v ∈ ℒ with respect to ℳ is min ∑ c i taken over all expressions v = ∑ c i m i , where m i ∈ ℳ and c i s are nonnegative integers. Define the growth series to be the generating function ∑ i ≥ 0 S ( i ) x i , where S ( i ) is the number of elements of ℒ having length equal to i . It is well known that the growth series can be expressed as a rational function of the form

∑ i ≥ 0 S ( i ) x i = h ( x ) ( 1 − x ) d ,

where the numerator h ( x ) is a polynomial of degree less than or equal to the rank d (see [15]). The polynomial h ( x ) is called the coordinator polynomial and it depends on both ℒ and ℳ (see [16]).

A root lattice of type X is an integral lattice generated by its corresponding root system which is denoted by ℳ X here. To be precise, let e i be the i -th unit coordinate vector in some Euclidean space E . If E is taken to be R n + 1 , the root lattice of type A n is generated as a monoid by the finite set of vectors (see, e.g., [12])

ℳ A n = { ± ( e i − e j ) ∣ 1 ≤ i < j ≤ n + 1 } .

Note that this set is precisely a root system. If E is taken to be R n , we obtain the root lattices of type B n , C n , and D n , which are generated as monoids by their respective root systems (see, e.g., [12])

ℳ B n = { ± e i ± e j ∣ 1 ≤ i < j ≤ n } ⊔ { ± e i ∣ 1 ≤ i ≤ n } , ℳ C n = { ± e i ± e j ∣ 1 ≤ i < j ≤ n } ⊔ { ± 2 e i ∣ 1 ≤ i ≤ n } , ℳ D n = { ± e i ± e j ∣ 1 ≤ i < j ≤ n } .

Denote the coordinator polynomial of the root lattice of type X by P X ( x ) . The coordinator polynomials of the respective root lattices are known to be (see [17,18])

P A n ( x ) = ∑ k = 0 n n k 2 x k , P B n ( x ) = 1 2 [ ( 1 + x ) 2 n + 1 + ( 1 − x ) 2 n + 1 ] − 2 n x ( 1 + x ) n − 1 = ∑ k = 0 n 2 n + 1 2 k − 2 k n k x k , P C n ( x ) = 1 2 [ ( 1 + x ) 2 n + ( 1 − x ) 2 n ] = ∑ k = 0 n 2 n 2 k x k , P D n ( x ) = 1 2 [ ( 1 + x ) 2 n + ( 1 − x ) 2 n ] − 2 n x ( 1 + x ) n − 2 = ∑ k = 0 n 2 n 2 k − 2 k ( n − k ) n − 1 n k x k .

Incidentally, the polynomials P A n ( x ) , P C n ( x ) , and P D n ( x ) are all palindromic, but P B n ( x ) is not palindromic. Besides, it is known that the polynomials P A n ( x ) , P C n ( x ) , and P D n ( x ) have only real zeros. However, P B n ( x ) is not real rooted in general since P B 16 ( x ) has 14 real roots and 2 nonreal roots (see [12]). Note that the coordinator polynomial of the root lattice of type B n is indeed the h -polynomial of any unimodular triangulation of the corresponding root polytope [17]. It is an important object in both combinatorics and the geometry of numbers.

In this article, we prove a refinement of strict ratio monotonicity, that is, a series of strict inequalities about the coefficients, of the coordinator polynomials of the root lattice of type B n interlaced by the ratios of natural numbers. Our results are much stronger than the log-concavity of the coordinator polynomial P B n ( x ) . The rest of this article is organized as follows. For the coordinator polynomial P B n ( x ) , we present a refined version of strict ratio monotonicity in Section 2, and a refined version of strict log-concavity in Section 3.

2 The main result and its proof

In this section, we present the following theorem that states a refinement of strict ratio monotonicity for the coordinator polynomial P B n ( x ) . For brevity, denote

b k ( n ) = 2 n + 1 2 k − 2 k n k > 0 .

Then P B n ( x ) = ∑ k = 0 n b k ( n ) x k . It is trivial that P B 0 ( x ) = 1 , b 0 ( n ) = 1 , and b n ( n ) = 1 for all n ≥ 1 .

Theorem 1

Let n ≥ 3 be an integer. The coordinator polynomials of the root lattice of type B n possess a refinement of strict ratio monotonicity, except for the first and the last terms. To be precise, the coefficients of the coordinator polynomials of the root lattice of type B n satisfy

(3) 1 n < b 1 ( n ) b n − 1 ( n ) < ⋯ < k n − k + 1 < b k ( n ) b n − k ( n ) < k + 1 n − k < ⋯ < b n − 1 2 ( n ) b n − n − 1 2 ( n ) < n − 1 2 + 1 n − n − 1 2

and

(4) b n ( n ) b 1 ( n ) < 1 n < ⋯ < k − 1 n − k + 2 < b n − k + 1 ( n ) b k ( n ) < k n − k + 1 < ⋯ < b n − n 2 + 1 ( n ) b n 2 ( n ) < n 2 n − n 2 + 1 .

Let us first point out that, by taking the inversion x n P B n 1 x , the chains (3) and (4) of inequalities in Theorem 1 match, respectively, with the chains (1) and (2), where the ratio b 0 ( n ) b n ( n ) is excluded, regardless of ratios of the natural numbers between 1 and n .

Specifically, let P ˆ B n ( x ) = ∑ k = 0 n b ˆ k ( n ) x k be the inversion of the polynomial P B n ( x ) , i.e., b ˆ k ( n ) = b n − k ( n ) for 0 ≤ k ≤ n . Therefore, the chains (3) and (4) of inequalities in Theorem 1 are equivalent to the following two chains:

1 n < b ˆ n − 1 ( n ) b ˆ 1 ( n ) < ⋯ < k n − k + 1 < b ˆ n − k ( n ) b ˆ k ( n ) < k + 1 n − k < ⋯ < b ˆ n − n − 1 2 ( n ) b ˆ n − 1 2 ( n ) < n − 1 2 + 1 n − n − 1 2 < 1 ( 3 ′ )

and

b ˆ 0 ( n ) b ˆ n − 1 ( n ) < 1 n < ⋯ < k − 1 n − k + 2 < b ˆ k − 1 ( n ) b ˆ n − k ( n ) < k n − k + 1 < ⋯ < b ˆ n 2 − 1 ( n ) b ˆ n − n 2 ( n ) < n 2 n − n 2 + 1 < 1 . ( 4 ′ )

Then the chains ( 3 ′ ) and ( 4 ′ ) match respectively with the chains (1) and (2) considering the rightmost 1 instead of ratios of the natural numbers between 1 and n .

In the sequel, we will provide a proof for Theorem 1.

Assume that n ≥ 3 . To prove Theorem 1, it suffices to show

for 1 ≤ k ≤ n − 1 2 ,
k n − k + 1 < b k ( n ) b n − k ( n ) < k + 1 n − k ;
for 1 ≤ k ≤ n 2 ,
k − 1 n − k + 2 < b n − k + 1 ( n ) b k ( n ) < k n − k + 1 .

We will divide the proof of Theorem 1 into several propositions, where some stronger results are obtained.

Proposition 1

For 1 ≤ k ≤ n − 1 2 ,

b k ( n ) b n − k ( n ) < k + 1 n − k .

Proof

Note that

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

Denote

H ( n , k ) = ( n + 1 ) 2 n + 1 2 k + 1 2 ( 2 n − 2 k + 1 ) ( n − k ) n k .

To prove the desired inequality, it suffices to show that H ( n , k ) > 1 for 1 ≤ k ≤ n − 1 2 .

Indeed, we have

1 < H ( n , 1 ) < H ( n , 2 ) < ⋯ < H n , n − 1 2 ,

because for n ≥ 3 ,

H ( n , 1 ) = ( n + 1 ) ( 2 n + 1 ) 6 ( n − 1 ) > 1 ,

and for 1 ≤ k ≤ n − 1 2 − 1 ,

H ( n , k + 1 ) H ( n , k ) = 2 n − 2 k + 1 2 k + 3 ⋅ n − k n − k − 1 > 1 .

The proof is completed.□

Proposition 2

For 1 ≤ k ≤ n − 1 2 ,

k n − k + 1 < b k ( n ) b n − k ( n ) .

Proof

We will prove a stronger result:

2 k 2 n − 2 k + 1 < b k ( n ) b n − k ( n ) for 1 ≤ k ≤ n − 1 .

Note that

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

From

( 2 n − 2 k + 1 ) 2 n + 1 2 k − 2 k 2 n + 1 2 k + 1 = 2 n + 1 2 k + 1 ,

it follows that

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

Then we complete the proof by verifying that for 1 ≤ k ≤ n − 1 ,

□ 2 n + 1 2 k + 1 2 k n k = 1 2 k ∏ i = 1 k + 1 2 n + 3 − 2 i 2 k + 3 − 2 i = 2 n + 1 2 k ⋅ 2 n − 1 2 k + 1 2 n − 3 2 k − 1 ⋯ 2 n + 1 − 2 k 3 > 1 .

The following corollary is immediate, which will be used in Section 3.

Corollary 1

For 2 ≤ k ≤ n − 1 ,

k n − k + 1 < b k ( n ) b n − k ( n ) .

Proposition 3

For 1 ≤ k ≤ n 2 ,

b n − k + 1 ( n ) b k ( n ) < k n − k + 1 .

Proof

Throughout the rest of this article, we note that

b n − k + 1 ( n ) b k ( n ) = k n − k + 1 ⋅ α − β k α − γ k ,

where

α ≔ ( 2 n + 1 ) ! n ! , β k ≔ ( 2 k − 1 ) ! ( 2 n − 2 k + 1 ) ! ( k − 1 ) ! ( n − k ) ! ⋅ 4 ( n − k + 1 ) , γ k ≔ ( 2 k − 1 ) ! ( 2 n − 2 k + 1 ) ! ( k − 1 ) ! ( n − k ) ! ⋅ 4 k .

This notation is valid for 1 ≤ k ≤ n , and it will be utilized repeatedly. Clearly, α − β k , α − γ k > 0 since b k ( n ) > 0 for 1 ≤ k ≤ n . Because α − β k < α − γ k for 1 ≤ k ≤ n 2 , we obtain the desired inequality immediately.□

Proposition 4

For 1 ≤ k ≤ n 2 ,

k − 1 n − k + 2 < b n − k + 1 ( n ) b k ( n ) .

Proof

We will prove a stronger result:

k − 1 n − k + 1 < b n − k + 1 ( n ) b k ( n ) for 1 ≤ k ≤ n 2 .

By the proof of Proposition 3, it is equivalent to prove that for 1 ≤ k ≤ n 2 ,

b n − k + 1 ( n ) b k ( n ) = k n − k + 1 ⋅ α − β k α − γ k > k − 1 n − k + 1 .

To this end, cancelling n − k + 1 , cross-multiplying, and subtracting, we will show that for 1 ≤ k ≤ n 2 ,

α > I ( n , k ) ,

where

I ( n , k ) ≔ k β k − ( k − 1 ) γ k = ( 2 k − 1 ) ! ( 2 n − 2 k + 1 ) ! ( k − 1 ) ! ( n − k ) ! ⋅ 4 k ( n − 2 k + 2 ) .

Note that I ( n , k ) > 0 for 1 ≤ k ≤ n 2 .

To prove α > I ( n , k ) for 1 ≤ k ≤ n 2 , it suffices to show that α > I ( n , 1 ) for n ≥ 3 , and the positive sequence { I ( n , k ) } is strictly decreasing over k = 1 , … , n 2 .

Indeed,

α = 2 n + 1 2 n ⋅ I ( n , 1 ) > I ( n , 1 ) for n ≥ 3 .

To prove the monotonicity of the sequence { I ( n , k ) } , we note that for n ≥ 4 and 1 ≤ k ≤ n 2 − 1 ,

I ( n , k + 1 ) I ( n , k ) = ( k + 1 ) ( 2 k + 1 ) ( n − 2 k ) k ( 2 n − 2 k + 1 ) ( n − 2 k + 2 ) .

Let p and q , respectively, denote the numerator and denominator of the right-hand side of the aforementioned equation. To show p / q < 1 , a longwinded computation reveals:

q − p = k ( 2 n − 2 k + 1 ) ( n − 2 k + 2 ) − ( k + 1 ) ( 2 k + 1 ) ( n − 2 k ) = 2 ( k − 1 ) ( n − 2 k + 1 ) 2 + 2 n − 10 k − 3 4 2 + 28 k 2 + 12 k + 7 8 ,

which is positive. Thus, I ( n , k + 1 ) I ( n , k ) < 1 , as desired. The proof is completed.□

The following corollary will be used in Section 3.

Corollary 2

For 1 ≤ k ≤ n ,

k − 1 n − k + 2 < b n − k + 1 ( n ) b k ( n ) .

Proof

It is easy to verify that α > I ( n , k ) ≥ 0 for 1 ≤ k ≤ n + 2 2 since the sequence { I ( n , k ) } is still strictly decreasing over k = 1 , … , n + 2 2 . Note that I ( n , k ) = 0 when n is even and k = n + 2 2 .

Moreover, because I ( n , k ) < 0 for n + 2 2 + 1 ≤ k ≤ n , we have α > I ( n , k ) for n + 2 2 + 1 ≤ k ≤ n . Then we conclude that α > I ( n , k ) for 1 ≤ k ≤ n . Therefore,

k − 1 n − k + 1 < b n − k + 1 ( n ) b k ( n ) for 1 ≤ k ≤ n ,

and then

□ k − 1 n − k + 2 < b n − k + 1 ( n ) b k ( n ) for 1 ≤ k ≤ n .

Proof of Theorem 1

By combining Propositions 1 and 2, we construct chain (3). Similarly, chain (4) follows immediately from Propositions 3 and 4. This completes the proof.□

It is routine that b 1 2 ( n ) > b 0 ( n ) b 2 ( n ) for n ≥ 1 . Then we have the following corollary by Theorem 1.

Corollary 3

The coordinator polynomials of the root lattice of type B n are strictly log-concave for n ≥ 1 .

A polynomial P ( x ) = ∑ i = 0 n a i x i of degree n is said to be alternatingly increasing (see [19]) if

a 0 ≤ a n ≤ a 1 ≤ a n − 1 ≤ ⋯ ≤ a ⌊ n + 1 2 ⌋ .

This property is a variant of the spiral property. The next corollary follows directly from Theorem 1.

Corollary 4

The coordinator polynomials of the root lattice of type B n are alternatingly increasing for n ≥ 1 .

3 Concluding remarks

Theorem 1 illustrates that

b k − 1 ( n ) b n − k + 1 ( n ) < k n − k + 1 < b n − k ( n ) b k + 1 ( n ) for 2 ≤ k ≤ n − 1 2

and

b n − k + 1 ( n ) b k ( n ) < k n − k + 1 < b k ( n ) b n − k ( n ) for 1 ≤ k ≤ ⌊ n 2 ⌋ − 1 .

The aforementioned inequalities can be extended as follows, and they interestingly yield a refined log-concavity.

Proposition 5

For n ≥ 3 and 2 ≤ k ≤ n − 1 ,

(5) b k − 1 ( n ) b n − k + 1 ( n ) < k n − k + 1 < b n − k ( n ) b k + 1 ( n )

and

(6) b n − k + 1 ( n ) b k ( n ) < 2 k 2 n − 2 k + 1 < b k ( n ) b n − k ( n ) .

As a consequence, we have a property that refines strict log-concavity: for 2 ≤ k ≤ n − 1 ,

b k − 1 ( n ) b k + 1 ( n ) < b n − k ( n ) b n − k + 1 ( n ) < b k ( n ) 2 .

Proof

Chain (5): By changing k in Corollary 1 into n − k + 1 and reversing the two ratios therein, we obtain

b k − 1 ( n ) b n − k + 1 ( n ) < k n − k + 1 for 2 ≤ k ≤ n − 1 .

Similarly, by changing k in Corollary 2 into k + 1 and letting the updated k start from 2 and end at n − 1 , we obtain

k n − k + 1 < b n − k ( n ) b k + 1 ( n ) for 2 ≤ k ≤ n − 1 .

Chain (6): We showed in the proof of Proposition 2 that for 2 ≤ k ≤ n − 1 ,

2 k 2 n − 2 k + 1 < b k ( n ) b n − k ( n ) .

In what follows, we will prove that for 2 ≤ k ≤ n − 1 ,

b n − k + 1 ( n ) b k ( n ) < 2 k 2 n − 2 k + 1 .

That is, for 2 ≤ k ≤ n − 1 , and recalling the notation from the proof of Proposition 3,

b n − k + 1 ( n ) b k ( n ) = k n − k + 1 ⋅ α − β k α − γ k < 2 k 2 n − 2 k + 1 .

Again, canceling k , cross-multiplying, and subtracting, this is equivalent to show that for 2 ≤ k ≤ n − 1 ,

α > J ( n , k ) ,

where

J ( n , k ) ≔ ( 2 n − 2 k + 2 ) γ k − ( 2 n − 2 k + 1 ) β k = ( 2 k − 1 ) ! ( 2 n − 2 k + 1 ) ! ( k − 1 ) ! ( n − k ) ! ⋅ 4 ( n − k + 1 ) ( 4 k − 2 n − 1 ) .

Clearly, J ( n , k ) < 0 for 2 ≤ k ≤ 2 n + 1 4 , and J ( n , k ) > 0 for 2 n + 1 4 + 1 ≤ k ≤ n − 1 . Hence, to prove α > J ( n , k ) for 2 ≤ k ≤ n − 1 , it suffices to show that α > J ( n , n − 1 ) , and the sequence { J ( n , k ) } is strictly increasing over k = 2 n + 1 4 + 1 , … , n − 1 (note that the sequence { J ( n , k ) } has a single term when n = 3 , 4 ).

Indeed,

α = 4 n 2 − 1 12 ( 2 n − 5 ) J ( n , n − 1 ) > J ( n , n − 1 ) for n ≥ 3 .

To prove the monotonicity of the sequence { J ( n , k ) } , we note that for n ≥ 5 and 2 n + 1 4 + 1 ≤ k ≤ n − 2 ,

J ( n , k + 1 ) J ( n , k ) = ( 2 k + 1 ) ( − 2 n + 4 k + 3 ) ( n − k ) ( − 2 n + 2 k − 1 ) ( − 2 n + 4 k − 1 ) ( − n + k − 1 ) .

Let p and q , respectively, denote the numerator and denominator on the right-hand side of the equality above. To show p / q > 1 , by a straightforward computation, we have

p − q = − 16 k 3 + 4 k 2 ( 8 n + 1 ) − 10 k ( 2 n 2 + n + 1 ) + 4 n 3 + 6 n 2 + 8 n + 1 = ( 1 − 2 k ) + 2 ( n − k ) n + 2 ( n − k ) [ 2 ( n − 2 k ) 2 + 2 ( n − k ) + 4 ] ≥ ( 1 − 2 k ) + 2 ( n − k ) n .

Next we assume that 2 n + 1 4 + 1 ≤ k ≤ n − 1 , where the range of k is slightly broader. Then p − q > 0 since for k ≤ n − 1 ,

p − q ≥ ( 1 − 2 k ) + 2 ( n − k ) n ≥ 1 − 2 k + 2 ⋅ 1 ⋅ n = 2 n − 2 k + 1 ≥ 3 .

So J ( n , k + 1 ) J ( n , k ) > 1 , as desired. The proof is completed.□

It would be interesting to find more examples of combinatorial sequences sharing the refined version of log-concavity.

Acknowledgments

We sincerely appreciate the anonymous referees’ feedback for pointing out the problems in the proofs and in the writings. The anonymous referees have spent considerable time and effort in giving many constructive comments. These comments have been very helpful during revision and significantly improved the presentation of the article.

Funding information: This work is partially supported by the National Natural Science Foundation of China (No. 11871304), the Outstanding Youth Foundation of Rizhao City (No. RZ2021ZR3), and the Taishan Scholar Project of Shandong Province (No. tsqn202103060).
Conflict of interest: The authors declare no conflict of interest.
Data availability statement: No data, models, or code was generated or used during the study.

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Received: 2022-08-23

Revised: 2023-01-15

Accepted: 2023-01-21

Published Online: 2023-02-14

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2022-0555

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ratio monotonicity; coordinator polynomials; log-concavity; spiral property

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