The q-log-concavity of q-ballot numbers

Abstract

Carlitz and Riordan introduced the q-analogue f_q\left(n,k\right) of ballot numbers. In this paper, using the combinatorial interpretation of f_q\left(n,k\right) and constructing injections, we prove that f_q\left(n,k\right) is q-log-concave with respect to n and k, i.e., all coefficients of the polynomials f_q{\left(n,k\right)}^2-f_q\left(n+1,k\right)f_q\left(n-1,k\right) and f_q{\left(n,k\right)}^2-f_q\left(n,k+1\right)f_q\left(n,k-1\right) are non-negative for .

Graphical abstract

Keywords

q-log-concavity / q-ballot number / lattice path / inversion

1 Introduction

Let \{a_k{\}}_{k\ge 0} be a non-negative, infinite sequence of real numbers. If {\left(a_k\right)}^2\ge a_{k-1}{a}_{k+1} (or {\left(a_k\right)}^2\le a_{k-1}{a}_{k+1}) holds for any arbitrary k>0, then the sequence \{a_k{\}}_k\ge 0 is said to be log-concave (or log-convex).

Assume q is an indeterminate and given two real polynomials f\left(q\right) and g\left(q\right). If g\left(q\right)-f\left(q\right) possesses non-negative coefficient (or non-positive coefficients) as the polynomial of q, then it is denoted by g\left(q\right){\ge }_{q}f\left(q\right) (or g\left(q\right){\le }_{q}f\left(q\right)). Stanley proposed that let \{f_n\left(q\right){\}}_{n\ge 0} be a non-negative real polynomial sequence, if for any arbitrary i\ge 1, f_i^2\left(q\right)-f_{i-1}\left(q\right)f_{i+1}\left(q\right){\ge }_{q}0, then the polynomial sequence \{f_n\left(q\right){\}}_{n\ge 0} is said to be q-log-concave. For more relevant results on q-log-concavity, please refer to [8, 9, 11].

The q-analogue of some well-known combinatorial numbers is q-log-concave. Butler [1] proved the q-log-concavity of q-binomial coefficient with inversion. Chen et al. [5] used Schur positivity to prove the q-log-concavity of q-Narayana numbers. Ji [7] further proved the q-log-concavity of q-Kaplansky numbers based on Butler [1].

Chapoton and Zeng [4] proposed the combinatorial interpretation of ballot numbers. Let {\mathcal{P}}_{n,k} be a set of lattice path p from the start point (0, 0) moving to the end point \left(n+1,k\right), in a manner of horizontal (1, 0) and vertical (0, 1) steps, ensuring that the last step is a horizontal (1, 0) movement and p does not go beyond the line y=x. The count of such path is the ballot number f\left(n,k\right)=\left|{\mathcal{P}}_{n,k}\right|. Comtet [6] presented its explicit expression:

f\left(n,k\right)=\frac{n-k+1}{n+1}\left(\genfrac{}{}{0pt}{}{n+k}{k}\right),n\ge k\ge 0.

It is evident that ballot number is a log-concave sequence with respect to k.

Carlitz and Riordan [3] introduced the q-analogue of the ballot number and provided the combinatorial interpretation

f_q\left(n,k\right)=\sum _{\gamma \in {\mathcal{P}}_{n,k}}{q}^{\mathcal{A}\left(\gamma \right)},

where \mathcal{A}\left(\gamma \right) represents the “lattice number” below \gamma and above x-axis.

For example, when n=2,k=1, f_q\left(2,1\right)=q+q^2, as shown in Fig.1.

Define all lattice paths in the set {\mathcal{P}}_{n,k} with the last horizontal step (1, 0) removed as set {\mathcal{B}}_{n,k}, where {\mathcal{B}}_{n,k} is the lattice paths from the start point (0, 0) moving to the end point \left(n,k\right), in a manner of horizontal (1, 0) and vertical (0, 1) steps, without crossing the line y=x. Then the q-ballot number also equals

f_q\left(n,k\right)=\sum _{\gamma \in {\mathcal{B}}_{n,k}}{q}^{\mathcal{B}\left(\gamma \right)+k},

where \mathcal{B}\left(\gamma \right) represents the “lattice number” below \gamma and above x-axis.

Later on, Carlitz [2] discovered the recursive relation of f_q\left(n,k\right) to be f_q\left(n,k\right); when , f_q\left(n,k\right)=0, and f_q\left(0,0\right)=1.

Next, we will introduce the definition of a ballot permutation and inversion.

If permutation p=p_1{p}_2\dots p_n, and for 1\le i\le n-1, if p_i>p_{i+1} (or ), then i is called a descendance (or ascendance), and the count of descending numbers is denoted by des\left(p\right) while the count of ascending numbers is denoted by asc\left(p\right). h\left(p\right)=asc\left(p_1{p}_2\cdots p_n\right)-des\left(p_1{p}_2\cdots p_n\right) is taken as the height for permutation p. A permutation p is a ballot permutation if and only if the height for any prefix of permutation p is non-negative; that is, for all 1\le i\le n, there is h\left(p_1{p}_2\cdots p_i\right)\ge 0. For relevant results on ballot permutation, please refer to [10, 12, 13].

If permutation p=p_1{p}_2\cdots p_n, and if and p_i>p_j, then the pair \left(p_i,p_j\right) is said to be an inverted sequence. The inversion of permutation p is the count of inverted sequences in it, denoted by inv\left(p\right).

For the lattice path in set {\mathcal{B}}_{n,k}, we denote horizontal (1, 0) movement by 1 and vertical (0, 1) movement by 2. Then the lattice path in set {\mathcal{B}}_{n,k} can be expressed by a permutation containing digits 1 and 2, and it is guaranteed that the count of any prefix 1 is always greater than or equal to 2. For permutation p=p_1{p}_2\cdots p_n comprised of 1 and 2, the length of p is denoted by l\left(p\right); l_1\left(p\right) is the count of digit 1 in p, while l_2\left(p\right) is the count of digit 2 in p. For example, when p=111221, l\left(p\right)=6,l_1\left(p\right)=4,l_2\left(p\right)=2. The permutation p comprised of digits 1 and 2 corresponding to every lattice path in set {\mathcal{B}}_{n,k} satisfies that for all 1\le i\le n, l_1\left(p_1{p}_2\cdots p_i\right)\ge l_2\left(p_1{p}_2\cdots p_i\right), namely, the permutation corresponding to every lattice path in set {\mathcal{B}}_{n,k} is a ballot permutation comprised of digits 1 and 2. And \mathcal{B}\left(\gamma \right) formed by every path in k exactly equals the corresponding inversion of the ballot permutation, then

f_q\left(n,k\right)=\sum _{\gamma \in B_{n,k}}{q}^{\mathcal{B}\left(\gamma \right)+k}=\sum _{\gamma \in B_{n,k}}{q}^{\mathrm{i}\mathrm{n}\mathrm{v}\left(\gamma \right)+k}=q^k\sum _{\gamma \in B_{n,k}}{q}^{\mathrm{i}\mathrm{n}\mathrm{v}\left(\gamma \right)}.

This paper proves that the q-ballot number f_q\left(n,k\right) is q-log-concave with respect to n and k through constructing injection; the coefficients of the polynomials f_q{\left(n,k\right)}^2-f_q\left(n+1,k\right)f_q\left(n-1,k\right) and f_q{\left(n,k\right)}^2-f_q\left(n,k+1\right)f_q\left(n,k-1\right) are non-negative for .

2 Main results

Theorem 2.1　 f_q\left(n,k\right) is a q-log-concave sequence with respect to n.

Proof　To prove that f_q\left(n,k\right) is a q-log-concave sequence with respect to n, we have to demonstrate the polynomial with respect to q:

According to f_q\left(n,k\right)=q^k\sum _{\gamma \in {\mathcal{B}}_{n,k}}{q}^{\mathrm{i}\mathrm{n}\mathrm{v}\left(\gamma \right)}, we get

\begin{array}{rl}{f}_q{\left(n,k\right)}^2& -f_q\left(n+1,k\right)f_q\left(n-1,k\right)\\ & =q^{2k}{\left(\sum _{\gamma \in {\mathcal{B}}_{n,k}}{q}^{\mathrm{i}\mathrm{n}\mathrm{v}\left(\gamma \right)}\right)}^2-q^{2k}\sum _{\gamma \in {\mathcal{B}}_{n+1,k}}{q}^{\mathrm{i}\mathrm{n}\mathrm{v}\left(\gamma \right)}\sum _{\gamma \in {\mathcal{B}}_{n-1,k}}{q}^{\mathrm{i}\mathrm{n}\mathrm{v}\left(\gamma \right)}\\ & =q^{2k}\left({\left(\sum _{\gamma \in {\mathcal{B}}_{n,k}}{q}^{\mathrm{i}\mathrm{n}\mathrm{v}\left(\gamma \right)}\right)}^2-\sum _{\gamma \in {\mathcal{B}}_{n+1,k}}{q}^{\mathrm{i}\mathrm{n}\mathrm{v}\left(\gamma \right)}\sum _{\gamma \in {\mathcal{B}}_{n-1,k}}{q}^{\mathrm{i}\mathrm{n}\mathrm{v}\left(\gamma \right)}\right).\end{array}

So, we only need to prove the polynomial with respect to q

We define a mapping

\phi :{\mathcal{B}}_{n+1,k}\times {\mathcal{B}}_{n-1,k}\to {\mathcal{B}}_{n,k}\times {\mathcal{B}}_{n,k}.

Let \left(\pi ,\sigma \right)\in {\mathcal{B}}_{n+1,k}\times {\mathcal{B}}_{n-1,k}. Then the path \pi ,\sigma can be expressed as

\begin{array}{c}\pi ={\pi }_1{\pi }_2\cdots {\pi }_{n+k-i}{\pi }_{n+k-i+1}\cdots {\pi }_{n+k+1},\\ \sigma ={\sigma }_1{\sigma }_2\cdots {\sigma }_{n+k-i-1}{\sigma }_{n+k-i}\cdots {\sigma }_{n+k-1}.\end{array}

Case 1　When {\pi }_{n+k+1}=1, let \phi \left(\pi ,\sigma \right)=\left(w,v\right), where

\begin{array}{c}w={\pi }_1{\pi }_2\cdots {\pi }_{n+k},\\ v={\sigma }_1{\sigma }_2\cdots {\sigma }_{n+k-1}{\pi }_{n+k+1}.\end{array}

For example, when \pi ={\pi }_1{\pi }_2\cdots {\pi }_{n+k+1}=11211221,\sigma ={\sigma }_1{\sigma }_2\cdots {\sigma }_{n+k-1}=121212, then

\begin{array}{c}\left(\pi ,\sigma \right)=\left(11211221,121212\right),\\ \left(w,v\right)=\left(1121122,1212121\right).\end{array}

The combinatorial meaning is shown in Fig.2.

We can obtain

\begin{array}{c}\mathrm{i}\mathrm{n}\mathrm{v}\left(w\right)=\mathrm{i}\mathrm{n}\mathrm{v}\left(\pi \right)-k,\\ \mathrm{i}\mathrm{n}\mathrm{v}\left(v\right)=\mathrm{i}\mathrm{n}\mathrm{v}\left(\sigma \right)+k.\end{array}

Then \mathrm{i}\mathrm{n}\mathrm{v}\left(w\right)+\mathrm{i}\mathrm{n}\mathrm{v}\left(v\right)-\mathrm{i}\mathrm{n}\mathrm{v}\left(\pi \right)-\mathrm{i}\mathrm{n}\mathrm{v}\left(\sigma \right)-0, and here w and v satisfy the definition of a ballot permutation.

Case 2　When {\pi }_{n+k+1}=2, for the sake of convenience, we denote {\pi }_1{\pi }_2\cdots {\pi }_{n+k-i},{\pi }_{n+k-i+1}{\pi }_{n+k-i+2}\cdots {\pi }_{n+k+1},{\sigma }_1{\sigma }_2\cdots {\sigma }_{n+k-i-1}, and {\sigma }_{n+k-i}{\sigma }_{n+k-i+1}\cdots {\sigma }_{n+k-1} as {\pi }_L,{\pi }_R,{\sigma }_L, and {\sigma }_R. We can then get \pi ={\pi }_L{\pi }_R and \sigma ={\sigma }_L{\sigma }_R.

Let i be the minimum that satisfies the following three conditions:

(1) l_1\left({\pi }_R\right)=l_1\left({\sigma }_R\right)+1,

(2) l\left({\pi }_R\right)=l\left({\sigma }_R\right)+1,

(3) l\left({\sigma }_R\right)=i.

Let \phi (\pi ,\sigma )=(w,v)=\left({\pi }_L{\sigma }_R,{\sigma }_L{\pi }_R\right). Then

\begin{array}{c}w={\pi }_1{\pi }_2\cdots {\pi }_{n+k-i}{\sigma }_{n+k-i}{\sigma }_{n+k-i+1}\cdots {\sigma }_{n+k-1},\\ v={\sigma }_1{\sigma }_2\cdots {\sigma }_{n+k-i-1}{\pi }_{n+k-i+1}{\pi }_{n+k-i+2}\cdots {\pi }_{n+k+1.}\end{array}

For example, when \pi ={\pi }_1{\pi }_2\cdots {\pi }_{n+k+1}=11121122,\sigma ={\sigma }_1{\sigma }_2\cdots {\sigma }_{n+k-1}=121212, then i=3. So, {\pi }_L=1112,{\sigma }_L=121,{\pi }_R=1122 and {\sigma }_R=212. Thus,

\begin{array}{c}\left(\pi ,\sigma \right)=\left(11121122,121212\right),\\ (w,v)=(1112212,1211122).\end{array}

The combinatorial meaning is shown in Fig.3.

Assuming l_1\left({\sigma }_R\right)=a, we obtain the count of digits 1 and 2 in {\pi }_L,{\pi }_R,{\sigma }_L, and {\sigma }_R, as shown in Tab.2.

It is further proved in the following that w and v remain ballot permutations.

Since l_1\left({\pi }_L\right)>l_1\left({\sigma }_L\right),l_2\left({\pi }_L\right)=l_2\left({\sigma }_L\right), and \sigma is a ballot permutation, it can be deduced that w={\pi }_L{\sigma }_R remains a ballot permutation.

Since \pi and \sigma are ballot permutations, it is evident that {\pi }_L and {\sigma }_L remain as ballot permutations.

Establish an array \left(\genfrac{}{}{0pt}{}{{\pi }_{n+k-i+1}}{{\sigma }_{n+k-i}}\right),\left(\genfrac{}{}{0pt}{}{{\pi }_{n+k-i+2}}{{\sigma }_{n+k-i+1}}\right),\dots ,\left(\genfrac{}{}{0pt}{}{{\pi }_{n+k}}{{\sigma }_{n+k-1}}\right) at the corresponding locations in permutations {\pi }_R and {\sigma }_R. Since the values of {\pi }_i and {\sigma }_i are either 1 or 2, all arrays that can possibly be formed are \left(\genfrac{}{}{0pt}{}{1}{1}\right),\left(\genfrac{}{}{0pt}{}{2}{2}\right),\left(\genfrac{}{}{0pt}{}{1}{2}\right), or \left(\genfrac{}{}{0pt}{}{2}{1}\right). Due to the determining rules of i, it is determined that \left(\genfrac{}{}{0pt}{}{{\pi }_{n+k-i+1}}{{\sigma }_{n+k-i}}\right)=\left(\genfrac{}{}{0pt}{}{1}{2}\right). Moreover, when the array order is \left(\genfrac{}{}{0pt}{}{{\pi }_{n+k-i+1}}{{\sigma }_{n+k-i}}\right),\left(\genfrac{}{}{0pt}{}{{\pi }_{n+k-i+2}}{{\sigma }_{n+k-i+1}}\right),\dots ,\left(\genfrac{}{}{0pt}{}{{\pi }_{n+k}}{{\sigma }_{n+k-1}}\right), and the count of any prefix array \left(\genfrac{}{}{0pt}{}{1}{2}\right) always remains greater than that of array \left(\genfrac{}{}{0pt}{}{2}{1}\right), l_1({\pi }_{n+k-i+1},{\pi }_{n+k-i+2},\dots ,{\pi }_{j+1})>l_1\left({\sigma }_{n+k-i},{\sigma }_{n+k-i+1},\dots ,{\sigma }_j\right), when n+k-i\le j\le n+k-1. Given \sigma is a ballot permutation, then for any arbitrary n+k-i\le j\le n+k-1, permutation {\sigma }_1,{\sigma }_2,\dots ,{\sigma }_j is a ballot permutation. Hence, v={\sigma }_L{\pi }_R is also a ballot permutation.

Next, we calculate the inversion of \pi ,\sigma ,w and v:

\begin{array}{c}\mathrm{i}\mathrm{n}\mathrm{v}\left(\pi \right)=\mathrm{i}\mathrm{n}\mathrm{v}\left({\pi }_L\right)+\mathrm{i}\mathrm{n}\mathrm{v}\left({\pi }_R\right)+l_2\left({\pi }_L\right)l_1\left({\pi }_R\right),\\ \mathrm{i}\mathrm{n}\mathrm{v}\left(\sigma \right)=\mathrm{i}\mathrm{n}\mathrm{v}\left({\sigma }_L\right)+\mathrm{i}\mathrm{n}\mathrm{v}\left({\sigma }_R\right)+l_2\left({\sigma }_L\right)l_1\left({\sigma }_R\right),\\ \mathrm{i}\mathrm{n}\mathrm{v}\left(w\right)=\mathrm{i}\mathrm{n}\mathrm{v}\left({\pi }_L\right)+\mathrm{i}\mathrm{n}\mathrm{v}\left({\sigma }_R\right)+l_2\left({\pi }_L\right)l_1\left({\sigma }_R\right),\\ \mathrm{i}\mathrm{n}\mathrm{v}\left(v\right)=\mathrm{i}\mathrm{n}\mathrm{v}\left({\sigma }_L\right)+\mathrm{i}\mathrm{n}\mathrm{v}\left({\pi }_R\right)+l_2\left({\sigma }_L\right)l_1\left({\pi }_R\right).\end{array}

According to l_2\left({\pi }_L\right)=l_2\left({\sigma }_L\right), it follows

\begin{array}{rl}\mathrm{i}\mathrm{n}\mathrm{v}& \left(w\right)+\mathrm{i}\mathrm{n}\mathrm{v}\left(v\right)-\mathrm{i}\mathrm{n}\mathrm{v}\left(\pi \right)-\mathrm{i}\mathrm{n}\mathrm{v}\left(\sigma \right)\\ & =l_2\left({\pi }_L\right)l_1\left({\sigma }_R\right)+l_2\left({\sigma }_L\right)l_1\left({\pi }_R\right)-l_2\left({\pi }_L\right)l_1\left({\pi }_R\right)-l_2\left({\sigma }_L\right)l_1\left({\sigma }_R\right)\\ & =\left(l_2\left({\pi }_L\right)-l_2\left({\sigma }_L\right)\right)\left(l_1\left({\sigma }_R\right)-l_1\left({\pi }_R\right)\right)\\ & =0.\end{array}

Then we prove the mapping \phi is injective. Paths w,v are expressed as

\begin{array}{c}w=w_1{w}_2\cdots w_{n+k},\\ v=v_1{v}_2\cdots v_{n+k}.\end{array}

Assume there exist {\pi }^{\prime }\in {\mathcal{B}}_{n+1,k},{\sigma }^{\prime }\in {\mathcal{B}}_{n-1,k} such that \phi \left({\pi }^{\prime },{\sigma }^{\prime }\right)=\left(w,v\right).

When v_{n+k}=1, we define path \pi =w_1{w}_2\cdots w_{n+k}1and \sigma =v_1{v}_2\cdots v_{n+k-1}

When v_{n+k}=2, let i be the minimum that satisfies the following three conditions:

(1) l_1\left(v_R\right)=l_1\left(w_R\right)+1,

(2) l\left(v_R\right)=l\left(w_R\right)+1,

(3) l\left(w_R\right)=i.

Paths \pi ,\sigma are expressed as

\begin{array}{c}\pi =w_1{w}_2\cdots w_{n+k-i}{v}_{n+k-i}{v}_{n+k-i+1}\cdots v_{n+k},\\ \sigma =v_1{v}_2\cdots v_{n+k-i-1}{w}_{n+k-i+1}{w}_{n+k-i+2}\cdots w_{n+k}.\end{array}

Then {\pi }^{\prime }=\pi ,{\sigma }^{\prime }=\sigma, so mapping \phi is injective.

Theorem 2.2　 f_q\left(n,k\right) is a q-log-concave sequence with respect to k.

Proof　To prove that f_q\left(n,k\right) is a q-log-concave sequence with respect to k, we need to prove the polynomial with respect to q

We define a mapping

\varphi :{\mathcal{B}}_{n,k+1}\times {\mathcal{B}}_{n,k-1}\to {\mathcal{B}}_{n,k}\times {\mathcal{B}}_{n,k}.

Let \left(b,c\right)\in {\mathcal{B}}_{n,k+1}\times {\mathcal{B}}_{n,k-1}. Then paths b,c can be expressed as

\begin{array}{rl}& b=b_1{b}_2\cdots b_{n+k-i}{b}_{n+k-i+1}\cdots b_{n+k+1},\\ & c=c_1{c}_2\cdots c_{n+k-i-1}{c}_{n+k-i}\cdots c_{n+k-1}.\end{array}

Case 1　 b_{n+k+1}=2. Let \varphi \left(b,c\right)=\left(h,k\right). Then

\begin{array}{c}h=b_1{b}_2\cdots b_{n+k},\\ k=c_1{c}_2\cdots c_{n+k-1}{b}_{n+k+1}.\end{array}

For example, when b=b_1{b}_2\cdots b_{n+k+1}=112211212,c=c_1{c}_2\cdots c_{n+k-1}=1121112, then

\begin{array}{rl}\left(b,c\right)& =\left(112211212,1121112\right),\\ \left(h,k\right)& =\left(11221121,11211122\right).\end{array}

The combinatorial meaning is shown in Fig.4.

Case 2　When b_{n+k+1}=1, let i be the minimum that satisfies the following three conditions:

(1) l_2\left(b_R\right)=l_2\left(c_R\right)+1,

(2) l\left(b_R\right)=l\left(c_R\right)+1,

(3) l\left(c_R\right)=i.

Let \varphi (b,c)=(h,k). Then

\begin{array}{c}h=b_1{b}_2\cdots b_{n+k-i}{c}_{n+k-i}{c}_{n+k-i+1}\cdots c_{n+k-1},\\ k=c_1\cdots c_{n+k-i-1}{b}_{n+k-i+1}{b}_{n+k-i+2}\cdots b_{n+k+1}.\end{array}

For example, when b=b_1{b}_2\cdots b_{n+k+1}=112212121,c=c_1{c}_2\cdots c_{n+k-1}=1121112, then i=3. Thus, b_L=11221,c_L=1121,b_R=2121, and c_R=112.

\begin{array}{rl}(b,c)& =\left(112212121,1121112\right),\\ (h,k)& =\left(11221112,11212121\right).\end{array}

The combinatorial meaning is shown in Fig.5.

The proof is similar to Theorem 2.1. We can establish that h,k are ballot permutations, and the mapping \varphi is injective.

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