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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# The log-concavity of the q-derangement numbers of type B

Eric H. Liu
• Corresponding author
• School of Statistics and Information, Shanghai University of International Business and Economics, Shanghai, 201620, China
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• De Gruyter OnlineGoogle Scholar
/ Wenjing Du
Published Online: 2018-02-23 | DOI: https://doi.org/10.1515/math-2018-0009

## Abstract

Recently, Chen and Xia proved that for n ≥ 6, the q-derangement numbers Dn(q) are log-concave except for the last term when n is even. In this paper, employing a recurrence relation for $\begin{array}{}{D}_{n}^{B}\left(q\right)\end{array}$ discovered by Chow, we show that for n ≥ 4, the q-derangement numbers of type B $\begin{array}{}{D}_{n}^{B}\left(q\right)\end{array}$ are also log-concave.

MSC 2010: 05A15; 05A19; 05A20

## 1 Introduction

Let 𝓓n denote the set of derangements on {1, 2, …, n} and let D(π) := {i|1 ≤ in – 1, π(i) > π(i + 1)} denote the descent set of a permutation π. Define the major index of π by

$maj(π):=∑i∈D(π)i.$(1)

The q-derangement number Dn(q) is defined by

$Dn(q):=∑π∈Dnqmaj(π).$(2)

Gessel [1] (see also [2]) discovered the following formula

$Dn(q):=[n]!∑k=0n(−1)kqk21[k]!,$(3)

where [n] = 1 + q + q2 + … + qn – 1 and [n]! = [1][2] … [n]. Combinatorial proofs of (3) have been found by Wachs [3] and Chen and Xu [4]. Chen and Rota [5] showed that the q-derangement numbers are unimodal, and conjectured that the maximum coefficient appears in the middle. Zhang [6] confirmed this conjecture by showing that the q-derangement numbers satisfy the spiral property. Recently, Chen and Xia [7] introduced the notion of ratio monotonicity for polynomials with nonnegative coefficients, and they proved that, for n ≥ 6, the q-derangement numbers Dn(q) are strictly ratio monotone except for the last term when n is even. The ratio monotonicity implies the spiral property and log-concavity.

Let Bn denote the hyperoctahedral group of rank n, consisting of the signed permutations of {1, 2, …, n}. Let $\begin{array}{}{\mathcal{D}}_{n}^{B}\end{array}$ denote the set of derangements on Bn, which is defined as

$DnB:={π|π∈Bn,π(i)≠ifor alli∈{1,2,…,n}}.$(4)

Let N(π) := #{i|1 ≤ in, π(i) < 0} be the number of negative letters of π and let maj(π) be defined as before. In [8], Chow considered the q-derangement number of type B $\begin{array}{}{D}_{n}^{B}\left(q\right)\end{array}$ which is defined as

$DnB(q):=∑π∈DnBqfmaj(π),$(5)

where fmaj(π) := 2 maj(π) + N(π). Chow [8] (see also [9]) established the following formula

$DnB(q):=[2][4]⋯[2n]∑k=0n(−1)kq2k2[2][4]⋯[2k],$(6)

where [n] is defined as before. Furthermore, Chow [8] discovered that for all integers n ≥ 1,

$Dn+1B(q)=(1+q+⋯+q2n+1)DnB(q)+(−1)n+1qn2+n.$(7)

Chen and Wang [10] proved the normality of the limiting distribution of the coefficients of the usual q-derangement numbers of type B.

Recall that a positive sequence a0, a1, …, an or the polynomial a0 + a1x + … + anxn is called log-concave if the ratios

$a0a1,a1a2,…,an−1an$(8)

form an increasing sequence. Clearly, if a positive sequence is log-concavity, then it is unimodality. In this paper, we prove that for n ≥ 4, the q-derangement numbers of type B $\begin{array}{}{D}_{n}^{B}\left(q\right)\end{array}$ are log-concave.

Suppose that n is given. It is easy to prove that the degree of $\begin{array}{}{D}_{n}^{B}\left(q\right)\end{array}$ is n2 and the coefficient of qn2 is 1. Set

$DnB(q)=Bn(1)q+Bn(2)q2+⋯+Bn(n2)qn2.$(9)

The log-concavity of $\begin{array}{}{D}_{n}^{B}\left(q\right)\end{array}$ can be stated as the following theorem.

#### Theorem 1.1

For all integers n ≥ 4, the q-derangement numbers of type B $\begin{array}{}{D}_{n}^{B}\left(q\right)\end{array}$ are log-concave, namely,

$Bn(1)Bn(2)(10)

For example, by (6), we have

$D4B(q)=q+4q2+8q3+13q4+18q5+22q6+26q7+28q8+28q9+25q10+21q11+17q12+11q13+7q14+3q15+q16.$

It is easy to check that

$14<48<813<1318<1822<2226<2628<2828<2825<2521<2117<1711<117<73<31.$

## 2 Some lemmas

To prove Theorem 1.1, we first present some lemmas. By (7), it is easy to check that

#### Lemma 2.1

For n ≥ 4,

$Bn+1(k)=∑i=1kBn(i),1≤k≤2n+2,∑i=k−2n−1kBn(i),2n+2(11)

Based on recurrence relation (11), it is easy to verify the following lemma.

#### Lemma 2.2

Let n4 be an integer. Then Bn(i) are positive integers for 1in2 and

$Bn(n2)=1,Bn(n2−1)=n−1,$(12)

$Bn(n2−2)=n2−n+22,Bn(n2−3)=n3+5n−186.$(13)

To prove Theorem 1.1, we require the following lemma.

#### Lemma 2.3

For positive integers a1, a2, …, ak+1, ak+2 (k ≥ 1) satisfying

$aiai+1(14)

$∑i=1kai∑i=1k+1ai<∑i=1k+1ai∑i=1k+2ai,$(15)

$∑i=1kai∑i=1k+1ai<∑i=1k+1ai∑i=2k+2ai,$(16)

$∑i=1kai∑i=2k+1ai<∑i=2k+1ai∑i=3k+2ai,$(17)

$∑i=1kai∑i=2k+1ai<∑i=2k+1ai∑i=3k+1ai,$(18)

$∑i=1kai∑i=2kai<∑i=2kai∑i=3kai.$(19)

#### Proof

We only prove (15). The rest can be proved similarly and the details are omitted. Based on (14),

$aiak+2

and

$ak+2(a1+a2+⋯+ak)

Therefore,

$(a1+a2+⋯+ak)(a1+a2+⋯+ak+ak+1+ak+1)=(a1+a2+⋯+ak)2+ak+1(a1+a2+⋯+ak)+ak+2(a1+a2+⋯+ak)<(a1+a2+⋯+ak)2+ak+1(a1+a2+⋯+ak)+ak+1(a2+a3+⋯+ak+1)<(a1+a2+⋯+ak)2+ak+1(a1+a2+⋯+ak)+ak+1(a2+a3+⋯+ak+1)+a1ak+1=(a1+a2+⋯+ak+ak+1)2,$

which yields (15). This completes the proof of this lemma.

## 3 Proof of Theorem 1.1

We prove Theorem 1.1 by induction on n. It is easy to check that Theorem 1.1 holds for 4 ≤ n ≤ 12. Thus, we always assume that n ≥ 13 in the following proof. Suppose that Theorem 1.1 holds for n = m, namely,

$Bm(i)Bm(i+1)(20)

We proceed to show that Theorem 1.1 holds for n = m + 1, that is,

$Bm+1(k)Bm+1(k+1)(21)

Employing (11), (15) and (20), we see that (21) holds for 1 ≤ k ≤ 2m. It follows from (11), (16) and (20) that (21) is true for the case k = 2m + 1. In view of (11), (17) and (20), we find that (21) holds for 2m + 2 ≤ km2 – 2. From (11), (18) and (20), we deduce that (21) is true for the case k = m2 – 1. By (11), (19) and (20), we can prove that (21) holds for m2km2 + m – 3 and m2 + m + 1 ≤ k ≤ (m + 1)2 – 2.

Now, special attentions should be paid to three cases k = m2 + m – 2, k = m2 + m – 1 and k = m2 + m.

By (12) and (13), it is easy to check that for m ≥ 4,

$Bm(m2−2)+Bm(m2−1)+Bm(m2)−m−3Bm(m2−1)+Bm(m2)+1−Bm(m2−3)Bm(m2−2)=m4−13m2+72m−1326(m−3)(m2−m+2)>0.$(22)

In view of (20) and (22),

$Bm(m2−m−3)Bm(m2−m−2)(23)

From (11), it is easy to prove that for m ≥ 4,

$Bm(m2−m−2)>Bm(m2−m−1)>⋯>Bm(m2−1)>Bm(m2).$(24)

Thanks to (23) and (24),

$Bm(m2−m−3)(Bm(m2−1)+Bm(m2)+(−1)m+1)+(−1)m+1∑i=0m+2Bm(m2−i)(25)

By (20),

$Bm(m2−m−3)∑i=2m+1Bm(m2−i)(26)

Combining (25) and (26) yields

$∑i=m2−m−3m2Bm(i)∑i=m2−m−2m2Bm(i)<∑i=m2−m−2m2Bm(i)∑i=m2−m−1m2Bm(i)+(−1)m+1,$(27)

which can be rewritten as

$Bm+1(m2+m−2)Bm+1(m2+m−1)(28)

Therefore, (21) holds for the case k = m2 + m – 2.

Based on (12) and (13), we deduce that for m ≥ 13,

$Bm(m2−2)+Bm(m2−1)+Bm(m2)−2(m+2)Bm(m2−1)+Bm(m2)−Bm(m2−3)Bm(m2−2)=m4−12m3−13m2+36m−366m(m2−m+2)>0.$(29)

By (20) and (29),

$Bm(m2−m−2)Bm(m2−m−1)(30)

It follows from (24) and (30) that

$Bm(m2−m−2)(Bm(m2−1)+Bm(m2))(31)

In view of (20),

$Bm(m2−m−2)∑i=2mBm(m2−i)(32)

It follows from (31) and (32) that

$∑i=m2−m−2m2Bm(i)∑i=m2−m−1m2Bm(i)+(−1)m+1<∑i=m2−m−1m2Bm(i)+(−1)m+1∑i=m2−mm2Bm(i).$(33)

By (11), we can rewrite (33) as follows

$Bm+1(m2+m−1)Bm+1(m2+m)(34)

which implies that (21) holds for the case k = m2 + m – 1.

In view of (12) and (13), we see that for m ≥ 4,

$Bm(m2−2)+Bm(m2−1)+Bm(m2)−mBm(m2−1)+Bm(m2)−Bm(m2−3)Bm(m2−2)=(m+1)(m3−7m2+12m+12)6m(m2−m+2)>0.$(35)

By (20) and (35), we find that for m ≥ 4,

$Bm(m2−m−1)Bm(m2−m)(36)

It follows from (24) and (36) that

$Bm(m2−m−1)(Bm(m2−1)+Bm(m2))+(−1)m+1∑i=0m−1Bm(m2−i)(37)

By (20),

$Bm(m2−m−1)∑i=2m−1Bm(m2−i)(38)

In view of (37) and (38), we can prove that

$∑i=m2−m−1m2Bm(i)+(−1)m+1∑i=m2−mm2Bm(i)<∑i=m2−mm2Bm(i)∑i=m2−m+1m2Bm(i).$(39)

By (11), we can rewrite (39) as follows

$Bm+1(m2+m)Bm+1(m2+m+1)(40)

which implies that (21) holds for the case k = m2 + m. This completes the proof.

## Acknowledgement

The authors would like to thank the anonymous referee for valuable corrections and comments. This work was supported by the National Science Foundation of China (11701362).

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## About the article

Accepted: 2018-01-11

Published Online: 2018-02-23

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 127–132, ISSN (Online) 2391-5455,

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