The ratio monotonicity of the q -derangement numbers

Discrete Mathematics 311 (2011) 393–397

Contents lists available at ScienceDirect

Discrete Mathematics journal homepage: www.elsevier.com/locate/disc

Note

The ratio monotonicity of the q-derangement numbers William Y.C. Chen, Ernest X.W. Xia ∗ Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, PR China

article

abstract

info

Article history: Received 10 March 2009 Received in revised form 20 November 2010 Accepted 25 November 2010 Available online 28 December 2010

We introduce the notion of ratio monotonicity for polynomials with nonnegative coefficients, and we show that, for n ≥ 6, the q-derangement numbers Dn (q) are strictly ratio monotone except for the last term when n is even. This property implies the spiral property and log-concavity. © 2010 Elsevier B.V. All rights reserved.

Keywords: q-derangement number Spiral property Log-concavity Ratio monotone property

1. Introduction Let Dn be the set of derangements on {1∑ , 2, . . . , n}, and let maj(π ) denote the major index of a permutation π . The maj(π) q-derangement number Dn (q) is defined as . The following formula is due to Gessel [5] (see also Gessel and π ∈Dn q Reutenauer [6]): Dn (q) = [n]!

  n − k 1 , (−1)k q 2 [k]! k=0

(1.1)

where [n] = 1 + q + q2 + · · · + qn−1 and [n]! = [1][2] · · · [n]. Combinatorial proofs of (1.1) have been found by Wachs [7] and by Chen and Xu [3]. The polynomials Dn (q) satisfy the following recurrence relation: Dn (q) = [n]Dn−1 (q) + (−1)n q( 2 ) , n

n ≥ 2,

(1.2)

with D1 (q) = 0. Note that the q-derangement polynomials of type B have been introduced and studied independently by Chen et al. [2] and Chow [4]. Chen and Rota [1] showed that the q-derangement numbers are unimodal, and conjectured that the maximum coefficient appears in the middle. Zhang [8] confirmed this conjecture by showing that the q-derangement numbers satisfy the spiral property. For example, we have D6 (q) = q + 4q2 + 9q3 + 16q4 + 24q5 + 32q6 + 37q7 + 38q8 + 35q9

+ 28q10 + 20q11 + 12q12 + 6q13 + 2q14 + q15 . Observe that D6 (q) has the following spiral property: 1 < 2 < 4 < 6 < 9 < 12 < 16 < 20 < 24 < 28 < 32 < 35 < 37 < 38,

∗

Corresponding author. E-mail addresses: [email protected] (W.Y.C. Chen), [email protected] (E.X.W. Xia).

0012-365X/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2010.11.012

394

W.Y.C. Chen, E.X.W. Xia / Discrete Mathematics 311 (2011) 393–397

where the last term is not taken into consideration. We say that a sequence a(1), a(2), . . . , a(n) of positive numbers is ratio monotone if a(1) a(n)

a(2)

≤

a(⌊n/2⌋)

≤ ··· ≤

a(n − 1)

a(⌈n/2⌉ + 1)

≤1

(1.3)

≤ 1,

(1.4)

and a(n) a(2)

a(n − 1)

≤

a(⌊n/2⌋ + 2)

≤ ··· ≤

a(3)

a(⌈n/2⌉)

where ⌊x⌋ and ⌈x⌉ are the floor function and the ceiling function. In the case that all the inequalities are strict, we say that the sequence is strictly ratio monotone. It can be easily seen that the ratio monotone property implies the spiral property and log-concavity. For example, for the case of D6 (q) without the last term, we see that 1 2 2 4

< <

4 6 6 9

< <

9 12 12 16

< <

16 20 20 24

< <

24

32

<

28 28

35 35

<

32

37

<

37 38

< 1,

(1.5)

< 1.

(1.6)

In the next section, we shall show that the q-derangement numbers Dn (q) are strictly ratio monotone for n ≥ 6 except for the last term when n is even. 2. The ratio monotonicity In order to state the main result, we first introduce some notation. Set Bn (q) = Let δn =



Dn (q) − q( 2 ) , Dn (q), n

if n is even, if n is odd.

(2.1)

− 1 denote the degree of Bn (q). Then (1.2) can be recast as  [n]Bn−1 (q), if n is even,   n−1 Bn (q) = 2 [n]Bn−1 (q) + [n − 1]q , if n is odd. n 2

(2.2)

Write Bn (q) = bn (1)q + bn (2)q2 + · · · + bn (δn )qδn . The ratio monotone property of Bn (q) can be stated as follows. Theorem 2.1. For n ≥ 6, we have bn (1) bn (δn ) bn (δn ) bn (2)

< <

bn (2) bn (δn − 1) bn (δn − 1) bn (3)

< ··· < < ··· <

bn (⌈ bn (⌊ bn (⌈

n

/2⌉ − 1)

< 1,

(2.3)

/2⌉ + 1) n < 1. /2⌋) 2

(2.4)

 2n  2

/2⌋ + 1)

n 2

bn (⌊

Theorem 2.1 implies the log-concavity of Bn (q). Corollary 2.2. For n ≥ 6, the polynomials Bn (q) are log-concave; that is, bn (1) bn (2)

<

bn (2) bn (3)

< ··· <

bn (δn − 2) bn (δn − 1)

<

bn (δn − 1) bn (δn )

.

To prove Theorem 2.1, we need the following lemmas. Lemma 2.3. For positive numbers c1 , c2 , . . . , ck+1 , d1 , d2 , . . . , dk+1 satisfying d1 c1 we have

<

d2 c2

< ··· <

dk ck

<

dk+1 ck+1

,

(2.5)

W.Y.C. Chen, E.X.W. Xia / Discrete Mathematics 311 (2011) 393–397

d1 + d2 + · · · + dk c1 + c2 + · · · + ck d1 + d2 + · · · + dk c1 + c2 + · · · + ck

<

d1 + d2 + · · · + dk + dk+1

<

d2 + · · · + dk + dk+1

c1 + c2 + · · · + ck + ck+1 c2 + · · · + ck + ck+1

d1 + d2 + · · · + dk + dk+1 c1 + c2 + · · · + ck + ck+1

<

,

395

(2.6)

,

(2.7)

d2 + · · · + dk + dk+1 c2 + · · · + ck + ck+1

.

(2.8)

The proof of Lemma 2.3 is straightforward, and the details are omitted. Using recurrence relation (2.2), it is easy to verify the following lemma. Lemma 2.4. If n ≥ 3, we have bn (1) = 1, bn (2) = n − 2, and bn (3) = bn (4) =

n(n − 3) 2

,

(n2 − 4)(n − 3) 6

,

bn (δn ) = ⌈n/2⌉ − 1, bn (δn − 1) = ⌈n2 /4 − n/2⌉, bn (δn − 2) = ⌈n3 /12 − n2 /8 − n/12⌉ − 1. Corollary 2.5. For n ≥ 3, bn (1) bn (δn ) + 1

<

bn (2) bn (δn − 1)

.

(2.9)

.

(2.10)

For n ≥ 4 and n even, bn (δn ) + 1 bn (2)

<

bn (δn − 1) bn (3)

For n ≥ 6, bn (δn ) + bn (δn − 1) + 1 bn (2) + bn (3)

<

bn (δn − 2) bn (4)

.

(2.11)

We are now ready to present the proof of Theorem 2.1. Proof of Theorem 2.1. We claim that for n ≥ 6 the coefficients of Bn (q) satisfy the following relations: bn (1) bn (δn ) bn (δn ) bn (2)

< <

bn (2) bn (δn − 1) bn (δn − 1) bn (3)

< ··· < < ··· <

bn (δn − 1) bn (2) bn (3) bn (δn − 1)

bn (δn )

<

bn (1) bn (2)

<

bn (δn )

,

(2.12)

.

(2.13)

To prove the above assertion, we use induction on n. For n = 6, 7, it is easy to check that the claim is valid. Suppose that the claim holds for n. We now proceed to show that it holds for n + 1; that is, bn+1 (i) bn+1 (δn+1 − i + 1) bn+1 (δn+1 − i + 1) bn+1 (i + 1)

< <

bn+1 (i + 1) bn+1 (δn+1 − i) bn+1 (δn+1 − i) bn+1 (i + 2)

,

1 ≤ i ≤ δn+1 − 1,

(2.14)

,

1 ≤ i ≤ δn+1 − 2.

(2.15)

We only consider the case when n is even, since the case when n is odd can be dealt with by using the same argument. Assume that n ≥ 6 and n is even. From (2.2), we get the following recurrence relation for bn (k):

bn+1 (k) =

 k −    bn (i),  

1≤k<

i=k−n

k −    1 + bn (i),   i=k−n

n 2

n 2

≤k<

,



n+1

with the convention that bn (i) = 0 if i < 1 or i > δn .

2



(2.16)

,

396

W.Y.C. Chen, E.X.W. Xia / Discrete Mathematics 311 (2011) 393–397

By (2.12), (2.13) and Corollary 2.5, we obtain that bn (1)

bn (2)

<

bn (δn ) + 1 bn (δn ) + 1

bn (δn − 1) bn (δn − 1)

<

bn (2) + 1

bn (3)

< ··· < < ··· <

bn (δn − 1) bn (2) bn (3) bn (δn − 1)

< <

bn (δn ) + 1 bn (1) bn (2) + 1 bn (δn ) + 1

,

(2.17)

.

(2.18)

Thus, (2.14) can be deduced from (2.17) and (2.16) because of Lemma 2.3. For 1 ≤ i ≤ n, use (2.6). For n < i ≤ δn − 1, use (2.7). For δn − 1 < i ≤ δn+1 − 1, use (2.8). Again, according to Lemma 2.3 and Corollary 2.5, (2.15) can be deduced from (2.18) and (2.16). For 1 ≤ i < n, apply (2.6). For n < i < δn − 1, apply (2.7). For δn − 1 < i ≤ δn+1 − 2, apply (2.8). Now, special attention should be paid to the case i = n. It follows from (2.11) and (2.13) that bn (δn ) + bn (δn − 1) + 1

bn (δn − n)

<

bn (2) + bn (3)

bn (n + 2)

.

(2.19)

By (2.13), we find that n −1

∑

bn (δn − i)

i=2 n +1

∑

<

bn (δn − n) bn (n + 2)

bn (i)

.

(2.20)

i=4

In view of (2.12) and (2.7), we obtain n+1

∑

n ∑

bn (i)

i=2 n ∑

<

bn (δn − i)

i=1 n +1

bn (δn − i)

∑

i =1

, bn (i)

i=2

which yields that n +1 −

bn (i) <

i=2

n −

bn (δn − i).

(2.21)

i=1

From (2.19)–(2.21), we conclude that n −1

1+

∑ i =0

n+1

1+

n ∑

bn (δn − i)

∑

<

bn (δn − i)

i=0 n +2

bn (i)

∑

i=2

, bn (i)

i =2

which can be restated as the case i = n of (2.15) based on (2.16). The case i = δn − 1 can be verified by the same argument, and hence the above claim is confirmed. b (r ) b (u) In order to prove (2.3) and (2.4), it remains to verify that bn (s) < 1 and bn (v) < 1, where n

r = ⌈n(n − 1)/4⌉ − 1,

s = ⌊n(n − 1)/4⌋ + 1,

n

u = ⌈n(n − 1)/4⌉ + 1,

If n ≡ 0, 1 (mod 4), then r + 1 = s − 1. By (2.12), we see that By (2.12), we get

bn (r ) bn (s)

<

bn (r +1) bn (s−1)

By (2.13), we find that the proof. 

<

bn (r +1) bn (s−1)

= 1. If n ≡ 2, 3 (mod 4), then r = s − 1.

<

<

If n ≡ 0, 1 (mod 4), then u = v + bn (u) bn (v)

<

b (r ) Hence, in either case, we have bn (s) 1, and so (2.3) holds. n bn (u) bn (u−1) bn (v) . If n 2 3 mod 4 , then u 1 1. 1. By (2.13), we see that b (v) bn (v+1) bn (u) n bn (u−1) bn (u) 1. Consequently, we have b (v) 1 in either case, and so (2.4) holds. This completes bn (v+1) n

=

bn (s) . bn (r )

bn (r ) bn (s)

v = ⌊n(n − 1)/4⌋ .

=

=

≡ , (

)

− = v+

<

Acknowledgements The authors would like to thank the referees for helpful suggestions leading to an improvement of an earlier version. We would also thank Francesco Brenti and Michelle Wachs for valuable comments. This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education, and the National Science Foundation of China.

W.Y.C. Chen, E.X.W. Xia / Discrete Mathematics 311 (2011) 393–397

397

References [1] W.Y.C. Chen, G.-C. Rota, q-analogs of the inclusion–exclusion principle and permutations with restricted position, Discrete Math. 104 (1992) 7–22. [2] W.Y.C. Chen, R.L. Tang, A.F.Y. Zhao, Derangement polynomials and excedances of type B, Electron. J. Combin. 16 (2) (2009). Special volume in honor of Anders Björner, Research Paper 15, p. 16. [3] W.Y.C. Chen, D.H. Xu, Labeled partitions and the q-derangement numbers, SIAM J. Discrete Math. 22 (2008) 1099–1104. [4] C.-O. Chow, On derangement polynomials of type B. II, J. Combin. Theory Ser. A 116 (2009) 816–830. [5] I.M. Gessel, Counting permutations by descents, greater index, and cycle structure, 1981, unpublished manuscript. [6] I.M. Gessel, C. Reutenauer, Counting permutations with given cycle structure and descent set, J. Combin. Theory Ser. A 64 (1993) 189–215. [7] M.L. Wachs, On q-derangement numbers, Proc. Amer. Math. Soc. 106 (1989) 273–278. [8] X.D. Zhang, Note on the spiral property of the q-derangement numbers, Discrete Math. 159 (1996) 295–298.