Page 1 of 7

APPLIED & INTERDISCIPLINARY MATHEMATICS | SHORT COMMUNICATION

A recovery of two determinantal representations for derangement numbersFeng Qi, Jing-Lin Wang and Bai-Ni Guo

Cogent Mathematics (2016), 3: 1232878

Qi et al., Cogent Mathematics (2016), 3: 1232878http://dx.doi.org/10.1080/23311835.2016.1232878

APPLIED & INTERDISCIPLINARY MATHEMATICS | SHORT COMMUNICATION

A recovery of two determinantal representations for derangement numbersFeng Qi1,2,3*, Jing-Lin Wang3 and Bai-Ni Guo1

Abstract: In the paper, the authors recover, correct, and extend two representations for derangement numbers in terms of a tridiagonal determinant.

Subjects: Advanced Mathematics; Analysis - Mathematics; Calculus; Combinatorics; DiscreteMathematics; Mathematics & Statistics; Number Theory; Real Functions; Science; Special Functions

Keywords: derangement number; determinantal representation; tridiagonal determinant; generating function

2010 Mathematics subject classifications: 05A05; 05A15; 11B83; 11C20; 15A15

1. IntroductionIn combinatorics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. The number of derangements of a set of size n is called the derange-ment number and sometimes denoted by !n. The problem of counting derangements was first con-sidered in 1708 and solved in 1713 by Pierre Raymond de Montmort, as did Nicholas Bernoulli at about the same time. Derangement numbers !n arise naturally in many different contexts. More generally, the number of derangements in various families of transitive permutation groups has been studied extensively in recent years. For more information on !n, please refer to Aigner (2007), Andreescu and Feng (2004), Wilf (1994, 2006) and plenty of references therein.

*Corresponding author: Feng Qi, Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province 454010, China; College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region 028043, China; Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City 300387, China E-mails: qifeng618@gmail.com, qifeng618@hotmail.com

Reviewing editor:Prasanna K. Sahoo, University of Louisville, USA

Additional information is available at the end of the article

ABOUT THE AUTHORFeng Qi received his PhD degree of Science in Mathematics from University of Science and Technology of China. He is being a full professor in mathematics at Henan Polytechnic University (HPU) and Tianjin Polytechnic University. He is the founder of School of Mathematics and Informatics at HPU. He was visiting professors at Victoria University in Australia and University of Hong Kong in China. He was part-time professors at Henan University, Henan Normal University, and Inner Mongolia University for Nationalities. He visited Copenhagen University and seven universities in South Korea. He attended an academic conference held in April 2016 at Antalya by Ağrı İbrahim Çeçen University in Turkey. He is or was editors of over 20 international journals. Since 1993, he published over 600 articles. His current research interests include the analytic combinatorics, computational number theory, special functions, integral transforms, and complex functions See https://qifeng618.wordpress.com.

PUBLIC INTEREST STATEMENTA derangement is a permutation of elements of a set, such that no element appears in its original position. The number of derangements of a set is called the derangement number. The problem of counting derangements was first considered in 1708 and solved in 1713. Derangement numbers arise naturally in many different contexts. The number of derangements in various families of transitive permutation groups has been studied extensively in recent years.

In the paper, by virtue of an old formula for computing derivatives of a ratio between two differentiable functions in terms of the Hessenberg determinants, the authors recover, correct, and extend two representations for derangement numbers in terms of a tridiagonal determinant.

Received: 07 April 2016Accepted: 31 August 2016Published: 23 September 2016

© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.

Page 2 of 7

Feng Qi

Page 3 of 7

Qi et al., Cogent Mathematics (2016), 3: 1232878http://dx.doi.org/10.1080/23311835.2016.1232878

The first ten derangement numbers !n for 0 ≤ n ≤ 9 are

In Kittappa (1993, p. 216, Example 2), it was given that

In Janjić (2012, p. 8, 5◦) and Janjić (2011, p. 5, 5◦), it was deduced that

By the determinantal expression (3), we figure out that !2 = 1, !3 = 2, !4 = 6, and !5 = 24. It is clear that the latter two values 6 and 24 do not coincide with the numbers 9 and 44 in (1). Therefore, the expression (3) appeared in Janjić (2011, 2012) is slightly wrong.

It is known in Comtet (1974, p. 182, Theorem B) that derangement numbers !n have an exponen-tial generating function

The aim of this paper is, by computing the nth derivative of the exponential generating function D(x), to recover, correct, and extend the above determinantal representations (2) and (3) for derange-ment numbers !n.

Theorem 1 For n ∈ {0} ∪ ℕ, derangement numbers !n can be represented by a tridiagonal (n + 1) × (n + 1) determinant

(1)0, 1, 2, 9, 44, 265, 1854, 14833, 133496.

(2)!(n + 2) =

||||||||||||||||

2 −1 0 ⋯ 0 0 0

3 3 −1 ⋯ 0 0 0

0 4 4 ⋯ 0 0 0

⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮

0 0 0 ⋯ n − 1 −1 0

0 0 0 ⋯ n n −1

0 0 0 ⋯ 0 n + 1 n + 1

||||||||||||||||n×n

, n ∈ ℕ.

(3)!(r + 1) =

||||||||||||||||

1 1 0 0 ⋯ 0 0

−1 1 2 0 ⋯ 0 0

0 −1 2 3 ⋯ 0 0

0 0 −1 3 ⋯ 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

0 0 0 0 ⋯ r − 1 r

0 0 0 0 ⋯ −1 r

||||||||||||||||

, r ∈ ℕ.

(4)D(x) =e−x

1 − x=

∞∑

n=0

!nxn

n!.

(5)!n = −

||||||||||||||||||||||||||

−1 1 0 0 0 … 0 0 0

0 0 1 0 0 … 0 0 0

0 −1 1 1 0 … 0 0 0

0 0 −2 2 1 … 0 0 0

0 0 0 −3 3 … 0 0 0

⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮

0 0 0 0 0 … n − 3 1 0

0 0 0 0 0 … −(n − 2) n − 2 1

0 0 0 0 0 … 0 −(n − 1) n − 1

||||||||||||||||||||||||||

= −|eij|(n+1)×(n+1)

,

Page 4 of 7

Qi et al., Cogent Mathematics (2016), 3: 1232878http://dx.doi.org/10.1080/23311835.2016.1232878

where

2. LemmaIn order to recover Theorem 1, we need the following lemma which was reformulated in Qi (2015, Section 2.2, p. 849), Qi and Chapman (2016, p. 94), and Wei and Qi (2015, Lemma 2.1) from Bourbaki (2004, p. 40, Exercise 5).

Lemma 1 Let u(x) and v(x) ≠ 0 be differentiable functions, let U(n+1)×1(x) be an (n + 1) × 1 matrix whose

elements uk,1(x) = u(k−1)

(x) for 1 ≤ k ≤ n + 1, let V(n+1)×n(x) be an (n + 1) × n matrix whose elements

for 1 ≤ i ≤ n + 1 and 1 ≤ j ≤ n, and let |||W

(n+1)×(n+1)(x)||| denote the determinant of the (n + 1) × (n + 1)

matrix

Then the nth derivative of the ratio u(x)v(x)

can be computed by

3. Proof of Theorem 1Now we are in a position to prove Theorem 1.

Applying u(x) = ex and v(x) = 1 + x in Lemma 1 gives

as x → 0 for 1 ≤ k ≤ n + 1 and

as x → 0 for 1 ≤ i ≤ n + 1 and 1 ≤ j ≤ n. Consequently, employing (6) reveals

eij =

⎧

⎪

⎪

⎨

⎪

⎪⎩

1, i − j = −1;

i − 2, i − j = 0;

2 − i, i − j = 1;

0, i − j ≠ 0,±1.

vi,j(x) =

⎧

⎪

⎨

⎪⎩

�

i − 1

j − 1

�

v(i−j)(x), i − j ≥ 0

0, i − j < 0

W(n+1)×(n+1)(x) =

[

U(n+1)×1(x) V

(n+1)×n(x)]

.

(6)dn

dxn

[

u(x)

v(x)

]

= (−1)n|||W

(n+1)×(n+1)(x)|||

vn+1(x).

uk,1 = (ex)(k−1) = ex → 1

vi,j=

�

i − 1

j − 1

�

(1 + x)(i−j)

=

⎧

⎪

⎪

⎨

⎪

⎪⎩

�

i − 1

j − 1

�

(1 + x), i − j = 0

�

i − 1

j − 1

�

, i − j = 1

0, i − j ≠ 0, 1

=

⎧

⎪

⎨

⎪⎩

1 + x, i − j = 0

i − 1, i − j = 1

0, i − j ≠ 0, 1

→

⎧

⎪

⎨

⎪⎩

1, i − j = 0

i − 1, i − j = 1

0, i − j ≠ 0, 1

Page 5 of 7

Qi et al., Cogent Mathematics (2016), 3: 1232878http://dx.doi.org/10.1080/23311835.2016.1232878

as x → 0. From (4), it follows that

This implies that

Subtracting the nth row from the (n + 1)th row, then the (n − 1)th row from the nth row, ..., then the 1st row from the 2nd row of the above determinant leads to

which can be readily rearranged as the formula (5). The proof of Theorem 1 is complete.

Remark 1 On 10 May 2016, Dr Wiwat Wanicharpichat at Naresuan University in Thailand told the first author that the matrix

dnD(−x)

dxn=

(−1)n

(1 + x)n+1

|||||||||||||||||||

ex 1 + x 0 0 ⋯ 0 0

ex 1 1 + x 0 ⋯ 0 0

ex 0 2 1 + x ⋯ 0 0

ex 0 0 3 ⋯ 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

ex 0 0 0 ⋯ 1 + x 0

ex 0 0 0 ⋯ n − 1 1 + x

ex 0 0 0 ⋯ 0 n

|||||||||||||||||||

→ (−1)n

|||||||||||||||||||

1 1 0 0 ⋯ 0 0

1 1 1 0 ⋯ 0 0

1 0 2 1 ⋯ 0 0

1 0 0 3 ⋯ 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

1 0 0 0 ⋯ 1 0

1 0 0 0 ⋯ n − 1 1

1 0 0 0 ⋯ 0 n

|||||||||||||||||||

D(−x) =ex

1 + x=

∞∑

k=0

(−1)n!nxn

n!.

(7)(−1)n!n = lim

x→0

dnD(−x)

dxn= (−1)n

|||||||||||||||||||

1 1 0 0 ⋯ 0 0

1 1 1 0 ⋯ 0 0

1 0 2 1 ⋯ 0 0

1 0 0 3 ⋯ 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

1 0 0 0 ⋯ 1 0

1 0 0 0 ⋯ n − 1 1

1 0 0 0 ⋯ 0 n

|||||||||||||||||||

.

!n =

|||||||||||||||||||

1 1 0 0 ⋯ 0 0

0 0 1 0 ⋯ 0 0

0 −1 1 1 ⋯ 0 0

0 0 −2 2 ⋯ 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

0 0 0 0 ⋯ 1 0

0 0 0 0 ⋯ n − 2 1

0 0 0 0 ⋯ −(n − 1) n − 1

|||||||||||||||||||

Page 6 of 7

Qi et al., Cogent Mathematics (2016), 3: 1232878http://dx.doi.org/10.1080/23311835.2016.1232878

is known as the “population projection matrix”. See (Kirkland & Neumann, 2013,  p. 48, Equation (4.1)).

Remark 2 In the paper (Qi, 2016), an alternative proof of Theorem 1 was given.

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜⎝

1 1 0 0 ⋯ 0 0

1 1 1 0 ⋯ 0 0

1 0 2 1 ⋯ 0 0

1 0 0 3 ⋯ 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

1 0 0 0 ⋯ 1 0

1 0 0 0 ⋯ n − 1 1

1 0 0 0 ⋯ 0 n

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟⎠

AcknowledgementsThe authors thank Dr Sophie Moufawad at IFP Energies nouvelles in France, Dr Yuri S. Semenov at Moscow State University of Railway Engineering in Russia, Dr Wiwat Wanicharpichat at Naresuan University in Thailand, and several other mathematicians for their observations and discussion on the determinant in (7) at the ResearchGate website http://www.researchgate.net/post/What_is_the_name_of_the_matrix_or_determinant_showed_by_the_picture. The authors appreciate several anonymous mathematicians for their valuable comments on the original version of this paper, especially for their recommending the papers (Janjić, 2011, 2012, Kittappa, 1993) and related results in them. The authors appreciate the anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

FundingThe authors received no direct funding for this research.

Author detailsFeng Qi1,2,3

E-mails: qifeng618@gmail.com, qifeng618@hotmail.comORCID ID: http://orcid.org/0000-0001-6239-2968Jing-Lin Wang3

E-mail: jing-lin.wang@hotmail.comORCID ID: http://orcid.org/0000-0001-6725-533XBai-Ni Guo1

E-mails: bai.ni.guo@gmail.com, bai.ni.guo@hotmail.comORCID ID: http://orcid.org/0000-0001-6156-25901 Institute of Mathematics, Henan Polytechnic University,

Jiaozuo City, Henan Province 454010, China.2 College of Mathematics, Inner Mongolia University for

Nationalities, Tongliao City, Inner Mongolia Autonomous Region 028043, China.

3 Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City 300387, China.

Citation informationCite this article as: A recovery of two determinantal representations for derangement numbers, Feng Qi, Jing-Lin Wang & Bai-Ni Guo, Cogent Mathematics (2016), 3: 1232878.

Cover imageSource: Author.

ReferencesAigner, M. (2007). A course in enumeration, Graduate texts in

Mathematics (Vol., 238). Berlin: Springer.Andreescu, T., & Feng, Z. (2004). A path to combinatorics

for undergraduates–counting strategies. Boston-Basel-Berlin: Birkhäuser.

Bourbaki, N. (2004). Elements of Mathematics: Functions of a Real Variable: Elementary Theory. In P. Spain Elements of Mathematics. Berlin: Springer-Verlag. doi:10.1007/978-3-642-59315-4

Comtet, L. (1974). Advanced Combinatorics: The Art of Finite and Infinite Expansions (Revised and Enlarged ed.). Dordrecht and Boston: D. Reidel Publishing.

Janjić, M. (2011). Recurrence relations and determinants. (arXiv preprint). Retrieved from http://arxiv.org/abs/1112.2466

Janjić, M. (2012). Determinants and recurrence sequences. Journal of Integer Sequences, 15, 21 p. (Article 12.3.5).

Kirkland, S. J., & Neumann, M. (2013). Group inverses of M-matrices and their applications. Boca Raton, FL: CRC Press Taylor & Francis Group.

Kittappa, R. K. (1993). A representation of the solution of the nth order linear difference equation with variable coefficients. Linear Algebra and its Applications, 193, 211–222. doi:10.1016/0024-3795(93)90278-V

Qi, F. (2015). Derivatives of tangent function and tangent numbers. Applied Mathematics and Computation, 268, 844–858. doi:10.1016/j.amc.2015.06.123

Qi, F. (2016). A determinantal representation for derangement numbers. Global Journal of Mathematical Analysis, 4, 17. doi:10.14419/gjma.v4i3.6574

Qi, F., & Chapman, R. J. (2016). Two closed forms for the Bernoulli polynomials. Journal of Number Theory, 159, 89–100. doi:10.1016/j.jnt.2015.07.021

Wei, C.-F., & Qi, F. (2015). Several closed expressions for the Euler numbers. Journal of Inequalities and Applications, 219, 8. doi:10.1186/s13660-015-0738-9

Wilf, H. S. (1994). Generatingfunctionology (2nd ed.). Boston, MA: Academic Press.

Wilf, H. S. (2006). Generatingfunctionology (3rd ed.). Wellesley, MA: A K Peters.

Page 7 of 7

Qi et al., Cogent Mathematics (2016), 3: 1232878http://dx.doi.org/10.1080/23311835.2016.1232878

© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.You are free to: Share — copy and redistribute the material in any medium or format Adapt — remix, transform, and build upon the material for any purpose, even commercially.The licensor cannot revoke these freedoms as long as you follow the license terms.

Under the following terms:Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.

Cogent Mathematics (ISSN: 2331-1835) is published by Cogent OA, part of Taylor & Francis Group. Publishing with Cogent OA ensures:• Immediate, universal access to your article on publication• High visibility and discoverability via the Cogent OA website as well as Taylor & Francis Online• Download and citation statistics for your article• Rapid online publication• Input from, and dialog with, expert editors and editorial boards• Retention of full copyright of your article• Guaranteed legacy preservation of your article• Discounts and waivers for authors in developing regionsSubmit your manuscript to a Cogent OA journal at www.CogentOA.com