RELATION BETWEEN STIRLING’S NUMBERS OF THE SECOND KIND AND TRIBONACCI MATRIX

  Bulletin of Mathematics

  ISSN Printed: 2087-5126; Online: 2355-8202

Vol. 10, No. 01 (2018), pp. 33–39 https://talenta.usu.ac.id/index.php/bullmath

RELATION BETWEEN STIRLING’S

NUMBERS OF THE SECOND KIND AND

  

TRIBONACCI MATRIX

Fifitriani Rasmi, Sri Gemawati, Kartini and M. D. H.

_Abstract.

Gamal

The Stirling’s matrix of the second kind is expressed by S n

(2) with each

entry being a second type Stirling’s number. The tribonacci matrix is represented

by T n with each entry being a tribonacci number. In this article we discuss the

Stirling’s matrix that is the relation between Stirling’s numbers of The second kind

and tribonacci matrix. Then from the relation between the two matrices we obtain

a new matrix is the matrix called D n . Then the matrix D n is expressed as S n _D

  (2) = T n n .

  1. INTRODUCTION Stirling’s number was proposed by James Stirling around the 18th century (1692-1770). Stirling’s number consists of two types: Stirling’s numbers of the first kind and of the second kind. This article deals only with the Stirling’s numbers of the second kind. The Stirling’s numbers of the second kind is the number of ways to arrange a partition of a set with n elements into k non-empty subsets denoted by S(n, k) [2, p.91].

  Received 04-08-2018, Accepted 20-08-2018. 2010 Mathematics Subject Classification : 11B73, 81U20 Key words and Phrases

  : Stirling’s numbers of the second kind, tribonacci numbers, Striling’s matrix of the second kind, tribonacci matrix

F. Rasmi, S. Gemawati, Kartini & M. D. H.Gamal–Stirling’s Numbers of Second Kind and Tribonacci Matrix

  Lee et al. [7] discuss the relation between Stirling’s matrix with Fibonacci matrix , Cheon and Kim [3] discuss the relation between Stir- ling’s matrix with Pascal matrix, Maltais and Gulliver [8] discuss Pascal matrix and Stirling’s matrix, Rennie and Dobson [9] addresses the second Stirling’s numbers, Sabeth et al. [10] discuss the matrix factorization of Pascal with tribonacci and other matrices. Lee et al. [7] discuss the relation between Stirling’s matrix with Fibonacci matrix. From the relation between Stirling’s matrix of the second kind and the Fibonacci matrix, the new ma- trix M n , is obtained so Stirling’s matrix of the second kind S n (2) can be

  M expressed by S n (2) = F n n . The tribonacci sequence is one of the generalizations of the Fibonacci sequence. The tribonacci sequence was originally studied by Feinbreg [5].

  Kuhapatanakul [6] discusses the generalization of the tribonacci sequence that is the tribonacci sequence to- emph p. The Hessenberg matrix discussed by Aktas and Kose [1] is a matrix with each element involving the Padovan numbers, the Perrin numbers and tribonacci numbers. Sabeth et al. [10] define a tribonacci matrix which is express by T n and discuss the relation between of Pascal matrix and tribonacci matrix.

  In this article discuss the relation between Stirling’s matrix of the second kind and tribonacci matrix. Using the two ideas of Lee et al. [7] and Sabeth et al. [10] we get a new matrix of D matrix which states the

  n relation between the Stirling’s matrix and the tribonacci matrix.

  2. STIRLING’S MATRIX AND TRIBONACCI MATRIX In this section we give the definition of the second Stirling’s matrix and the tribonacci matrix. The Stirling’s numbers of the second kind S(n, k) is the number of how to set up the partition of a set having n elements into k set of the nonempty parts [2, p.91]. The following theorem is obtained.

  Theorem 1.1 For every natural number n and k where n ≥ k satisfies the following recursive relation

  S (n, k) = S(n − 1, k − 1) + kS(k − 1). (1) Proof.

  The proof of this theorem can be seen in Bona [2, p.91]. The Stirling’s numbers of the second kind is represented into a square matrix n × n ie the second Stirling’s matrix S(i, j).[4] defines the second

  Stirling’s matrix S(i, j) as follows. Definition 1.1

  For each natural number n, Stirling’s matrix of the second F. Rasmi, S. Gemawati, Kartini & M. D. H.Gamal–Stirling’s Numbers of Second Kind and Tribonacci Matrix ₍

  S (i, j), if i ≥ j, S

  i,j =

  (2) others. 0,

  Similar to the Fibonacci numbers, the tribonacci is also expressed as a square matrix. Sabeth et al. [10] define the n × n tribonacci matrix as follows: Definition 1.2

  For every natural number n, the n × n tribonacci matrix T

  n = [t i,j ], ∀i, j = 1, 2, 3, · · · , n is given as ₍

  T i−j , jika i − j + 2 ≥ 0,

• 2

  t

  i,j =

  (3) 0, jika i − j + 2 < 0. From the equation (3) the matrix of the T is the lower triangular

  n

  matrix with the main diagonal being 1 and the determinant (det) value of the tribonacci matrix T n is the product of the diagonal entries so as to obtain det (T n ) = 1. Because det (T n ) 6= 0 then the tribonacci matrix T n has an inverse.

  From the calculation, the inverse of the tribonacci matrix T

  5 is obtained

  as follows: _{   −}

  1 _{−  }

  1 1 

1 T  − −  .

  =

  1

  1 1 (4)

  5 _{   }

  1

  1 − − −

  − 1 − 1 −

  1

  1

  1

  1 Based on the observation of equation (4) it can be concluded that for each entry of the inverse of the tribonacci matrix T applies, i.e. each

  5

  pattern of column entries [t i,j ] is worth 1, −1, −1, and −1. This is true for the tribonacci matrix of n × n where the column entry pattern [t ] will not

  i,j

  change. Thus, Sabeth et al. [10] define the n × n invers tribonacci matrix is set n × n, ∀n ∈ N with every entry from the invers of tribonacci matrix _{− ′}

1 T

  = [t ], ∀i, j = 1, 2, 3, · · · , n can be declared as

  n i,j _{ ′  } 1, if i = j, t = −

  (5) 1, if i − 3 ≤ j ≤ i − 1,

  i,j _{ } 0, others. F. Rasmi, S. Gemawati, Kartini & M. D. H.Gamal–Stirling’s Numbers of Second Kind and Tribonacci Matrix _{− −}

  1

  1 T T

  Because tribonacci matrix T n has an invers then T n = I n = T n

  n n can be applied. Thus, the tribonacci matrix T n is an invertible matrix.

  3. RELATION BETWEEN STIRLING’S MATRIX AND TRIBONACCI MATRIX

  In this section, we discuss the second type Stirling’s matrix and the tri- bonacci matrix. The relation between Stirling’s matrix of the second kind and the matrix of the new tribonacci matrix is obtained. With the two ideas of Lee et al . [7] and Sabeth et al. [10] the new matrix D n is defined as follows. Definition 1.3

  For every natural number n, an n ×n matrix D with D

  n n =

  [d i,j ], ∀i, j = 1, 2, 3, · · · , n is defined as d

  

i,j = S(i, j) − S(i − 1, j) − S(i − 2, j) − S(i − 3, j). (6)

  Furthermore from equation (6) we obtain, d = 1, d = 0, ∀j ≥

  1,1 1,2

  2; d = 0, d = 1, d = 0, ∀j ≥ 3; d = −1, d = 2, d = 1, d =

  2,1 2,2 2,j 3,1 3,2 3,3 3,j

  0, ∀j ≥ 4; d = −2, d = 3, d = 5, d = 1, d = 0 and for i, j ≥

  4,1 4,2 4,3 4,4 4,j 2, d i,j = d i− + j.d i− . 1,j−1 1,j

  From defining the matrix D n in equation (6) the following Theorem 1.2 can be derived: Theorem 1.2

  There is a matrix of D , so for every natural numbers n

  n

  with Stirling’s matrix of the second kind S

  n (2) defined in equation (2) and

  tribonacci matrix T which is defined in equation (3) can be express S (2) =

  n n T D . n n Proof.

  For every natural number n, the T n tribonacci matrix is the invert- ible matrix. We will prove that ₋

1 T S .

  n (2) = D n (7) n _{′ ′} Noticing the left side of the equation (7) if ∀i = 1 and ∀j ≥ 2, then t = t = 0. Then ∀i, j = 1 is subsequently obtained that i,j

F. Rasmi, S. Gemawati, Kartini & M. D. H.Gamal–Stirling’s Numbers of Second Kind and Tribonacci Matrix

  1

  1

  _ 1 ^{   } _{ } 

  6

  7

  1

  1

  3

  1

  1

  1

  1

  1 ^{       }

  1

  1 −

  1 −

  1 −

  1

  1 − 1 −

  1

  1 −

  2

  4 (2) = ^{  } _

  ] = ^{  } _

  1 ^{   } .

  5

  3

  2

  1 −

  2

  1

  1 −

  1

  ij

  1 −

  = [d

  4

  D

  4 , i.e.

  (8) Thus based on the matrix multiplication of the equation (8) we obtain the entries for the matrix D

  1 ^{   } .

  5

  3

  2

  1 −

  n _X k =1

  t ^′

  If ∀i = 1 and ∀j ≥ 2 then t ^′

  k,j

  S

  i,k

  t ^′

  n _X k =1

  = 0. Then ∀i = 1 and ∀j ≥ 2, we get

  1,j

  = 0 and S

  1,j

  1,1 .

  n _X k =1

  = 1 = d

  

1

  k,

  S

  1,k

  t ^′

  k,j = n _X k =1

  S

  i,k

  =

  t ^′

  4

  k,j

  1

  as follows: T ⁻

  4

  S n (2) = D n . Suppose that for n = 4, we get the entries for the matrix D

  1 n

  Thus it is proved that T ⁻

  i,j

  (0)S(i − 4, j) + · · · + (0)S(n, j), =d

  =S(i, j) − S(i − 1, j) − S(i − 2, j) − S(i − 3, j), =(1)S(i, j) + (−1)S(i − 1, j) + (−1)S(i − 2, j) + (−1)S(i − 3, j)+

  S

  1,k

  i,k

  t ^′

  n _X k =1

  Then from equations (5) and (2) ∀i ≥ 4 and ∀j ≥ 2 we obtain

  1,j .

  = 0 = d

  

1

  k,

  S

  . S

F. Rasmi, S. Gemawati, Kartini & M. D. H.Gamal–Stirling’s Numbers of Second Kind and Tribonacci Matrix

  4. CONCLUDING REMARKS In this paper the authors discuss the relation between Stirling’s matrix of the second kind and tribonacci matrix. Then from the relation between of two matrices we obtain a formula for a new matrix that is matrix D n . For future research it is necessary to think about the relation between Stirling’s matrix and tetranacci matrix as well as the relation between Stirling’s matrix and other n-nacci matrices.

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  [2.]

  M. Bona, A Walk Through Combinatorics, World Scientific Publishing, Singapore, (2006).

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  G. S. Cheon dan J. S. Kim, Stirling matrix via Pascal matrix, Linear Algebra and Its Applications , 329 (2001), 49–59.

  [4.]

  L. Comtet, Advanced Combinatorics, Reidel Publishing Company, Hol- land, (1974).

  [5.] M. Feinberg, Fibonacci-tribonacci, Fibonacci Quartely, 1 (1963), 70–74. [6.] K. Kuhapatanakul, The generalized tribonacci p-numbers and applica- tions, East-West Journal of Mathematical, 14 (2012), 144-153.

  [7.]

  G. Y. Lee, J. S. Kim and S. H. Cho, Some combinatorial identities via Fibonacci numbers, Discrete Applied Mathematics, 130 (2003), 527–534.

  [8.] P. Maltais and T. A. Gulliver, Pascal matrices and Stirling Numbers, Applied Mathematical , 2 (1997), 7–11. [9.]

  B. C. Rennie and A. J. Dobson, On Stirling Numbers of the Second Kind, Journal of Combinatorial Theory , 7 (1969), 116–121.

  [10.] N. Sabeth, S. Gemawati and H. Saleh, A factorization of the tribonacci

  matrix and the Pascal matrix, Applied Mathematical Sciences, 11 (2017), 489–497.

  Fifitriani Rasmi : Graduate Student, Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Riau, Bina Widya Campus, Pekan- baru 28293, Indonesia.

  E-mail: fifitrianirasmi26@gmail.com F. Rasmi, S. Gemawati, Kartini & M. D. H.Gamal–Stirling’s Numbers of Second Kind and Tribonacci Matrix Sri Gemawati

: Senior Lecturer, Department of Mathematics, Faculty of Mathe-

matics and Natural Sciences, University of Riau, Bina Widya Campus, Pekanbaru

  28293, Indonesia.

  E-mail: gemawati.sri@gmail.com

  Kartini : Senior Lecturer, Department of Mathematics, Education Faculty of Teach- ers Training and Education, University of Riau, Bina Widya Campus, Pekanbaru

  28293, Indonesia.

  E-mail: tin baa@yahoo.com

  M. D. H. Gamal : Associate Professor, Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Riau, Bina Widya Campus, Pekan- baru 28293, Indonesia.

  E-mail: mdhgamal@unri.ac.id