PROTEUS JOURNAL ISSN/eISSN: 0889-6348

On Some Properties of Narayana Numbers

Dr. R. Sivaraman Associate Professor, Department of Mathematics, D. G. Vaishnav College, Chennai, India National Awardee for popularizing mathematics among masses Email: [email protected]

Abstract Among several classes of numbers that exist in mathematics, Narayana numbers plays a significant role especially in several problems of combinatorics. In this paper, we will introduce Narayana Numbers through coefficients similar to binomial coefficients and prove that sums of row entries of Narayana triangle are precisely the well known Catalan numbers. Some of the interesting combinatorial properties concerning with Narayana numbers were discussed in this paper. This will add more insight about such numbers and will reach wider research audience.

Keywords: Narayana Numbers, Triangular Numbers, Factorials, Narayana Triangle, Catalan Numbers.

1. Introduction

Pascal and other notable mathematicians investigated Pascal’s triangle using binomial coefficients for natural numbers. During 1970’s mathematicians like Hansell and Hoggatt generalized the concept of binomial coefficients for triangular numbers. They proved that such numbers are always positive integers. Narayana numbers were first studied by famous British Combinatorist Percy Alexander Macmahon during 1915 and 1916. During 1955, Indian origin Canadian mathematician Tadepalli Venkata Narayana identified as T.V. Narayana rediscovered the Narayana numbers through the construction of a triangle named after him as Narayana triangle. This paper provides some of the interesting properties of Narayana numbers. We will first begin with some definitions.

2. Definitions

2.1 The triangular numbers are numbers which are sum of consecutive natural numbers. They form a special case of more general class of numbers called figurate numbers. The nth triangular n( n  1) number is given by t1  2  3  n  (2.1) n 2

2.2 The numbers formed through two integers n and r where n0,0  r n given by the n  n! expression    (2.2) are called binomial coefficients. They are called so because r  r! ( nr  )! these numbers form the coefficients of the binomial expansion (a b )n .

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2.3 The numbers formed through two integers integers n and r where n0,0  r n given by the n  t ! expression  n (2.3) where t is the nth triangular number is defined in (2.1) and   n r  tr! t nr !

tn ! is the factorial of the nth triangular number tn . In particular, the factorial of the nth triangular

number tn is given by ttttn! n  n1  n  2  ttt 3 2  1 (2.4) .We assume that t0 ! 1similar to n  n 0! 1. With this convention, from (2.3), we get    1. 0 n

Note that equation (2.4) is a direct analogue of usual factorial which is product of first n natural numbers. We also see that equation (2.3) in a sense acts like a binomial coefficient for triangular n  numbers. We call numbers generated through (2.3) of the form   as Narayana numbers. r 

1 2n  2.4 The nth Catalan number is given by the expression Cn    (2.5) n 1n 

3. Theorem 1

n  n1  n n 1  The Narayana numbers   are given by      (3.1) r  rn r 1  rr 1 

Proof: By definition (2.3), we have

n  tn! ttt nrnr !  1  nr 2  tttt n  1  nnn   1  tt nr  2  nr 1     r  ttrnr!! tt rnr !!   tttt rr  1 2 1 (nnnn 1) (  1) ( nr  2)( nr 1)    2 2 2 (r 1) rrr (  1) 3  2 2 1    2 2 2 2 (nnn 1)2  (  1) 2  ( nr  2) 2  ( nr  1)  (r 1)  rr2  (  1) 2  3 2  2 2  1 2

(1)n nn (1)  ( nr  2)(  nr  1) nn  (1)  ( nr  2)(  nr  1) 1    (1)(1)rrr 321 rr  (1) 321 nr  1 n1  n 1    . r1  r n r 1

n1  n n 1  Thus,      as required. This completes the proof. rn r 1  rr 1 

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n  We notice that equation (3.1) provides an explicit formula for Naryana numbers of the form   r  n  n  in terms of the binomial coefficients   . Some authors call   as Tribinomial coefficients r  r  since they represent binomial coefficients analogue of triangular numbers. We further know that nn  n 1  the binomial coefficients satisfy the recurrence relation    . The Narayana numbers rr  r 1  also satisfy the same relation which we will prove through following theorem.

3.1 Theorem 1 (Recurrence Relation of Narayana Numbers)

n  ntn  n 1  The Narayana numbers   satisfy the recurrence relation    (3.2) r  rtr  r 1 

Proof: Using equations (2.3) and (2.4), we have

n  tn! t n !   tr  t r r  ttrnr!! tt r 1 !!  nr

tn1 ! n 1   tn  t n tr1 ! t nr  ! r 1 

This proves (3.2) and thus completes the proof.

4. Narayana Triangle

n  We denote the Narayana numbers   by N(n,r). We now try to construct a triangle which we r  call Narayana Triangle whose entries are N(n,r), where n is any non-negative integer and r = 1,2,3,…,n. Thus, the entries of the Narayana triangle are zero for r > n.

n 0 For the first row, if n = 0, then r = 0 is the only choice and   1 Hence the first row r 0 entry of Narayana triangle is just 1.

1  11  1 2 For the second row, if n = 1 then r = 0,1. Hence by (3.1), we have   1 and      1 0  11  1 2 So the second row entries of Narayana triangle are 1, 1.

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2  21  2 3 For the third row, if n = 2, then r = 0, 1, 2. Hence by (3.1), we have   1,       3 0  12  1 2 2  and   1. Hence the third row entries of Narayana triangle are 1, 3, 1. 2 

3  For the fourth row, if n = 3, then r = 0, 1, 2, 3. Hence by (3.1), we have   1, 0  31  3 4  3 1  3 4  3  6,    6,  1.Hence the fourth row entries of Narayana 13  1 2  2 2  2 3  3 triangle are 1, 6, 6, 1.

Proceeding in this fashion, we get Narayana triangle displayed with first seven rows:

Figure 1: Narayana Triangle

The numbers forming the triangle when read from left to right row-wise gives 1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 20, 10, 1, 1, 15, 50, 50, 15, 1, . . . are called Narayana numbers. These numbers are obtained from (3.1). We can notice many similarities between Pascal’s triangle and Narayana n  n triangle. For example, since    1 each row of Narayana triangle (except the first) begin 0 n and end with 1 like in Pascal’s triangle. We knew that the entries of Pascal’s triangle are symmetrical about the vertical line through the middle. This is because of the fact that n n    . Similarly, from Figure 1 we notice that the entries of Narayana triangle also r nr  satisfy the same property.

We will now prove this property formally through the following theorem.

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4.1 Theorem 2

The entries of Narayana triangle are symmetrical about the central vertical line.

n  Proof: All we have to do is to show that for Narayana numbers of the form   , we should have r  n n    (4.1) . Now by the generation of Narayana numbers presented in (3.1), we get r nr 

n1  n n1  1  n (n  1)!        nrr1  nrnr 1  rnrr  1(1)!!  r 

1n (n  1)! 1  n n1 n       nr1r ( nr  )! ( r 1)! nr  1  rrr1

This completes the proof.

Thus from Theorem 2, we see that the Narayana numbers lie symmetrically about the central vertical line formed by the numbers 1, 3, 20, 175, . .. (See Figure 1). We call the numbers 1, 3, 20, 175, . . . as Central Narayana numbers or Central Tribinomial coefficients. In the following property, we assure that all entries of Narayana triangle namely Narayana numbers are integers. Hence if we extend the rows of construction of Narayana triangle we are sure enough to get only particular positive integers.

4.2 Theorem 3

n  The Narayana numbers   are always positive integers. r 

Proof: We consider evaluating the following 2 × 2 determinant of binomial coefficients.

n  n     r  r 1  n n1  n  n  1  n n 1 n r  n n  1            n1  n  1  rr1  r 1  rrr   1 n r 1  rr  1     r  r 1  1 n n1   n      n r 1rr1   r

Thus the Narayana numbers are precisely the determinant of four binomial coefficients which are integers. Since the determinant value of entries which are integers also result in an integer, it follows that the Narayana numbers are integers. Narayana numbers are positive because from

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(3.1), they are product of three terms in which two are binomial coefficients which are positive 1 and third term which is also positive since 0r  n . This completes the proof. n r 1

4.3 Getting one from another

Since the entries of Pascal’s triangle are binomial coefficients, Theorem 3 provides a way of generating entries of Narayana triangle directly through the entries of Pascal’s triangle. For 4  instance, we know that    6 . From this we can generate the corresponding entry of Narayana 2  4  triangle namely   using the method used in the proof of Theorem 3. In view of this, we get 2  n n n1  n  n  1  4  4 5  4 5        . So,         (6 10) (4 10) 20. Thus the rrr1  r  1  r  2  2 3  3 2 4  4  number    6 in Pascal’s triangle corresponds to    20 in Narayana triangle as shown in 2  2  Figure 2.

Figure 2

5. Properties of Narayana Numbers

In this section, we will prove some interesting properties of Narayana numbers apart from those mentioned through the following theorems.

5.1 Theorem 4

The second slant diagonal numbers of Narayana triangle are triangular numbers

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n  Proof: The second slant diagonal numbers of Narayana triangle are given by   . Now by 1  n1  n n1  n  1  n ( n  1) equation (3.1), we have       tn , where tn is the nth 1n 1  1  1 2  2  2 n  triangular number. Thus,    tn (5.1) . This completes the proof. 1 

It is interesting to note that in Pascal’s triangle, we find triangular numbers in the third slant diagonal where as in Narayana triangle we find them in second slant diagonal.

5.2 Theorem 5

The Central Narayana numbers are squares if and only if 2n  1 is a square.

2n  Proof: We first note that the Central Narayana numbers are numbers of the form   . Now n  using (3.1), we have

2n1  2 n 2 n 1  1  2 n (21)!n        n2n n 1  n n1  n  1  n (1)!! n  n 12n  1(2)!n   (2n 1) n1n  n  1!! n n

12n 1  2 n 2 (2n 1)    (2 nC 1) n n1n n  1  n

2n  2 Thus,   (2n  1) Cn (5.2) where Cn is the nth Catalan number defined in (2.5). n 

The Central Narayana numbers are given by (5.2). We notice that in the right hand side, among product of two terms the second is already a perfect square. Hence the Central Narayana number would be a perfect square if and only if the first factor namely 2n  1is a perfect square. This completes the proof.

5.3 Theorem 6

The Central Narayana numbers are odd if and only if either n = 0 or n is a Mersenne number.

Proof: By definition of Narayana numbers it is clear that the theorem is true for n = 0, since the top most entry of Narayana triangle is 1 which is odd. We also note that numbers of the form 2n  1are called Mersenne numbers. Now by (5.2) of Theorem 5, we know that the Central Narayana numbers are product of 2n  1and square of nth Catalan number.

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2n  2 That is,   (2n  1) Cn . Now, we know that for all n, 2n  1is always an odd number. Thus n  2 the Central Narayana number will be odd if and only if Cn is odd which happens if and only if

Cn is odd. Now we know that the nth Catalan number Cn will be odd if and only if n is a Mersenne number. (see [5]) for proof). Thus the Central Narayana numbers will be odd if and only if either n = 0 or n is a Mersenne number. This completes the proof.

5.4 Theorem 7

The six Narayana numbers satisfy the following hexagonal identity

n a  n  n  a  n  a  n a  n         (5.3) r a  r a  r  r  r  a  r a 

Proof: Using the definition of Narayana numbers as in (2.3), we get

n a  n  n  a  tn a! t n ! t n  a !        rara    r  tra!!! tt nr  ra   t nra  !! tt r  nra  !

tn a! t n  a ! t n !    ttr! nar !!!! t ra   tt nr  ra   t nra  ! n a  n  a  n       r  r a  r  a 

We now prove an important result which provides the connection between Narayana numbers and Catalan numbers.

6. Theorem 8

The sum of entries of each row in the Narayana triangle is a Catalan number. In particular, we n n   C (6.1) have    n1 where Cn1 is the (n+1)th Catalan number. r0 r 

Proof: To establish (6.1), we make use of the following two well known combinatorial identities concerning binomial coefficients.

pp  p 1     (6.2) qq  q 1 

p  q  p q      (6.3) . k a k  b  k  a  b 

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First, using (6.2), we have

1n1 1n  1  n 1  n     n1nr1 nnr  1 1  nr  nr  1  r

Now using equation (3.1) from Theorem 1 and the previous identity, we have

nn n1  n n1 n 1  n  1  n 1 1 n  n  1 n 1             rr0r 0n r1  rr1 r  0 n  1  nrr  1  1 n  1 r  0  rnr  1 1  1n n1 n 1  1  2 n  2  1 (2n  2)!       n1r0 1r n 1 r  n  1  n  2  n  1 ( n 2)! n ! 1 (2n  2)! 1 2n  2       Cn1 n  2 (n 1)! ( n 1)! n  2 n 1 

This completes the proof.

We present Figure 3 below to identify the concepts proved in Theorems 4 and 8.

Figure 3

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7. Conclusion

The concept of Narayana numbers and construction of Narayana triangle which is viewed as generalization of Pascal’s triangle is explained briefly in this paper. Using the definition of Narayana numbers and other combinatorial identities of binomial coefficients we have proved nine interesting properties through eight theorems discussed in this paper. These properties will be very useful for further exploration of Narayana numbers. Since the Narayana numbers in each row sum to Catalan number, Narayana numbers have plenty of connections with several combinatorial problems. One such example is that Narayana numbers represent the number of complete binary trees with n internal nodes and r – 1 jumps. Also, the number of monomials n r  with n operators and r variables is the Narayana number   . The number of non-crossing r  n  partitions with r blocks of a set with n elements is precisely the Narayana number N( nr , )    . r  It is wonderful to see simple generalization of definitions lead us to numbers which have profound applications both with in mathematics and to other branches of Science as well.

As further scope, it is interesting to note that the concept of Narayana numbers can be extended to negative integer values of n also. In such cases, we can prove many interesting properties similar to those which have been proved in this paper.

REFERENCES

[1] Tadepalli Venkata Narayana. Sur les treillis form_es par les partitions d'un entier et leurs applications _a la th_eorie des probabilit_es. C. R. Acad. Sci. Paris, 240:1188{1189, 1955.

[2] Paul Barry, On a generalization of the Narayana triangle, J. Integer Seq., 14(4):Article 11.4.5, 22, 2011.

[3] Paul Barry and Aoife Hennessy, A note on Narayana triangles and related polynomials, Riordan arrays, and MIMO capacity calculations. J. Integer Seq., 14(3):Article 11.3.8, 26, 2011.

[4] Nelson Y. Li and Tou_k Mansour, An identity involving Narayana numbers, European J. Combin., 29(3):672{675, 2008.

[5] R. Sivaraman, Two Interesting Results about Catalan Numbers, Journal of Scientific Computing, Volume 9, Issue 5, (2020), pp. 57 – 61.

[6] Tou_k Mansour and Yidong Sun. Identities involving Narayana polynomials and Catalan numbers. Discrete Math., 309(12):4079{4088, 2009.

[7] Richard P. Stanley. Enumerative combinatorics. Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.

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