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 t1 2 3 n (2.1) n 2 2.2 The numbers formed through two integers n and r where n0,0 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 . VOLUME 11 ISSUE 8 2020 http://www.proteusresearch.org/ Page No: 8 PROTEUS JOURNAL ISSN/eISSN: 0889-6348 2.3 The numbers formed through two integers integers n and r where n0,0 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 n1 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 1n 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 n1 n 1 . r1 r n r 1 n1 n n 1 Thus, as required. This completes the proof. rn r 1 rr 1 VOLUME 11 ISSUE 8 2020 http://www.proteusresearch.org/ Page No: 9 PROTEUS JOURNAL ISSN/eISSN: 0889-6348 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 tn1 ! n 1 tn t n tr1 ! 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. VOLUME 11 ISSUE 8 2020 http://www.proteusresearch.org/ Page No: 10 PROTEUS JOURNAL ISSN/eISSN: 0889-6348 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. VOLUME 11 ISSUE 8 2020 http://www.proteusresearch.org/ Page No: 11 PROTEUS JOURNAL ISSN/eISSN: 0889-6348 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 n1 1 n (n 1)! nrr1 nrnr 1 rnrr 1(1)!! r 1n (n 1)! 1 n n1 n nr1r ( nr )! ( r 1)! nr 1 rrr1 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 n1 n n 1 n n 1 n r n n 1 n1 n 1 rr1 r 1 rrr 1 n r 1 rr 1 r r 1 1 n n1 n n r 1rr1 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 VOLUME 11 ISSUE 8 2020 http://www.proteusresearch.org/ Page No: 12 PROTEUS JOURNAL ISSN/eISSN: 0889-6348 (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 0r 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.