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� 2018 � 2 � 5 � -8 � � � � � � � � � � Inspiration of The Pascal Triangle � � � � � � � � � � � � � � (Yidong Sun) ◭◭ ◮◮ ◭ ◮ A Joint Work with Ma Fei and Ma Luping � 2 of 61 � � College of Science of Dalian Maritime University � � � � � � � � � � � � � � � � �
Contents � Pascal Triangle � The Well-known Motivation � The First Starting Point and Problems � � � � � The Second Starting Point and Problems � � � � ◭◭ ◮◮ The Third Starting Point and Problems ◭ ◮ � New Discovery � 3 of 61 � � � References � � � � � Thanks � � � �
1. The Pascal Triangle: In mathematics, Pascal’s triangle is a triangular array of the binomial coeffi- cients. In much of the Western world, it is named after French mathematician Blaise Pascal (1623-1662), although other mathematicians studied it centuries before him in India, Iran, China, Germany, and Italy. � � � � n/k 0 1 2 3 4 5 � � � 0 1 ◭◭ ◮◮ ◭ ◮ 1 1 1 � 4 of 61 2 1 2 1 � � 3 1 3 3 1 � � � � 4 1 4 6 4 1 � � 5 1 5 10 10 5 1 � � Table 1. The Pascal triangle P for n and k up to 5 .
Pascal’s triangle was known in China in the early 11th century through the work of the Chinese mathematician Jia Xian (1010-1070). In the 13th century, Yang Hui (1238-1298) presented the triangle and hence it is still called Yang Hui’s triangle in China. � � � � � � � ◭◭ ◮◮ ◭ ◮ � 5 of 61 � � � � � � � � � � Figure 1. Yang Hui’s triangle P for n and k up to 9 .
2. The Well-known Motivation: (1) Gaussian binomial coefficient [ k ]![ n − k ]! = (1 − q n )(1 − q n − 1 ) · · · (1 − q n − k +1 ) � n � [ n ]! = (1 − q k )(1 − q k − 1 ) · · · (1 − q ) k q � � � � n � n � ( x + y ) n = � x k y n − k , xy = yx, � � � k k =0 ◭◭ ◮◮ n ◭ ◮ � n � ( x + y ) n = � x k y n − k , qxy = yx. � 6 of 61 k q k =0 � � • Vector Spaces on Finite Field, � � � � • The Theory of Partitions, � � • The Theory of Q-series. � �
(2) Binomial Inversion Formula n n � n � � n � � � ( − 1) n − k A n = B k ⇔ B n = A k . k k k =0 k =0 • Gould-Hsu Inversion Formula, • Krattenthaler Inversion Formula, � � � � • Ma Xinrong Inversion Formula. � � � ◭◭ ◮◮ ◭ ◮ (3) Congruence Properties of Binomial Coefficients � 7 of 61 • Lucas Congruence, � � • Wolstenholme Congruence, � � � � • Binomial Congruence. � � � �
(4) The Theory of Riordan Array � 1 t � P = 1 − t, 1 − t A Riordan array R = ( R n,k ) n ≥ k ≥ 0 is an infinite lower triangular matrix with nonzero entries on the main diagonal, such that R n,k = [ t n ] g ( t )( f ( t )) k for n ≥ k , namely, R n,k equals the coefficient of t n in the expansion of the se- ries g ( t )( f ( t )) k , where g ( t ) = 1 + g 1 t + g 2 t 2 + · · · and f ( t ) = f 1 t + f 2 t 2 + · · · � � � � with f 1 � = 0 are two formal power series. � � � ◭◭ ◮◮ ◭ ◮ R 0 , 0 � 8 of 61 R 1 , 0 R 1 , 1 � � R 2 , 0 R 2 , 1 R 2 , 2 � � � � R 3 , 0 R 3 , 1 R 3 , 2 R 3 , 3 � � ... · · · · · · · · · · · · � � g ( t ) g ( t ) f ( t ) g ( t ) f ( t ) 2 g ( t ) f ( t ) 3 · · ·
(5) Fibonomial Coefficient Analog � n � n ! F = k k ! F ( n − k )! F F where n ! F = F 1 F 2 · · · F n with F 0 = 0 , F 1 = 1 and F n = F n − 1 + F n − 2 . (6) Fractal � � � � � � � ◭◭ ◮◮ ◭ ◮ � 9 of 61 � � � � � � � � � � Figure 2. Sierpinski triangle.
(7) Unimodality of Binomial Coefficients � n � n � � n � � � � � n � n ≤ ≤ · · · ≤ ≥ ≥ · · · ≥ � � � � 0 1 [ n/ 2] [ n/ 2] − 1 n � � � • Log-Concavity ( a 2 k ≥ a k − 1 a k +1 ) , Log-Convexity ( a 2 k ≤ a k − 1 a k +1 ) , ◭◭ ◮◮ • q -Log-Concavity, q -Log-Convexity, ◭ ◮ • Infinite Log-Concavity, Infinite Log-Convexity. � 10 of 61 � � � � � � � � � �
2. The First Starting Points and Problems: 2.1 The Rule of David Star The Star of David rule, originally stated by Gould in 1972, is given by � n �� n + 1 �� n + 2 � � n �� n + 1 �� n + 2 � = , k k − 1 k + 1 k − 1 k + 1 k for any k and n , or equivalently which implies that � � � � � �� n + 1 �� n + 2 � � n �� n + 1 �� n + 2 � n = . � � � k + 1 k k + 2 k k + 2 k + 1 ◭◭ ◮◮ ◭ ◮ � 11 of 61 � � � � � � � � � �
In 2003, the author observed in his Master dissertation that if multiplying the above two identities and dividing by n ( n + 1)( n + 2) , one can arrive at N n,k +1 N n +1 ,k N n +2 ,k +2 = N n,k N n +1 ,k +2 N n +2 ,k +1 , �� n where N n,k = 1 � n � is the Narayana number. n k k − 1 � � � � n/k 0 1 2 3 4 5 � � � 0 1 ◭◭ ◮◮ 1 1 1 ◭ ◮ 2 1 3 1 � 12 of 61 3 1 6 6 1 � � 4 1 10 20 10 1 � � � � � � 5 1 15 50 50 15 1 � � Table 2.1. The Narayana triangle N for n and k up to 5 .
In the summer of 2006, the author asked Mansour for a combinatorial proof of the above Narayana identity to be found. Later, by Chen’s bijective algorithm for trees, Li and Mansour provided a combinatorial proof of a general identity � � � � � � � N n,k + m − 1 N n +1 ,k + m − 2 N n +2 ,k + m − 3 · · · N n + m − 2 ,k +1 N n + m − 1 ,k N n + m,k + m ◭◭ ◮◮ = N n,k N n +1 ,k + m N n +2 ,k + m − 1 · · · N n + m − 2 ,k +3 N n + m − 1 ,k +2 N n + m,k +1 . ◭ ◮ � 13 of 61 � � � � � � � � � �
This motivates the author to reconsider the Star of David rule and to propose a new concept called SDR -matrix which obeys the generalized rule of David star. � � Definition 0.1 Let A = A n,k n ≥ k ≥ 0 be an infinite lower triangular matrix, for � � � � any given integer m ≥ 3 , if there hold � � � p − r − 1 p − r − 1 r r ◭◭ ◮◮ � � � � A n + i,k + r − i A n + p − i,k + r + i +1 = A n + p − i,k + p − r + i A n + i,k + p − r − i − 1 , ◭ ◮ i =0 i =0 i =0 i =0 � 14 of 61 for all 2 ≤ p ≤ m − 1 and 0 ≤ r ≤ p − 1 , then A is called an SDR -matrix of � � order m . � � � � � � � �
In order to give a more intuitive view on the definition, we present a pictorial description of the generalized rule for the case m = 5 . � � � � � � � ◭◭ ◮◮ ◭ ◮ � 15 of 61 � � � � � � � � � �
SDR m : the set of SDR -matrices of order m ; SDR ∞ : the set of SDR -matrices A of order ∞ , that is A ∈ SDR m for any m ≥ 3 . � � � � �� n � � � �� By our notation, it is obvious that the Pascal triangle P = n ≥ k ≥ 0 and the k ◭◭ ◮◮ � � Narayana triangle N = N n +1 ,k +1 n ≥ k ≥ 0 are SDR -matrices of order 3 . ◭ ◮ In fact, both of them will be proved to be SDR -matrices of order ∞ . � 16 of 61 � � � � � � � � � �
2.2 The Basic Properties of SDR-matrices Lemma 0.1 For any A ∈ SDR m , B ∈ SDR m + i with i ≥ 0 , there hold A ◦ B ∈ SDR m , and A ◦ ( − 1) ∈ SDR m if it exists, where A ◦ B is the � � � � Hadamard product of A and B , A ◦ ( − 1) is the Hadamard inverse of A . � � � ◭◭ ◮◮ ◭ ◮ � � � � Lemma 0.2 For any A = A n,k n ≥ k ≥ 0 ∈ SDR m , then A n + i,k + j n ≥ k ≥ 0 ∈ � 17 of 61 SDR m for fixed i, j ≥ 0 . � � � � � � � � � �
Lemma 0.3 Given any sequence ( a n ) n ≥ 0 , let A n,k = a k , B n,k = b n − k and � � � � � � C n,k = c n for n ≥ k ≥ 0 , then A n,k n ≥ k ≥ 0 , B n,k n ≥ k ≥ 0 , C n,k n ≥ k ≥ 0 ∈ SDR ∞ . a 0 b 0 c 0 a 0 a 1 b 1 b 0 c 1 c 1 � � � � � � � , , a 0 a 1 a 2 b 2 b 1 b 0 c 2 c 2 c 2 ◭◭ ◮◮ a 0 a 1 a 2 a 3 b 3 b 2 b 1 b 0 c 3 c 3 c 3 c 3 ◭ ◮ · · · · · · · · · � 18 of 61 �� n �� • The Pascal triangle P = 0 ≤ k ≤ n ∈ SDR ∞ . � � k � 1 � n +1 �� n +1 �� • The Narayana triangle N = � � � � 0 ≤ k ≤ n ∈ SDR ∞ . n +1 k k +1 � �� ( n +1)! � � � n • The Lah triangle L = 0 ≤ k ≤ n ∈ SDR ∞ . ( k +1)! k � �
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