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Binomial Arrays and Generalized Vandermonde Identities Robert W. Donley, Jr. (CUNY-QCC) March 27, 2019 Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 1 / 37

Table of Contents 1 Catalan Numbers 2 Catalan Number Trapezoids 3 Pascal’s Triangle 4 Generalized Binomial Transform and Inverse 5 Binomial Arrays 6 Generalized Chu-Vandermonde Convolution 7 Hockey Stick Rules 8 New(-ish) Sequences Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 2 / 37

Binomial Coe ffi cients Non-negative numerator For n � 0 and 0  k  n , ✓ n ◆ n ! = k k !( n � k )! Negative numerator For n � 1 and k � 0 , ✓ � n ◆ ✓ n + k � 1 ◆ = ( � 1) k k k In all other cases, ✓ n ◆ = 0 k Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 3 / 37

Catalan Numbers C n Segner’s Recurrence for C n C 0 = 1 , and, for n � 1 , n X C n +1 = C i C n � i . i =0 Interpret: C n +1 = ( C 0 , C 1 , . . . , C n ) · ( C n , C n � 1 , . . . , C 0 ) . Direct Definition For n � 1 , ✓ 2 n 1 ✓ 2 n ◆ = 1 ◆ C n = n + 1 n + 1 n n Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 4 / 37

Catalan Numbers C n C 0 = 1 C 1 = 1 · 1 = 1 C 2 = (1 , 1) · (1 , 1) = 2 C 3 = (1 , 1 , 2) · (2 , 1 , 1) = 5 C 4 = (1 , 1 , 2 , 5) · (5 , 2 , 1 , 1) = 14 C 5 = (1 , 1 , 2 , 5 , 14) · (14 , 5 , 2 , 1 , 1) = 42 1 , 1 , 2 , 5 , 14 , 42 , 132 , 429 , 1430 , 4862 , 16796 , . . . Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 5 / 37

Interpretation Stanley’s List (2015): Over 200 examples as enumeration Euler: C n is the number of triangulations of a regular ( n + 2)-gon C n is the number of ordered lists with ( n + 1) +s and n � s such that all partial sums are positive. C 0 : + C 1 : + + � C 2 : + + + � � , + + � + � C 3 : + + + + � � � , + + + � + � � , + + + � � + � , + + � + � + � , + + � + + � � Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 6 / 37

Shapiro’s Catalan Triangle Shapiro’s Catalan triangle for B n , k n \ k 1 2 3 4 5 6 1 1 2 2 1 3 5 4 1 4 14 14 6 1 5 42 48 27 8 1 6 132 165 110 44 10 1 ✓ 2 n ◆ B n , k = k n � k n B n , k = B n � 1 , k � 1 + 2 B n � 1 , k + B n � 1 , k +1 Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 7 / 37

Shapiro’s Catalan Triangle Shapiro’s Catalan triangle for B n , k n \ k 1 2 3 4 5 6 1 1 2 2 1 3 5 4 1 4 14 14 6 1 5 42 48 27 8 1 6 132 165 110 44 10 1 Shapiro (1976): Dot product of any row with itself or two (adjacent) rows is another Catalan number Examples: (2 , 1 , 0 , 0) · (14 , 14 , 6 , 1) = 28 + 14 + 0 + 0 = 42 , (14 , 14 , 6 , 1) · (14 , 14 , 6 , 1) = 196 + 196 + 36 + 1 = 429 . Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 8 / 37

Kirillov-Melnikov’s Catalan triangle (1996) 2 3 1 1 1 1 1 1 1 1 1 · · · � 1 0 1 2 3 4 5 6 7 · · · 6 7 6 7 0 � 1 � 1 0 2 5 9 14 20 · · · 6 7 6 7 6 0 0 � 1 � 2 � 2 0 5 14 28 · · · 7 6 7 6 7 0 0 0 � 1 � 3 � 5 � 5 0 14 · · · 6 7 6 7 0 0 0 0 � 1 � 4 � 9 � 14 � 14 · · · 4 5 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Rule: 1 (1 , � 1 , 0 , 0 , . . . ) down left column, 2 1s along top row, and 3 capital L -summation to progress to the right. Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 9 / 37

Dot Product Rule 2 3 1 1 1 1 1 1 1 1 1 · · · -1 0 1 2 3 4 5 6 7 · · · 6 7 6 7 0 � 1 � 1 0 2 5 9 14 20 · · · 6 7 6 7 0 0 � 1 -2 � 2 0 5 14 28 · · · 6 7 6 7 6 0 0 0 -1 � 3 -5 � 5 0 14 · · · 7 6 7 6 7 0 0 0 0 � 1 -4 � 9 � 14 -14 · · · 4 5 · · · · · · · · · · · · 0 � 1 · · · · · · · · · · · · To recover/extend Shapiro’s formulas, 1 columns are skew-palindromes, and 2 use convolution (Segner’s Rule) to align correctly. (1 , 3 , 2 , � 2 , � 3 , � 1) · ( � 1 , � 3 , � 2 , 2 , 3 , 1) = � 2(14) (1 , 2 , 0 , � 2 , � 1 , 0) · ( � 4 , � 5 , 0 , 5 , 4 , 1) = � 2(14) Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 10 / 37

Shifting Columns for Dot Product Start with any column; convolution gives � 2 C n Unchanged if we telescope inwards or outwards (1 , 3 , 2 , � 2 , � 3 , � 1) · ( � 1 , � 3 , � 2 , 2 , 3 , 1) = � 28 (1 , 2 , 0 , � 2 , � 1 , 0) · ( � 4 , � 5 , 0 , 5 , 4 , 1) = � 28 (1 , 1 , � 1 , � 1 , 0 , 0) · ( � 9 , � 5 , � 1 , 1 , 5 , 9) = � 28 (1 , 0 , � 1 , 0 , 0 , 0) · ( � 14 , 0 , 14 , 14 , 6 , 1) = � 28 (1 , � 1 , 0 , 0 , 0 , 0) · ( � 14 , 14 , 0 , 0 , 0 , 0) = � 2(14) Point of talk: Explain this phenomenon in a general setting. Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 11 / 37

Pascal’s Triangle 2 1 1 1 1 1 1 1 1 1 · · · 3 0 1 2 3 4 5 6 7 8 · · · 6 7 6 7 0 0 1 3 6 10 15 21 28 · · · 6 7 6 7 0 0 0 1 4 10 20 35 56 · · · 6 7 6 7 0 0 0 0 1 5 15 35 70 · · · 6 7 6 7 0 0 0 0 0 1 6 21 28 · · · 4 5 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Rule: 1 (1 , 0 , 0 , . . . ) down left column, 2 1s along top row, and 3 capital L -summation to progress to the right. Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 12 / 37

Chu-Vandermonde Convolution 2 3 1 1 1 1 1 1 1 1 1 · · · 0 1 2 3 4 5 6 7 8 · · · 6 7 6 7 0 0 1 3 6 10 15 21 28 · · · 6 7 6 7 6 7 0 0 0 1 4 10 20 35 56 · · · 6 7 6 7 0 0 0 0 1 5 15 35 70 · · · 6 7 6 7 0 0 0 0 0 1 6 21 28 · · · 4 5 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (1 , 3 , 3 , 1) · (1 , 3 , 3 , 1) = (1 , 2 , 1 , 0) · (4 , 6 , 4 , 1) = (1 , 1 , 0 , 0) · (10 , 10 , 5 , 1) = (1 , 0 , 0 , 0) · (20 , 15 , 6 , 1) = 20 . Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 13 / 37

Chu-Vandermonde Convolution This is just the famous Chu-Vandermonde convolution ◆ ✓ n k ✓ m + n ◆ ✓ m ◆ X = k � i . k i i =0 Suppose X = A [ B is a disjoint union with | A | = m and | B | = n . If we choose k elements from X , the summation expresses this choice with respect to independent choices from A and B . In fact, one may allow negative m or n . Lagrange: k ◆ 2 ✓ 2 n ◆ ✓ n X = . k i i =0 Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 14 / 37

Generalized Binomial Transform 1 X a i x i { a i } 1 i =0 ! p ( x ) = i =0 Generalized Binomial Transform n ✓ n ◆ X B n a k = a k , n = a k � i i i =0 1 1 B n a k x k = (1 + x ) n X X a i x i i =0 k =0 ✓ n ◆ B n a n = a n , n = P n Usual binomial transform: a n � i i =0 i Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 15 / 37

Pascal’s Recurrence n = 1 : Ba k = a k � 1 + a k n = 2 : B 2 a k = a k � 2 + 2 a k � 1 + a k n = 3 : B 3 a k = a k � 3 + 3 a k � 2 + 3 a k � 1 + a k Pascal’s Recurrence (Capital L) B n +1 a k = B n a k + B n a k � 1 Pascal’s Identity ✓ n + 1 ◆ ✓ n ◆ ✓ ◆ n = + k � 1 k k Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 16 / 37

Matrix Implementation 1 Fourth quadrant matrix 2 { a i } down first column, a 0 along first row 3 Pascal’s recurrence: Capital- L summation 2 · · · 3 a 0 a 0 a 0 a 0 a 0 a 0 + a 1 2 a 0 + a 1 3 a 0 + a 1 4 a 0 + a 1 · · · a 1 6 7 6 7 a 2 a 1 + a 2 a 0 + 2 a 1 + a 2 3 a 0 + 3 a 1 + a 2 6 a 0 + 4 a 1 + a 2 · · · 6 7 6 7 a 2 + a 3 a 1 + 2 a 2 + a 3 · · · · · · · · · a 3 4 5 · · · · · · · · · · · · · · · · · · B n a k : row k + 1, column n + 1 (first row and column indexed to 0) Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 17 / 37

Pascal’s triangle ✓ n ◆ B n a k = a i = (1 , 0 , 0 , . . . ) ! k 1 1 1 1 1 1 1 · · · 2 3 0 1 2 3 4 5 6 · · · 6 7 6 7 PT = 0 0 1 3 6 10 15 · · · 6 7 6 7 0 0 0 1 4 10 20 · · · 4 5 · · · · · · · · · · · · · · · · · · · · · · · · 1 " : Sums to 2 n 2 % : Sums to F n (Fibonacci numbers) 3 Hockey Stick Summation : 1 + 2 + 3 + 4 + 5 = 15 Robert W. Donley, Jr. (CUNY-QCC) Binomial Arrays and Generalized Vandermonde Identities March 27, 2019 18 / 37