Eulerian Numbers Armin Straub 23-Apr 2007 Armin Straub Eulerian - PowerPoint PPT Presentation

Introduction Examples More Properties Eulerian Numbers Armin Straub 23-Apr 2007 Armin Straub Eulerian Numbers

Introduction Examples More Properties Outline Introduction Abstract Definition Simple Properties Examples Differentiating the Geometric Series Counting Points in Hypercubes Occurrence in Probability Theory More Properties Asymptotics Generating Functions Armin Straub Eulerian Numbers

Introduction Abstract Definition Examples Simple Properties More Properties Abstract Definition Definition (1 − q n ) 24 = ∆ sep � sepqθ ( q ) 24 sep � sepq � � τ ( n ) q n . n � 1 n � 1 Armin Straub Eulerian Numbers

Introduction Abstract Definition Examples Simple Properties More Properties Abstract Definition Definition (1 − q n ) 24 = ∆ sep � sepqθ ( q ) 24 sep � sepq � � τ ( n ) q n . n � 1 n � 1 Example Denote σ ∈ S n as [ σ (1) , . . . , σ ( n )] . [5 , 1 , 3 , 4 , 2] has 2 ascents [2 , 3 , 4 , 1 , 5] has 3 ascents Armin Straub Eulerian Numbers

Introduction Abstract Definition Examples Simple Properties More Properties Abstract Definition Definition (1 − q n ) 24 = ∆ sep � sepqθ ( q ) 24 sep � sepq � � τ ( n ) q n . n � 1 n � 1 Example Denote σ ∈ S n as [ σ (1) , . . . , σ ( n )] . [5 , 1 , 3 , 4 , 2] has 2 ascents [2 , 3 , 4 , 1 , 5] has 3 ascents Definition � n � The Eulerian number is the number of permutations in S n Armin Straub Eulerian Numbers k

Introduction Abstract Definition Examples Simple Properties More Properties Simple Properties ◮ Row Sums � n � � = n ! k k Armin Straub Eulerian Numbers

Introduction Abstract Definition Examples Simple Properties More Properties Simple Properties ◮ Row Sums � n � � = n ! k k ◮ Symmetry � n � � � n = k n − 1 − k Armin Straub Eulerian Numbers

Introduction Abstract Definition Examples Simple Properties More Properties Simple Properties ◮ Row Sums � n � � = n ! k k ◮ Symmetry � n � � � n = k n − 1 − k ◮ Recurrence � n � � n − 1 � � n − 1 � = ( k + 1) + ( n − k ) k − 1 k k Armin Straub Eulerian Numbers

Introduction Abstract Definition Examples Simple Properties More Properties Eulerian Triangle Note 1 1 1 The triangle starts 1 4 1 � 1 � 1 11 11 1 0 � 2 � 2 � � 1 26 66 26 1 0 1 1 57 302 302 57 1 Armin Straub Eulerian Numbers

Introduction Differentiating the Geometric Series Examples Counting Points in Hypercubes More Properties Occurrence in Probability Theory Differentiating the Geometric Series 1 x ( xD ) = (1 − x ) 2 1 − x Armin Straub Eulerian Numbers

Introduction Differentiating the Geometric Series Examples Counting Points in Hypercubes More Properties Occurrence in Probability Theory Differentiating the Geometric Series 1 x ( xD ) = (1 − x ) 2 1 − x 1 x ( xD ) 2 = (1 − x ) 3 (1 + x ) 1 − x 1 x ( xD ) 3 (1 − x ) 4 (1 + 4 x + x 2 ) = 1 − x 1 x (1 − x ) 5 (1 + 11 x + 11 x 2 + x 3 ) ( xD ) 4 = 1 − x Armin Straub Eulerian Numbers

Introduction Differentiating the Geometric Series Examples Counting Points in Hypercubes More Properties Occurrence in Probability Theory Differentiating the Geometric Series 1 x ( xD ) = (1 − x ) 2 1 − x 1 x ( xD ) 2 = (1 − x ) 3 (1 + x ) 1 − x 1 x ( xD ) 3 (1 − x ) 4 (1 + 4 x + x 2 ) = 1 − x 1 x (1 − x ) 5 (1 + 11 x + 11 x 2 + x 3 ) ( xD ) 4 = 1 − x . . . n − 1 1 � n � x � ( xD ) n x k = (1 − x ) n +1 1 − x k k =0 Armin Straub Eulerian Numbers

Introduction Differentiating the Geometric Series Examples Counting Points in Hypercubes More Properties Occurrence in Probability Theory Counting Points in Hypercubes ◮ 1 ≤ i ≤ x . � x � x 1 = 1 Armin Straub Eulerian Numbers

Introduction Differentiating the Geometric Series Examples Counting Points in Hypercubes More Properties Occurrence in Probability Theory Counting Points in Hypercubes ◮ 1 ≤ i ≤ x . � x � x 1 = 1 ◮ 1 ≤ i, j ≤ x . � x + 1 � � x � i ≤ j � x 2 = + j < i 2 2 Armin Straub Eulerian Numbers

Introduction Differentiating the Geometric Series Examples Counting Points in Hypercubes More Properties Occurrence in Probability Theory Counting Points in Hypercubes ◮ 1 ≤ i ≤ x . � x � x 1 = 1 ◮ 1 ≤ i, j ≤ x . � x + 1 � � x � x 2 = + 2 2 ◮ 1 ≤ i, j, k ≤ x . i j k ≤ ≤ i k < j ≤ � x + 2 � � x + 1 � � x � j < i k ≤ � x 3 = + 4 + j ≤ k < i 3 3 3 k < i j ≤ k < j < i Armin Straub Eulerian Numbers

Introduction Differentiating the Geometric Series Examples Counting Points in Hypercubes More Properties Occurrence in Probability Theory Counting Points in Hypercubes ◮ 1 ≤ i ≤ x . � x � x 1 = 1 ◮ 1 ≤ i, j ≤ x . � x + 1 � � x � x 2 = + 2 2 ◮ 1 ≤ i, j, k ≤ x . � x + 2 � � x + 1 � � x � x 3 = + 4 + 3 3 3 ◮ Generally, n − 1 � n �� x + k � x n = � k n k =0 Armin Straub Eulerian Numbers

Introduction Differentiating the Geometric Series Examples Counting Points in Hypercubes More Properties Occurrence in Probability Theory Counting Points in Hypercubes (Sums of Powers) Using n − 1 � n �� x + k � x n = � k n k =0 and � x + k � � x + k � ∆ x = n n − 1 we get � N n − 1 n − 1 N � n � x + k � � n �� N + k + 1 � x n = � � � � = n + 1 k n k x =0 k =0 x =0 k =0 Armin Straub Eulerian Numbers

Introduction Differentiating the Geometric Series Examples Counting Points in Hypercubes More Properties Occurrence in Probability Theory Occurrence in Probability Theory X j iid, uniformly distributed on [0 , 1] .   n 1 � n � � = P X j ∈ [ k, k + 1]   n ! k j =1 Armin Straub Eulerian Numbers

Introduction Asymptotics Examples Generating Functions More Properties Asymptotics k � n � � n + 1 � � ( − 1) j ( k + 1 − j ) n = k j j =0 Armin Straub Eulerian Numbers

Introduction Asymptotics Examples Generating Functions More Properties Asymptotics k � n � � n + 1 � � ( − 1) j ( k + 1 − j ) n = k j j =0 � n � 2 n − n − 1 = 1 � n � � n + 1 � 3 n − ( n + 1)2 n + = 2 2 Armin Straub Eulerian Numbers

Introduction Asymptotics Examples Generating Functions More Properties Asymptotics k � n � � n + 1 � � ( − 1) j ( k + 1 − j ) n = k j j =0 � n � 2 n − n − 1 = 1 � n � � n + 1 � 3 n − ( n + 1)2 n + = 2 2 . . . � n � ( k + 1) n as n → ∞ ∼ k Armin Straub Eulerian Numbers

Introduction Asymptotics Examples Generating Functions More Properties Generating Functions � n � ◮ Let A n,k = . k +1 x n y k 1 − y � 1 + A n,k = n ! 1 − ye (1 − y ) x k,n ≥ 1 Armin Straub Eulerian Numbers

Introduction Asymptotics Examples Generating Functions More Properties Generating Functions � n � ◮ Let A n,k = . k +1 x n y k 1 − y � 1 + A n,k = n ! 1 − ye (1 − y ) x k,n ≥ 1 � r + s +1 ◮ Let A [ r,s ] = � . r e x − e y x r y s � A [ r,s ] ( r + s + 1)! = xe y − ye x r,s ≥ 0 Armin Straub Eulerian Numbers