The Complementary Bell Numbers Explored via a Matrix Constructed - PowerPoint PPT Presentation

Introduction Construction of the R -Matrix Results Conclusion The Complementary Bell Numbers Explored via a Matrix Constructed with Rising Factorials Jonathan Broom, Stefan Hannie, Sarah Seger Ole Miss,ULL,LSU July 6, 2012 Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Construction of the R -Matrix Results Conclusion Introduction 1 Factorials Stirling Numbers Bell Numbers Construction of the R -Matrix 2 λ j ( x ) Basis Coefficients Matrices Results 3 Infinite Matrices Finite Matrices Conclusion 4 Conclusion Acknowledgements Works Cited Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Factorials The falling factorial is denoted ( x ) r Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Factorials The falling factorial is denoted ( x ) r ( x ) r = x ( x − 1)( x − 2) · · · ( x − r + 1) Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Factorials The falling factorial is denoted ( x ) r ( x ) r = x ( x − 1)( x − 2) · · · ( x − r + 1) The rising factorial is denoted x ( r ) Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Factorials The falling factorial is denoted ( x ) r ( x ) r = x ( x − 1)( x − 2) · · · ( x − r + 1) The rising factorial is denoted x ( r ) x ( r ) = x ( x + 1)( x + 2) · · · ( x + r − 1) Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Factorials The falling factorial is denoted ( x ) r ( x ) r = x ( x − 1)( x − 2) · · · ( x − r + 1) The rising factorial is denoted x ( r ) x ( r ) = x ( x + 1)( x + 2) · · · ( x + r − 1) Rising factorial example: Let x = 7 and r = 4 Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Factorials The falling factorial is denoted ( x ) r ( x ) r = x ( x − 1)( x − 2) · · · ( x − r + 1) The rising factorial is denoted x ( r ) x ( r ) = x ( x + 1)( x + 2) · · · ( x + r − 1) Rising factorial example: Let x = 7 and r = 4 7 (4) = 7(8)(9)(10) = 5040 Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Factorials The falling factorial is denoted ( x ) r ( x ) r = x ( x − 1)( x − 2) · · · ( x − r + 1) The rising factorial is denoted x ( r ) x ( r ) = x ( x + 1)( x + 2) · · · ( x + r − 1) Rising factorial example: Let x = 7 and r = 4 7 (4) = 7(8)(9)(10) = 5040 Note that both ( x ) r and x ( r ) are polynomials of degree r . Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Stirling Numbers of the Second Kind The Stirling Numbers of the Second Kind are denoted S ( n , k ). Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Stirling Numbers of the Second Kind The Stirling Numbers of the Second Kind are denoted S ( n , k ). They are the number of ways you can partition n elements into k non-empty blocks. Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Stirling Numbers of the Second Kind The Stirling Numbers of the Second Kind are denoted S ( n , k ). They are the number of ways you can partition n elements into k non-empty blocks. For example, take a set containing 3 items { a , b , c } S (3 , 1) = 1 Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Stirling Numbers of the Second Kind The Stirling Numbers of the Second Kind are denoted S ( n , k ). They are the number of ways you can partition n elements into k non-empty blocks. For example, take a set containing 3 items { a , b , c } S (3 , 1) = 1 {{ a , b , c }} Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Stirling Numbers of the Second Kind The Stirling Numbers of the Second Kind are denoted S ( n , k ). They are the number of ways you can partition n elements into k non-empty blocks. For example, take a set containing 3 items { a , b , c } S (3 , 2) = 3 S (3 , 1) = 1 {{ a , b , c }} Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Stirling Numbers of the Second Kind The Stirling Numbers of the Second Kind are denoted S ( n , k ). They are the number of ways you can partition n elements into k non-empty blocks. For example, take a set containing 3 items { a , b , c } S (3 , 2) = 3 S (3 , 1) = 1 {{ a } , { b , c }} {{ a , b , c }} Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Stirling Numbers of the Second Kind The Stirling Numbers of the Second Kind are denoted S ( n , k ). They are the number of ways you can partition n elements into k non-empty blocks. For example, take a set containing 3 items { a , b , c } S (3 , 2) = 3 S (3 , 1) = 1 {{ a } , { b , c }} {{ a , b , c }} {{ b } , { a , c }} Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Stirling Numbers of the Second Kind The Stirling Numbers of the Second Kind are denoted S ( n , k ). They are the number of ways you can partition n elements into k non-empty blocks. For example, take a set containing 3 items { a , b , c } S (3 , 2) = 3 S (3 , 1) = 1 {{ a } , { b , c }} {{ a , b , c }} {{ b } , { a , c }} {{ c } , { a , b }} Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Stirling Numbers of the Second Kind The Stirling Numbers of the Second Kind are denoted S ( n , k ). They are the number of ways you can partition n elements into k non-empty blocks. For example, take a set containing 3 items { a , b , c } S (3 , 2) = 3 S (3 , 1) = 1 S (3 , 3) = 1 {{ a } , { b , c }} {{ a , b , c }} {{ b } , { a , c }} {{ c } , { a , b }} Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Stirling Numbers of the Second Kind The Stirling Numbers of the Second Kind are denoted S ( n , k ). They are the number of ways you can partition n elements into k non-empty blocks. For example, take a set containing 3 items { a , b , c } S (3 , 2) = 3 S (3 , 1) = 1 S (3 , 3) = 1 {{ a } , { b , c }} {{ a , b , c }} {{ a } , { b } , { c }} {{ b } , { a , c }} {{ c } , { a , b }} Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Stirling Numbers of the Second Kind The Stirling Numbers of the Second Kind are denoted S ( n , k ). They are the number of ways you can partition n elements into k non-empty blocks. For example, take a set containing 3 items { a , b , c } S (3 , 2) = 3 S (3 , 1) = 1 S (3 , 3) = 1 {{ a } , { b , c }} {{ a , b , c }} {{ a } , { b } , { c }} {{ b } , { a , c }} {{ c } , { a , b }} Another example for S (3 , k ): Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

Introduction Factorials Construction of the R -Matrix Stirling Numbers Results Bell Numbers Conclusion Stirling Numbers of the Second Kind The Stirling Numbers of the Second Kind are denoted S ( n , k ). They are the number of ways you can partition n elements into k non-empty blocks. For example, take a set containing 3 items { a , b , c } S (3 , 2) = 3 S (3 , 1) = 1 S (3 , 3) = 1 {{ a } , { b , c }} {{ a , b , c }} {{ a } , { b } , { c }} {{ b } , { a , c }} {{ c } , { a , b }} Another example for S (3 , k ): Figure: S (3 , 1) = 1 Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers