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

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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,

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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

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Introduction Construction of the R-Matrix Results Conclusion

1

Introduction Factorials Stirling Numbers Bell Numbers

2

Construction of the R-Matrix λj(x) Basis Coefficients Matrices

3

Results Infinite Matrices Finite Matrices

4

Conclusion Conclusion Acknowledgements Works Cited

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

Factorials

The falling factorial is denoted (x)r

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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}} S(3, 2) = 3

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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}} S(3, 2) = 3 {{a}, {b, c}}

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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}} S(3, 2) = 3 {{a}, {b, c}} {{b}, {a, c}}

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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}} S(3, 2) = 3 {{a}, {b, c}} {{b}, {a, c}} {{c}, {a, b}}

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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}} S(3, 2) = 3 {{a}, {b, c}} {{b}, {a, c}} {{c}, {a, b}} S(3, 3) = 1

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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}} S(3, 2) = 3 {{a}, {b, c}} {{b}, {a, c}} {{c}, {a, b}} S(3, 3) = 1 {{a}, {b}, {c}}

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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}} S(3, 2) = 3 {{a}, {b, c}} {{b}, {a, c}} {{c}, {a, b}} S(3, 3) = 1 {{a}, {b}, {c}}

Another example for S(3, k):

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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}} S(3, 2) = 3 {{a}, {b, c}} {{b}, {a, c}} {{c}, {a, b}} S(3, 3) = 1 {{a}, {b}, {c}}

Another example for S(3, k):

Figure: S(3, 1) = 1

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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}} S(3, 2) = 3 {{a}, {b, c}} {{b}, {a, c}} {{c}, {a, b}} S(3, 3) = 1 {{a}, {b}, {c}}

Another example for S(3, k):

Figure: S(3, 1) = 1 Figure: S(3, 2) = 3

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

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}} S(3, 2) = 3 {{a}, {b, c}} {{b}, {a, c}} {{c}, {a, b}} S(3, 3) = 1 {{a}, {b}, {c}}

Another example for S(3, k):

Figure: S(3, 1) = 1 Figure: S(3, 2) = 3 Figure: S(3, 3) = 1

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

Stirling Numbers of the Second Kind

Similarly S(4, k):

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

Stirling Numbers of the Second Kind

Similarly S(4, k):

Figure: S(4, 1) = 1

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

Stirling Numbers of the Second Kind

Similarly S(4, k):

Figure: S(4, 1) = 1 Figure: S(4, 2) = 7

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

Stirling Numbers of the Second Kind

Similarly S(4, k):

Figure: S(4, 1) = 1 Figure: S(4, 2) = 7 Figure: S(4, 3) = 6

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

Stirling Numbers of the Second Kind

Similarly S(4, k):

Figure: S(4, 1) = 1 Figure: S(4, 2) = 7 Figure: S(4, 3) = 6 Figure: S(4, 4) = 1

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

Stirling Numbers of the Second Kind

Similarly S(4, k):

Figure: S(4, 1) = 1 Figure: S(4, 2) = 7 Figure: S(4, 3) = 6 Figure: S(4, 4) = 1

Note: From the examples, it is clear that S(n, 1) = S(n, n) = 1.

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

Growth

S7, 4 350 S6, 3 90 S8, 4 1701

2 4 6 8 k 500 1000 1500 n

k

The points labeled are the k values that yield the maximum S(n, k) for a given n.

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

Bell Numbers and Complementary Bell Numbers

The Bell Numbers are denoted B(n)

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

Bell Numbers and Complementary Bell Numbers

The Bell Numbers are denoted B(n) B(n) =

n

  • k=1

S(n, k)

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

Bell Numbers and Complementary Bell Numbers

The Bell Numbers are denoted B(n) B(n) =

n

  • k=1

S(n, k) The Complementary Bell Numbers are denoted B(n)

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

Bell Numbers and Complementary Bell Numbers

The Bell Numbers are denoted B(n) B(n) =

n

  • k=1

S(n, k) The Complementary Bell Numbers are denoted B(n)

  • B(n) =

n

  • k=1

(−1)kS(n, k)

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

  • B(n) Examples

Examples:

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

  • B(n) Examples

Examples:

  • B(2) = 0

Odd Even {a, b} {a}, {b}

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

  • B(n) Examples

Examples:

  • B(2) = 0

Odd Even {a, b} {a}, {b}

  • B(3) = 1

Odd Even

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

  • B(n) Examples

Examples:

  • B(2) = 0

Odd Even {a, b} {a}, {b}

  • B(4) = 1

Odd Even

  • B(3) = 1

Odd Even

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

  • B(n) Examples

Examples:

  • B(2) = 0

Odd Even {a, b} {a}, {b}

  • B(4) = 1

Odd Even

  • B(3) = 1

Odd Even

  • B(5) = −2

Odd Even

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

Complementary Bell Numbers

n

  • B(n)

1 1 −1 2 3 1 4 1 5 −2 6 −9 7 −9 8 50 9 267 10 413 11 −2180 12 −17731 13 −50533 14 110176 15 1966797 16 9938669 17 8638718 . . . . . .

  • 2

4 6 8 n 5 10 15 20 Bn1

Figure: | B(n)| for n ≤ 8

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion Factorials Stirling Numbers Bell Numbers

Wilf’s Conjecture

H.S. Wilf’s Conjecture:

  • B(n) = 0 for all n > 2

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The λj(x) Polynomials

There exist polynomials λj, for all n, j ≥ 0, that satisfy

  • B(n + j) =

n

  • k=0

(−1)kλj(k)S(n, k)

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The λj(x) Polynomials

There exist polynomials λj, for all n, j ≥ 0, that satisfy

  • B(n + j) =

n

  • k=0

(−1)kλj(k)S(n, k) λj(x) can be defined recursively as follows: λ0(x) = 1 λn+1(x) = xλn(x) − λn (x + 1)

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The λj(x) Polynomials

There exist polynomials λj, for all n, j ≥ 0, that satisfy

  • B(n + j) =

n

  • k=0

(−1)kλj(k)S(n, k) λj(x) can be defined recursively as follows: λ0(x) = 1 λn+1(x) = xλn(x) − λn (x + 1) Alternate Form: λ0(x) = 1 λn+1(x − 1) = (x − 1)λn(x − 1) − λn(x)

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The λj(x) Polynomials

There exist polynomials λj, for all n, j ≥ 0, that satisfy

  • B(n + j) =

n

  • k=0

(−1)kλj(k)S(n, k) λj(x) can be defined recursively as follows: λ0(x) = 1 λn+1(x) = xλn(x) − λn (x + 1) Alternate Form: λ0(x) = 1 λn+1(x − 1) = (x − 1)λn(x − 1) − λn(x)

Note that λn(x) is a monic polynomial of degree n.

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

Rising Factorials as a Basis for Pn

Theorem For each n ≥ 0, the set of rising factorials

  • x(k) : 0 ≤ k ≤ n
  • is a

basis for Pn, the vector space of polynomials of degree less than or equal to n.

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

Rising Factorials as a Basis for Pn

Theorem For each n ≥ 0, the set of rising factorials

  • x(k) : 0 ≤ k ≤ n
  • is a

basis for Pn, the vector space of polynomials of degree less than or equal to n. xn =

n

  • k=0

(−1)n+kS(n, k)x(k) for all n ≥ 0

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The Coefficients of the R-Matrix

By the previous theorem: λn(x) =

n

  • k=0

an(k)x(k)

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The Coefficients of the R-Matrix

By the previous theorem: λn(x) =

n

  • k=0

an(k)x(k) By the recurrence relation of λn(x): λn+1(x − 1) = (x − 1)λn(x − 1) − λn(x)

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The Coefficients of the R-Matrix

By the previous theorem: λn(x) =

n

  • k=0

an(k)x(k) By the recurrence relation of λn(x): λn+1(x − 1) = (x − 1)λn(x − 1) − λn(x) Therefore:

n+1

  • k=0

an+1(k) (x − 1)(k) =

n

  • k=0

an(k) (x − 1) (x − 1)(k) −

n

  • k=0

an(k)x(k)

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The Coefficients of the R-Matrix

By the previous theorem: λn(x) =

n

  • k=0

an(k)x(k) By the recurrence relation of λn(x): λn+1(x − 1) = (x − 1)λn(x − 1) − λn(x) Therefore:

n+1

  • k=0

an+1(k) (x − 1)(k) =

n

  • k=0

an(k) (x − 1) (x − 1)(k) −

n

  • k=0

an(k)x(k) Lemma For all n ≥ 0 and for all 0 ≤ k ≤ n + 1, an+1(k) − (k + 1) an+1(k + 1) = an(k − 1) − 2 (k + 1) an(k) + (k + 1)2 an(k + 1)

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The A-Matrix and B-Matrix

Let Aan+1 = Ban, then A, B are infinite matrices whose entries are defined by

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The A-Matrix and B-Matrix

Let Aan+1 = Ban, then A, B are infinite matrices whose entries are defined by

A(i, j) =      1 if j = i − (i + 1) if j = i + 1 if j < i or j > i + 1

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The A-Matrix and B-Matrix

Let Aan+1 = Ban, then A, B are infinite matrices whose entries are defined by

A(i, j) =      1 if j = i − (i + 1) if j = i + 1 if j < i or j > i + 1 A =        1 −1 . . . 1 −2 . . . 1 −3 . . . 1 . . . . . . . . . . . . . . . ...       

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The A-Matrix and B-Matrix

Let Aan+1 = Ban, then A, B are infinite matrices whose entries are defined by

A(i, j) =      1 if j = i − (i + 1) if j = i + 1 if j < i or j > i + 1 B(i, j) =          1 if j = i − 1 −2 (i + 1) if j = i (i + 1)2 if j = i + 1 if |i − j| > 1 A =        1 −1 . . . 1 −2 . . . 1 −3 . . . 1 . . . . . . . . . . . . . . . ...       

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The A-Matrix and B-Matrix

Let Aan+1 = Ban, then A, B are infinite matrices whose entries are defined by

A(i, j) =      1 if j = i − (i + 1) if j = i + 1 if j < i or j > i + 1 B(i, j) =          1 if j = i − 1 −2 (i + 1) if j = i (i + 1)2 if j = i + 1 if |i − j| > 1 A =        1 −1 . . . 1 −2 . . . 1 −3 . . . 1 . . . . . . . . . . . . . . . ...        B =        −2 1 . . . 1 −4 4 . . . 1 −6 9 . . . 1 −8 . . . . . . . . . . . . . . . ...       

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The A−1-Matrix

Taking Aan+1 = Ban, we solve for an+1. Therefore: an+1 = A−1Ban

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The A−1-Matrix

Taking Aan+1 = Ban, we solve for an+1. Therefore: an+1 = A−1Ban A−1(i, j) = j!

i!

if j ≥ i if j < i

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The A−1-Matrix

Taking Aan+1 = Ban, we solve for an+1. Therefore: an+1 = A−1Ban A−1(i, j) = j!

i!

if j ≥ i if j < i A−1 =          1 1 2 6 24 . . . 1 2 6 24 . . . 1 3 12 . . . 1 4 . . . 1 . . . . . . . . . . . . . . . . . . ...         

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-60
SLIDE 60

Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The R-Matrix

Taking an+1 = A−1Ban, we call R = A−1B. Therefore: an+1 = Ran

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-61
SLIDE 61

Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The R-Matrix

Taking an+1 = A−1Ban, we call R = A−1B. Therefore: an+1 = Ran

R(i, j) =          − j!

i!

if j > i −(i + 1) if j = i 1 if j = i − 1 if j < i − 1

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-62
SLIDE 62

Introduction Construction of the R-Matrix Results Conclusion λj (x) Basis Coefficients Matrices

The R-Matrix

Taking an+1 = A−1Ban, we call R = A−1B. Therefore: an+1 = Ran

R(i, j) =          − j!

i!

if j > i −(i + 1) if j = i 1 if j = i − 1 if j < i − 1 R =          −1 −1 −2 −6 −24 . . . 1 −2 −2 −6 −24 . . . 1 −3 −3 −12 . . . 1 −4 −4 . . . 1 −5 . . . . . . . . . . . . . . . . . . ...         

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-63
SLIDE 63

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Lower Section

Lemma For each n ∈ N, the nth power of R is defined and Rn(i, j) = 0 if j < i − n.

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-64
SLIDE 64

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Lower Section

Lemma For each n ∈ N, the nth power of R is defined and Rn(i, j) = 0 if j < i − n.

R =          −1 −1 −2 −6 −24 . . . 1 −2 −2 −6 −24 . . . 1 −3 −3 −12 . . . 1 −4 −4 . . . 1 −5 . . . . . . . . . . . . . . . . . . ...          Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-65
SLIDE 65

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Lower Section

Lemma For each n ∈ N, the nth power of R is defined and Rn(i, j) = 0 if j < i − n.

R =          −1 −1 −2 −6 −24 . . . 1 −2 −2 −6 −24 . . . 1 −3 −3 −12 . . . 1 −4 −4 . . . 1 −5 . . . . . . . . . . . . . . . . . . ...          R2 =          1 4 18 96 . . . −3 1 2 12 72 . . . 1 −5 4 3 24 . . . 1 −7 9 4 . . . 1 −9 16 . . . . . . . . . . . . . . . . . . ...          Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-66
SLIDE 66

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Lower Section

Lemma For each n ∈ N, the nth power of R is defined and Rn(i, j) = 0 if j < i − n.

R =          −1 −1 −2 −6 −24 . . . 1 −2 −2 −6 −24 . . . 1 −3 −3 −12 . . . 1 −4 −4 . . . 1 −5 . . . . . . . . . . . . . . . . . . ...          R2 =          1 4 18 96 . . . −3 1 2 12 72 . . . 1 −5 4 3 24 . . . 1 −7 9 4 . . . 1 −9 16 . . . . . . . . . . . . . . . . . . ...          R3 =          1 2 4 6 −24 . . . 4 3 10 30 96 . . . −6 13 −1 24 96 . . . 1 −9 28 −17 44 . . . 1 −12 49 −51 . . . . . . . . . . . . . . . . . . ...          Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-67
SLIDE 67

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Lower Section

Lemma For each n ∈ N, the nth power of R is defined and Rn(i, j) = 0 if j < i − n.

R =          −1 −1 −2 −6 −24 . . . 1 −2 −2 −6 −24 . . . 1 −3 −3 −12 . . . 1 −4 −4 . . . 1 −5 . . . . . . . . . . . . . . . . . . ...          R2 =          1 4 18 96 . . . −3 1 2 12 72 . . . 1 −5 4 3 24 . . . 1 −7 9 4 . . . 1 −9 16 . . . . . . . . . . . . . . . . . . ...          R3 =          1 2 4 6 −24 . . . 4 3 10 30 96 . . . −6 13 −1 24 96 . . . 1 −9 28 −17 44 . . . 1 −12 49 −51 . . . . . . . . . . . . . . . . . . ...          R4 =          1 −1 −12 −78 −504 . . . −1 −14 −96 −648 . . . 19 −21 13 −39 −312 . . . −10 45 −85 76 −76 . . . 1 −14 83 −217 249 . . . . . . . . . . . . . . . . . . ...          Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-68
SLIDE 68

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Row

Lemma For all n ≥ 1 and j ≥ 0, the (0, j)th entry of Rn is divisible by j!.

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-69
SLIDE 69

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Row

Lemma For all n ≥ 1 and j ≥ 0, the (0, j)th entry of Rn is divisible by j!.

j j! 1 1 1 2 2 3 6 4 24 5 120 6 720 . . . . . .

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-70
SLIDE 70

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Row

Lemma For all n ≥ 1 and j ≥ 0, the (0, j)th entry of Rn is divisible by j!.

j j! 1 1 1 2 2 3 6 4 24 5 120 6 720 . . . . . .

R =         −1 −1 −2 −6 −24 −120 −720 . . . 1 −2 −2 −6 −24 −120 −720 . . . 1 −3 −3 −12 −60 −360 . . . 1 −4 −4 −20 −120 . . . . . . . . . . . . . . . . . . . . . . . . ...         Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-71
SLIDE 71

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Row

Lemma For all n ≥ 1 and j ≥ 0, the (0, j)th entry of Rn is divisible by j!.

j j! 1 1 1 2 2 3 6 4 24 5 120 6 720 . . . . . .

R =         −1 −1 −2 −6 −24 −120 −720 . . . 1 −2 −2 −6 −24 −120 −720 . . . 1 −3 −3 −12 −60 −360 . . . 1 −4 −4 −20 −120 . . . . . . . . . . . . . . . . . . . . . . . . ...         R4 =         1 −1 −12 −78 −504 36840 −953280 . . . −1 −14 −96 −648 35640 −923760 . . . 19 −21 13 −39 −312 17700 −443880 . . . −10 45 −85 76 −76 6000 −141000 . . . . . . . . . . . . . . . . . . . . . . . . ...         Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-72
SLIDE 72

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Row

Lemma For all n ≥ 1 and j ≥ 0, the (0, j)th entry of Rn is divisible by j!.

j j! 1 1 1 2 2 3 6 4 24 5 120 6 720 . . . . . .

R =         −1 −1 −2 −6 −24 −120 −720 . . . 1 −2 −2 −6 −24 −120 −720 . . . 1 −3 −3 −12 −60 −360 . . . 1 −4 −4 −20 −120 . . . . . . . . . . . . . . . . . . . . . . . . ...         R4 =         1 −1 −12 −78 −504 36840 −953280 . . . −1 −14 −96 −648 35640 −923760 . . . 19 −21 13 −39 −312 17700 −443880 . . . −10 45 −85 76 −76 6000 −141000 . . . . . . . . . . . . . . . . . . . . . . . . ...        

For R4:

−78 3!

= −13, −504

4!

= −21, 36840

5!

= 307, −953280

6!

= −1324

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-73
SLIDE 73

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Left Entry

Theorem For all n ∈ N, B(n) = Rn(0, 0).

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-74
SLIDE 74

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Left Entry

Theorem For all n ∈ N, B(n) = Rn(0, 0).

n

  • B(n)

1 −1 2 3 1 4 1 5 −2 6 −9 . . . . . .

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-75
SLIDE 75

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Left Entry

Theorem For all n ∈ N, B(n) = Rn(0, 0).

n

  • B(n)

1 −1 2 3 1 4 1 5 −2 6 −9 . . . . . .

R =         −1 −1 −2 −6 . . . 1 −2 −2 −6 . . . 1 −3 −3 . . . 1 −4 . . . . . . . . . . . . . . . ...         Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-76
SLIDE 76

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Left Entry

Theorem For all n ∈ N, B(n) = Rn(0, 0).

n

  • B(n)

1 −1 2 3 1 4 1 5 −2 6 −9 . . . . . .

R =         −1 −1 −2 −6 . . . 1 −2 −2 −6 . . . 1 −3 −3 . . . 1 −4 . . . . . . . . . . . . . . . ...         R2 =         1 4 18 . . . −3 1 2 12 . . . 1 −5 4 3 . . . 1 −7 9 . . . . . . . . . . . . . . . ...         Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-77
SLIDE 77

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Left Entry

Theorem For all n ∈ N, B(n) = Rn(0, 0).

n

  • B(n)

1 −1 2 3 1 4 1 5 −2 6 −9 . . . . . .

R =         −1 −1 −2 −6 . . . 1 −2 −2 −6 . . . 1 −3 −3 . . . 1 −4 . . . . . . . . . . . . . . . ...         R5 =         −2 −11 −42 −156 . . . 1 −13 −52 −216 . . . −40 36 −74 −183 . . . 55 −165 261 −335 . . . . . . . . . . . . . . . ...         R2 =         1 4 18 . . . −3 1 2 12 . . . 1 −5 4 3 . . . 1 −7 9 . . . . . . . . . . . . . . . ...         Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-78
SLIDE 78

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Left Entry

Theorem For all n ∈ N, B(n) = Rn(0, 0).

n

  • B(n)

1 −1 2 3 1 4 1 5 −2 6 −9 . . . . . .

R =         −1 −1 −2 −6 . . . 1 −2 −2 −6 . . . 1 −3 −3 . . . 1 −4 . . . . . . . . . . . . . . . ...         R5 =         −2 −11 −42 −156 . . . 1 −13 −52 −216 . . . −40 36 −74 −183 . . . 55 −165 261 −335 . . . . . . . . . . . . . . . ...         R2 =         1 4 18 . . . −3 1 2 12 . . . 1 −5 4 3 . . . 1 −7 9 . . . . . . . . . . . . . . . ...         R6 =         −9 −18 −4 40644 . . . −14 −27 −36 40548 . . . 76 −106 47 20286 . . . −220 536 −898 7473 . . . . . . . . . . . . . . . ...         Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-79
SLIDE 79

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Row

Lemma For all m, n ≥ 1 and for each 0 ≤ j ≤ 2m − 1, Rn

m(0, j) ≡ Rn(0, j)

mod 22m−1.

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-80
SLIDE 80

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Row

Lemma For all m, n ≥ 1 and for each 0 ≤ j ≤ 2m − 1, Rn

m(0, j) ≡ Rn(0, j)

mod 22m−1.

R4 =         1 −1 −12 −78 . . . −1 −14 −96 . . . 19 −21 13 −39 . . . −10 45 −85 76 . . . . . . . . . . . . . . . ...         Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-81
SLIDE 81

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Row

Lemma For all m, n ≥ 1 and for each 0 ≤ j ≤ 2m − 1, Rn

m(0, j) ≡ Rn(0, j)

mod 22m−1.

R4 =         1 −1 −12 −78 . . . −1 −14 −96 . . . 19 −21 13 −39 . . . −10 45 −85 76 . . . . . . . . . . . . . . . ...         R4

1 =

  • 1

1 1

  • Jonathan Broom, Stefan Hannie, Sarah Seger

The Complementary Bell Numbers

slide-82
SLIDE 82

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Row

Lemma For all m, n ≥ 1 and for each 0 ≤ j ≤ 2m − 1, Rn

m(0, j) ≡ Rn(0, j)

mod 22m−1.

R4 =         1 −1 −12 −78 . . . −1 −14 −96 . . . 19 −21 13 −39 . . . −10 45 −85 76 . . . . . . . . . . . . . . . ...         R4

1 =

  • 1

1 1

  • R4

2 =

    1 7 4 2 7 2 3 7 1 5 6 1 3 4     Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-83
SLIDE 83

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Row

Lemma For all m, n ≥ 1 and for each 0 ≤ j ≤ 2m − 1, Rn

m(0, j) ≡ Rn(0, j)

mod 22m−1.

R4 =         1 −1 −12 −78 . . . −1 −14 −96 . . . 19 −21 13 −39 . . . −10 45 −85 76 . . . . . . . . . . . . . . . ...         R4

1 =

  • 1

1 1

  • R4

2 =

    1 7 4 2 7 2 3 7 1 5 6 1 3 4     R5 =         −2 −11 −42 −156 . . . 1 −13 −52 −216 . . . −40 36 −74 −183 . . . 55 −165 261 −335 . . . . . . . . . . . . . . . ...         Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-84
SLIDE 84

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Row

Lemma For all m, n ≥ 1 and for each 0 ≤ j ≤ 2m − 1, Rn

m(0, j) ≡ Rn(0, j)

mod 22m−1.

R4 =         1 −1 −12 −78 . . . −1 −14 −96 . . . 19 −21 13 −39 . . . −10 45 −85 76 . . . . . . . . . . . . . . . ...         R4

1 =

  • 1

1 1

  • R4

2 =

    1 7 4 2 7 2 3 7 1 5 6 1 3 4     R5 =         −2 −11 −42 −156 . . . 1 −13 −52 −216 . . . −40 36 −74 −183 . . . 55 −165 261 −335 . . . . . . . . . . . . . . . ...         R5

1 =

  • 1

1 1

  • Jonathan Broom, Stefan Hannie, Sarah Seger

The Complementary Bell Numbers

slide-85
SLIDE 85

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Row

Lemma For all m, n ≥ 1 and for each 0 ≤ j ≤ 2m − 1, Rn

m(0, j) ≡ Rn(0, j)

mod 22m−1.

R4 =         1 −1 −12 −78 . . . −1 −14 −96 . . . 19 −21 13 −39 . . . −10 45 −85 76 . . . . . . . . . . . . . . . ...         R4

1 =

  • 1

1 1

  • R4

2 =

    1 7 4 2 7 2 3 7 1 5 6 1 3 4     R5 =         −2 −11 −42 −156 . . . 1 −13 −52 −216 . . . −40 36 −74 −183 . . . 55 −165 261 −335 . . . . . . . . . . . . . . . ...         R5

1 =

  • 1

1 1

  • R5

2 =

    6 5 6 4 1 3 4 4 6 5 3 3 5 5     Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-86
SLIDE 86

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Left Entry

Theorem For all n, m ∈ N,

  • B(n) ≡ Rn

m(0, 0)(mod 22m−1)

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-87
SLIDE 87

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Left Entry

Theorem For all n, m ∈ N,

  • B(n) ≡ Rn

m(0, 0)(mod 22m−1) n

  • B(n)

1 1 −1 2 3 1 4 1 5 −2 6 −9 7 −9 8 50 9 267 . . . . . .

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-88
SLIDE 88

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Left Entry

Theorem For all n, m ∈ N,

  • B(n) ≡ Rn

m(0, 0)(mod 22m−1) n

  • B(n)

1 1 −1 2 3 1 4 1 5 −2 6 −9 7 −9 8 50 9 267 . . . . . .

R5

2 =

    22 −323 1422 −1884 25 −301 1124 −1008 −28 −96 382 −243 59 −205 373 −283     Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-89
SLIDE 89

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Left Entry

Theorem For all n, m ∈ N,

  • B(n) ≡ Rn

m(0, 0)(mod 22m−1) n

  • B(n)

1 1 −1 2 3 1 4 1 5 −2 6 −9 7 −9 8 50 9 267 . . . . . .

R5

2 =

    22 −323 1422 −1884 25 −301 1124 −1008 −28 −96 382 −243 59 −205 373 −283     −2 ≡ 22 ≡ 6 mod 8 Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-90
SLIDE 90

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Left Entry

Theorem For all n, m ∈ N,

  • B(n) ≡ Rn

m(0, 0)(mod 22m−1) n

  • B(n)

1 1 −1 2 3 1 4 1 5 −2 6 −9 7 −9 8 50 9 267 . . . . . .

R5

2 =

    22 −323 1422 −1884 25 −301 1124 −1008 −28 −96 382 −243 59 −205 373 −283     −2 ≡ 22 ≡ 6 mod 8 R9

2 =

    46203 −112360 161308 −139686 31762 −66157 80710 −76050 9756 −18293 24253 −36750 10181 −20787 33462 −30421     Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

slide-91
SLIDE 91

Introduction Construction of the R-Matrix Results Conclusion Infinite Matrices Finite Matrices

The Top Left Entry

Theorem For all n, m ∈ N,

  • B(n) ≡ Rn

m(0, 0)(mod 22m−1) n

  • B(n)

1 1 −1 2 3 1 4 1 5 −2 6 −9 7 −9 8 50 9 267 . . . . . .

R5

2 =

    22 −323 1422 −1884 25 −301 1124 −1008 −28 −96 382 −243 59 −205 373 −283     −2 ≡ 22 ≡ 6 mod 8 R9

2 =

    46203 −112360 161308 −139686 31762 −66157 80710 −76050 9756 −18293 24253 −36750 10181 −20787 33462 −30421     267 ≡ 46203 ≡ 3 mod 8 Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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SLIDE 92

Introduction Construction of the R-Matrix Results Conclusion Conclusion Acknowledgements Works Cited

Conclusion

In Conclusion:

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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SLIDE 93

Introduction Construction of the R-Matrix Results Conclusion Conclusion Acknowledgements Works Cited

Conclusion

In Conclusion: Additional Results

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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SLIDE 94

Introduction Construction of the R-Matrix Results Conclusion Conclusion Acknowledgements Works Cited

Conclusion

In Conclusion: Additional Results Alternate Bases

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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SLIDE 95

Introduction Construction of the R-Matrix Results Conclusion Conclusion Acknowledgements Works Cited

Acknowledgements

We would like to thank LSU for hosting the SMILE Program. Thank you NSF for funding the VIGRE program. Thank you to Dr. De Angelis for spending his summer with us. Thank you to Simon Pfeil for mentoring us.

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers

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SLIDE 96

Introduction Construction of the R-Matrix Results Conclusion Conclusion Acknowledgements Works Cited

Works Cited

  • T. Amdeberhan, V. De Angelis, and V.H. Moll.

Complementary Bell Numbers: Arithmetical Properties and Wilf’s Conjecture. 2011. http://www-history.mcs.st- and.ac.uk/Miscellaneous/StirlingBell/stirling2.html

Jonathan Broom, Stefan Hannie, Sarah Seger The Complementary Bell Numbers