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Combinatorial proofs of linear algebraic identities

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Author(s): Melanie Dennis
Series: PhD thesis at Dartmouth College
Year: 2019

Language: English

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction 1
1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Lewis Carroll Identity . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Dodgson/Muir Identity . . . . . . . . . . . . . . . . . . . . . . 8
1.3 The Matrix Tree Theorem . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Involution Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Lewis Carroll and the Red Hot Potato 16
2.1 Connecting Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 The Red Hot Potato algorithm . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Proof of φ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Identity Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Connection to Zeilberger 41
3.1 Zeilberger’s Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Alternate Definition of φ 1 . . . . . . . . . . . . . . . . . . . . . . . . 48
4 Dodgson/Muir Identity 59
4.1 Connecting Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Generalized Red Hot Potato algorithm . . . . . . . . . . . . . . . . . 69
4.2.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.2 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2.3 Generalized Red Hot Potato algorithm . . . . . . . . . . . . . 74
4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 Proof that φ 1 is a sign-reversing involution . . . . . . . . . . . . . . . 78
4.5 Proofs of Forest and Lewis Carroll Identities . . . . . . . . . . . . . . 84
5 Future Work 87
5.1 Jacobi’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Other Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
References 91