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Finding Canonically Represented Theorems

Finding Canonically Represented Theorems. Sara Billey University of Washington Reproducibility in Computational and Experimental Math ICERM December 13, 2012. The Standard Inquiry. “Do you know anything about the following math problem?”

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Finding Canonically Represented Theorems

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  1. Finding Canonically Represented Theorems Sara Billey University of Washington Reproducibility in Computational and Experimental Math ICERM December 13, 2012

  2. The Standard Inquiry “Do you know anything about the following math problem?” How many permutations of {1,2,…,n} have exactly two peaks in positions 2 and n-1? For n=5 there are 16 such permutations including 12 with peaks 4,5 such as [1 4 2 5 3] 4 with peaks 3,5 such as [1 3 2 5 4]

  3. The Standard Reply “Probably not!” But in this case I know where to find out if it is known. Let’s check OEIS! ” Calculate a few terms: n=4 0 n=5 16 n=6 80 n=7 288

  4. Experimentation to Theorem Theorem [Billey-Burdzy-Sagan 2012] The number of permutations in Sn with peak set S is given by 2(n-|S|) times a fixed polynomial depending on S provided n is larger than the maximum in S.

  5. Theorem Finders The OEIS is a “Theorem Finder”. The theorems in the OEIS all have a canonical form represented by terms in an integer sequence that allow lookup. What other types of theorems have canonical forms? • Permutation Patterns (Tenner’s Database) • Graphs • Hypergeometric Series • Finite Simple Groups

  6. Theorem Finders Goal: Each canonical theorem class should have a Theorem Finder database doing Community+Machine Learning. Why should we do this? • Enhances experimental mathematics. • Makes unexpected connections between fields. • Improves refereeing of articles. See my Theorem Finder Homepage for more info.

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