• DOI:10.7282/T3BK1FKF
• Corpus ID: 125635724

Experimental methods in permutation patterns and bijective proof

@inproceedings{Shar2016ExperimentalMI,
  title={Experimental methods in permutation patterns and bijective proof},
  author={Nathaniel Shar},
  year={2016}
}

• Nathaniel Shar
• Published 2016
• Mathematics

OF THE DISSERTATION Experimental Methods in Permutation Patterns and Bijective Proof by Nathaniel Shar Dissertation Director: Doron Zeilberger Experimental mathematics is the technique of developing conjectures and proving theorems through the use of experimentation; that is, exploring finitely many cases and detecting patterns that can then be rigorously proved. This thesis applies the techniques of experimental mathematics to several problems. First, we generalize the translation method of Wood and Zeilberger [49] to algebraic proofs, and as an example, produce (by computer) the first bijective proof of Franel’s recurrence for a (3) n = ∑n k=0 ( n k )3 . Next, we apply the method of enumeration schemes to several problems in the field of patterns on permutations and words. Given a word w on the alphabet [n] and σ ∈ Sk, we say that w contains the pattern σ if some subsequence of the letters of w is orderisomorphic to σ. First, we find an enumeration scheme that allows us to count the words containing r copies of each letter that avoid the pattern 123. Then we look at the case where w is in fact a permutation in Sn. A repeating permutation is one that is the direct sum of several copies of a smaller permutation. We produce an enumeration scheme to count permutations avoiding repeating patterns of low codimension, and show that for each repeating pattern, the problem belongs to the eventually polynomial ansatz.