Generating Permutations with Restricted Containers

We investigate a generalization of stacks that we call $\mathcal{C}$-machines. We show how this viewpoint rapidly leads to functional equations for the classes of permutations that $\mathcal{C}$-machines generate, and how these systems of functional equations can frequently be solved by either the kernel method or, much more easily, by guessing and checking. General results about the rationality, algebraicity, and the existence of Wilfian formulas for some classes generated by $\mathcal{C}$-machines are given. We also draw attention to some relatively small permutation classes which, although we can generate thousands of terms of their enumerations, seem to not have D-finite generating functions

Experimental methods in permutation patterns and bijective proof

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 an(3)=Σnk=0(nk)3. Next, we apply the method of enumeration schemes to several problems in the fieldof 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.Ph.D.Includes bibliographical referencesby Nathaniel Sha