Eric Rowland

Automatic congruences for diagonals of rational functions

In this paper we use the framework of automatic sequences to study combinatorial sequences modulo prime powers. Given a sequence whose generating function is the diagonal of a rational power series, we provide a method, based on work of Denef and Lipshitz, for computing a nite automaton for the sequence modulo p , for all but nitely many primes p. This method gives completely automatic proofs of known results, establishes a number of new theorems for well-known sequences, and allows us to resolve some conjectures regarding the Ap ery numbers. We also give a second method, which applies to all algebraic sequences, but is signicantly slower. Finally, we show that a broad range of multidimensional sequences possess Lucas products modulo p.

A case study in meta-automation: automatic generation of congruence automata for combinatorial sequences

In this paper, which may be considered a sequel to a recent article by Eric Rowland and Reem Yassawi, we present yet another approach for the automatic generation of automata (and an extension that we call congruence linear schemes) for the fast (log-time) determination of congruence properties, modulo small (and not so small!) prime powers, for a wide class of combinatorial sequences. Even more interesting than the new results that could be obtained is the illustrated methodology, that of designing ‘meta-algorithms’ that enable the computer to develop algorithms, that it (or another computer) can then proceed to use to actually prove (potentially!) infinitely many new results. This paper is accompanied by a Maple package, AutoSquared, and numerous sample input and output files, that readers can use as templates for generating their own, thereby proving many new ‘theorems’ about congruence properties of many famous (and, of course, obscure) combinatorial sequences.

Toward a language theoretic proof of the four color theorem

This paper considers the problem of showing that every pair of binary trees with the same number of leaves parses a common word under a certain simple grammar. We enumerate the common parse words for several infinite families of tree pairs and discuss several ways to reduce the problem of finding a parse word for a pair of trees to that for a smaller pair. The statement that every pair of trees has a common parse word is equivalent to the statement that every planar graph is four-colorable, so the results are a step toward a language theoretic proof of the four color theorem.

Structure and Enumeration of (3+1)-Free Posets

A poset is (3+1)-free if it does not contain the disjoint union of chains of lengths 3 and 1 as an induced subposet. These posets play a central role in the (3+1)-free conjecture of Stanley and Stembridge. Lewis and Zhang have enumerated (3+1)-free posets in the graded case by decomposing them into bipartite graphs, but until now the general enumeration problem has remained open. We give a finer decomposition into bipartite graphs which applies to all (3+1)-free posets and obtain generating functions which count (3+1)-free posets with labelled or unlabelled vertices. Using this decomposition, we obtain a decomposition of the automorphism group and asymptotics for the number of (3+1)-free posets.

k -Automatic sets of rational numbers

The notion of a k-automatic set of integers is well-studied. We develop a new notion -- the k-automatic set of rational numbers -- and prove basic properties of these sets, including closure properties and decidability.

A characterization of p-automatic sequences as columns of linear cellular automata

We show that a sequence over a finite field F q of characteristic p is p-automatic if and only if it occurs as a column of the spacetime diagram, with eventually periodic initial conditions, of a linear cellular automaton with memory over F q . As a consequence, the subshift generated by a length-p substitution can be realized as a topological factor of a linear cellular automaton.

ITERATED PRIMITIVES OF LOGARITHMIC POWERS

The evaluation of iterated primitives of powers of logarithms is expressed in closed form. The expressions contain polynomials with coefficients given in terms of the harmonic numbers and their generalizations. The logconcavity of these polynomials is established.

WHAT'S IN YOUR WALLET?!

We use Markov chains and numerical linear algebra — and several CPU hours — to determine the most likely set of coins in your wallet under reasonable spending assumptions. We also compute a number of additional statistics. In particular, the expected number of coins carried by a person in the United States in our model is 10.

THE ITERATED INTEGRALS OF ln(1 + xn)

For a polynomial P, we consider the sequence of iterated integrals of ln P(x). This sequence is expressed in terms of the zeros of P(x). In the special case of ln(1 + x2), arithmetic properties of certain coefficients arising are described. Similar observations are made for ln(1 + x3).

Avoiding 3/2-powers over the natural numbers

In this paper, we answer the following question: what is the lexicographically least sequence over the natural numbers that avoids 32-powers?