Motzkin numbers and related sequences modulo powers of 2

We show that the generating function $\sum_{n\ge0}M_n\,z^n$ for Motzkin numbers $M_n$, when coefficients are reduced modulo a given power of $2$, can be expressed as a polynomial in the basic series $\sum _{e\ge0} ^{} {z^{4^e}}/( {1-z^{2\cdot 4^e}})$ with coefficients being Laurent polynomials in $z$ and $1-z$. We use this result to determine $M_n$ modulo $8$ in terms of the binary digits of~$n$, thus improving, respectively complementing earlier results by Eu, Liu and Yeh [Europ. J. Combin. 29 (2008), 1449-1466] and by Rowland and Yassawi [J. Th\'eorie Nombres Bordeaux 27 (2015), 245-288]. Analogous results are also shown to hold for related combinatorial sequences, namely for the Motzkin prefix numbers, Riordan numbers, central trinomial coefficients, and for the sequence of hex tree numbers.Comment: 28 pages, AmS-LaTeX; minor typos correcte

Enumeration of idempotents in planar diagram monoids

We classify and enumerate the idempotents in several planar diagram monoids: namely, the Motzkin, Jones (a.k.a. Temperley-Lieb) and Kauffman monoids. The classification is in terms of certain vertex- and edge-coloured graphs associated to Motzkin diagrams. The enumeration is necessarily algorithmic in nature, and is based on parameters associated to cycle components of these graphs. We compare our algorithms to existing algorithms for enumerating idempotents in arbitrary (regular *-) semigroups, and give several tables of calculated values.Comment: Majorly revised (new title, new abstract, one additional author), 24 pages, 6 figures, 8 tables, 5 algorithm

Meixner polynomials of the second kind and quantum algebras representing su(1,1)

We show how Viennot's combinatorial theory of orthogonal polynomials may be used to generalize some recent results of Sukumar and Hodges on the matrix entries in powers of certain operators in a representation of su(1,1). Our results link these calculations to finding the moments and inverse polynomial coefficients of certain Laguerre polynomials and Meixner polynomials of the second kind. As an immediate consequence of results by Koelink, Groenevelt and Van Der Jeugt, for the related operators, substitutions into essentially the same Laguerre polynomials and Meixner polynomials of the second kind may be used to express their eigenvectors. Our combinatorial approach explains and generalizes this "coincidence".Comment: several correction

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 finite automaton for the sequence modulo pα, for all but finitely 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 Apéry numbers. We also give a second method, which applies to an algebraic sequence modulo pα for all primes p, but is significantly slower. Finally, we show that a broad range of multidimensional sequences possess Lucas products modulo p