A refinement of Dyck paths: A combinatorial approach

Local maxima and minima of a Dyck path are called peaks and valleys, respectively. A Dyck path is called restricted[Formula: see text]-Dyck if the difference between any two consecutive valleys is at least [Formula: see text] (right-hand side minus left-hand side) or if it has at most one valley. In this paper, we use several techniques to enumerate some statistics over this new family of lattice paths. For instance, we use the symbolic method, the Chomsky–Schűtzenberger methodology, Zeilberger’s creative telescoping method, recurrence relations, and bijective relations. We count, for example, the number of paths of length [Formula: see text], the number of peaks, the number of valleys, the number of peaks of a fixed height, and the area under the paths. We also give a bijection between the restricted [Formula: see text]-Dyck paths and a family of binary words.

Telescoping method and congruences for double sums

In recent years, Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double summations of hypergeometric terms. Using the telescoping method and certain mathematical software packages, we transform such a double summation into a single sum. With this new approach, we confirm several open conjectures of Sun.

THE NON-COMMUTATIVE A-POLYNOMIAL OF TWIST KNOTS

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form [Formula: see text] given a recursion relation for [Formula: see text] and the hypergeometric kernel c(n, k). As an application of our method, we explicitly compute the non-commutative A-polynomial for twist knots with -15 and 15 crossings. The non-commutative A-polynomial of a knot encodes the monic, linear, minimal order q-difference equation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to q = 1 is conjectured to be the better-known A-polynomial of a knot, which encodes important information about the geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easily computable for knots with 50 crossings, the A-polynomial is harder to compute and already unknown for some knots with 12 crossings.