Related papers: Humps for Dyck and for Motzkin paths

For any simple digraph $D$ we offer a new proof for the intersection number of its middle digraph, $M(D)$; while doing so we also solve for the intersection number when $D$ has loops. In addition, a new transformation, the union of $D$ and…

Recently, Griffin, Ono, and Tsai examined the distribution of the number of $t$-hooks in partitions of $n$, which was later followed by the work of Craig, Ono, and Singh on the distribution of the number of $t$-hooks in self-conjugate…

Noncrossing linked partitions arise in the study of certain transforms in free probability theory. We explore the connection between noncrossing linked partitions and colored Motzkin paths. A (3,2)-Motzkin path can be viewed as a colored…

A new approach is presented to evaluate multi-loop integrals, which appear in the calculation of cross-sections in high-energy physics. It relies on a fully numerical method and is applicable to a wide class of integrals with various mass…

We present a simple algorithm for inverting the sweep map on rational $(m,n)$-Dyck paths for a co-prime pair $(m,n)$ of positive integers. This work is inspired by Thomas-Williams work on the modular sweep map. A simple proof of the…

We define a map from subspaces to Motzkin paths and show that the inverse image of every path is a disjoint union of symmetric Boolean subsets yielding an explicit symmetric Boolean decomposition of the subspace lattice.

In this article, we study hook lengths of ordinary partitions and $t$-regular partitions. We establish hook length biases for the ordinary partitions and motivated by them we find a few interesting hook length biases in $2$-regular…

Motzkin paths of order-$\ell$ are a generalization of Motzkin paths that use steps $U=(1,1)$, $L=(1,0)$, and $D_i=(1,-i)$ for every positive integer $i \leq \ell$. We further generalize order-$\ell$ Motzkin paths by allowing for various…

The known bijections on Dyck paths are either involutions or have notoriously intractable cycle structure. Here we present a size-preserving bijection on Dyck paths whose cycle structure is amenable to complete analysis. In particular, each…

We construct a combinatorial function F which computes the number of oriented Hamiltonian paths of any given type, in a transitive tournament. We also study many properties of F that arise, and reach some observations.

Using only a symmetric p-core partition and p-quotient, we give an explicit formula for the set of diagonal hook lengths of the associated symmetric partition.

Here we give two bijections, one to show that the number of UUU-free Dyck n-paths is the Motzkin number M_n, the other to obtain the (known) distributions of the parameters "number of UDUs" and "number of DDUs" on Dyck n-paths. The first…

Motzkin paths with air pockets (MAP) of the first kind are defined as a generalization of Dyck paths with air pockets. They are lattice paths in $\mathbb{N}^2$ starting at the origin made of steps $U=(1,1)$, $D_k=(1,-k)$, $k\geq 1$ and…

Cover-inclusive Dyck tilings are tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths, in which tiles are no larger than the tiles they cover. These tilings arise in the study of certain statistical physics models and…

A {\em k-generalized Dyck path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $\mathbb{Z}\times\mathbb{Z}$ consisting of horizontal-steps $(k, 0)$ for a given integer $k\geq 0$, up-steps $(1,1)$, and…

Multiple harmonic sums appear in the perturbative computation of various quantities of interest in quantum field theory. In this article we introduce a class of Hopf algebras that describe the structure of such sums, and develop some of…

We count the number of occurrences of restricted patterns of length 3 in permutations with respect to length and the number of cycles. The main tool is a bijection between permutations in standard cycle form and weighted Motzkin paths.

Recently, many authors have investigated how various partition statistics distribute as the size of the partition grows. In this work, we look at a particular statistic arising from the recent rejuvenation of MacMahon's partition analysis.…

Dyck paths with air pockets are obtained from ordinary Dyck paths by compressing maximal runs of down-steps into giant down-steps of arbitrary size. Using the kernel method, we consider partial Dyck paths with air pockets, both, from left…

In the present article a new method of deriving integral representations of combinations and partitions in terms of harmonic products has been established. This method may be relevant to statistical mechanics and to number theory.