Down-step statistics in generalized Dyck paths

Catalan, Motzkin, and Riordan numbers

We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operad obtained from the additive monoid. These involve various familiar combinatorial objects: parking functions, packed words, planar rooted trees, generalized Dyck paths, Schröder trees, Motzkin paths, integer compositions, directed animals, etc. We also retrieve some known operads: the magmatic operad, the commutative associative operad, and the diassociative operad.

The Catalan numbers form a sequence that counts over 200 combinatorial\nobjects. A remarkable property of the Catalan numbers, which extends to these\nobjects, is its recursive definition; that is, we can determine the $n^{th}$\nobject from previous ones. Matroids are combinatorial objects that generalize\nthe notion of linear independence and have connections with other fields of\nmathematics. A family of matroids, called unit interval positroids (UIP), are\nCatalan objects induced by the antiadjacency matrices of unit interval orders.\nAssociated to each UIP is the set of externally ordered bases, which due to Las\nVergnas, produces a lattice after adjoining a bottom element. We study the\nposet of externally ordered UIP bases and the implied Catalan-induced\nrecursion. Explicitly, we describe an algorithm for constructing the lattice of\na rank $n$ UIP from the lattice of lower ranks. Using their inherent\ncombinatorial structure, we define a simple formula to enumerate the bases for\na given UIP.\n

A binary matrix has the Consecutive Ones Property (C1P) if its columns can be ordered in such a way that all 1's on each row are consecutive. A Minimal Conflicting Set is a set of rows that does not have the C1P, but every proper subset has the C1P. Such submatrices have been considered in comparative genomics applications, but very little is known about their combinatorial structure and efficient algorithms to compute them. We first describe an algorithm that detects rows that belong to Minimal Conflicting Sets. This algorithm has a polynomial time complexity when the number of 1s in each row of the considered matrix is bounded by a constant. Next, we show that the problem of computing all Minimal Conflicting Sets can be reduced to the joint generation of all minimal true clauses and maximal false clauses for some monotone boolean function. We use these methods on simulated data related to ancestral genome reconstruction to show that computing Minimal Conflicting Set is useful in discriminating between true positive and false positive ancestral syntenies. We also study a dataset of yeast genomes and address the reliability of an ancestral genome proposal of the Saccharomycetaceae yeasts.

Random walks in octants, and related structures

Concept lattices (also called Galois lattices) have been applied in numerous areas, and several algorithms have been proposed to construct them. Generally, the input for lattice construction algorithms is a binary matrix with size |G||M| representing binary relation Ii¾? G×M. In this paper, we consider polynomial delay algorithms for building concept lattices. Although the concept lattice may be of exponential size, there exist polynomial delay algorithms for building them. The current best delay-time complexity is O(|G||M|2). In this paper, we introduce the notion of irregular concepts, the combinatorial structure of which allows us to develop a linear delay lattice construction algorithm, that is, we give an algorithm with delay time of O(|G||M|). Our algorithm avoids the union operation for the attribute set and does not require checking if new concepts are already generated. In addition, we propose a compact representation for concept lattices and a corresponding construction algorithm. Although we are not guaranteed to achieve optimal compression, the compact representation can save significant storage space compared to the full representation normally used for concept lattices.

We consider the standard broadcast setup with a single server broadcasting information to a number of clients, each of which contains local storage (called cache) of some size, which can store some parts of the available files at the server. The centralized coded caching framework, consists of a caching phase and a delivery phase, both of which are carefully designed in order to use the cache and the channel together optimally. In prior literature, various combinatorial structures have been used to construct coded caching schemes. In this work, we propose a binary matrix model to construct the coded caching scheme. The ones in such a caching matrix indicate uncached subfiles at the users. Identity submatrices of the caching matrix represent transmissions in the delivery phase. Using this model, we then propose several novel constructions for coded caching based on the various types of combinatorial designs. While most of the schemes constructed in this work (based on existing designs) have a high cache requirement (uncached fraction being Θ( √1K), K being the number of users), they provide a rate R that is upper bounded by a constant (R ≤ 1) with increasing K, and moreover require extremely small levels of subpacketization (being O(K)), which is an extremely important parameter in practical applications of coded caching.

Using a finite field approach, a novel algebraic construction of low-density parity-check (LDPC) convolutional codes with fast encoding property is proposed. According to the matrices of quasi-cyclic (QC) codes constructed based on the multiplicative groups of finite fields and the algebraic property that a binary circulant matrix is isomorphic to a finite ring, we first generate a polynomial-form parity-check matrix of an LDPC convolutional code under a given rate over a given finite field. Then some related modifications are made upon the original polynomial-form matrix to obtain the new one with fast encoding property. Simulation results show that the proposed LDPC convolutional codes have good performance with the iterative belief propagation decoding algorithm.

This paper concentrates on the set $$\mathcal {V}_n$$ of weighted Dyck paths of length 2n with special restrictions on the level of valleys. We first give its explicit formula of the counting generating function in terms of certain weight functions. When the weight functions are specialized, some connections are builded between $$\mathcal {V}_n$$ and other classical combinatorial structures such as (a, b)-Motzkin paths, q-Schröder paths, Delannoy paths and complete k-ary trees. Some bijections are also established between these settings and $$\mathcal {V}_n$$ subject to certain special weight functions.

On the number of anchored rectangle packings for a planar point set