Related papers: Arndt compositions: a generating functions approac…
Additive composition (Foltz et al, 1998; Landauer and Dumais, 1997; Mitchell and Lapata, 2010) is a widely used method for computing meanings of phrases, which takes the average of vector representations of the constituent words. In this…
Working with generating functions, the combinatorics of a recurrence relation can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan numbers for an ordered binary tree…
We study the $k$-Bonacci word over the infinite alphabet $\mathbb{N}$. Since the alphabet is infinite, the usual factor complexity is infinite and does not provide any information. We therefore investigate factor occurrence statistics in…
We describe a novel approach for generating music using a self-correcting, non-chronological, autoregressive model. We represent music as a sequence of edit events, each of which denotes either the addition or removal of a note---even a…
Sequence of numbers generated by the recurrence relation based on the Collatz conjecture is investigated. An arithmetic operation on the Collatz conjecture is called descending operation, and ascending operation is carried out reversely to…
Motivated by Khintchin's 1923 conjecture, refuted by Marstrand in 1970, we study the Khintchin class of functions associated to a given increasing sequence of integers. When the Khintchin class contains L^p(\mathbb{T}), we call the sequence…
We generalize recent work of Andrews, Just, and Simay on modular palindromic compositions and anti-palindromic compositions by viewing all compositions partially (modular) palindromic or anti-palindromic. More precisely, we enumerate…
In this paper, we study a family of generating functions whose coefficients are polynomials that enumerate partitions in lower order ideals of Young's lattice. Our main result is that this family satisfies a rational recursion and are…
In this paper, we extend the work of Andrews, Beck and Hopkins by considering partitions and compositions with bounded gaps between each pair of consecutive parts. We show that both their generating functions and two matrices determined by…
We introduce the notion of multi-pattern, a combinatorial abstraction of polyphonic musical phrases. The interest of this approach lies in the fact that this offers a way to compose two multi-patterns in order to produce a longer one. This…
We study the problem of generating interesting integer sequences with a combinatorial interpretation. For this we introduce a two-step approach. In the first step, we generate first-order logic sentences which define some combinatorial…
Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions…
In this paper we introduce a family of partitions of the set of natural numbers, Fibonacci-like partitions. In particular, we introduce a Fibonacci-like partition in a number of parts corresponding to the Fibonacci numbers, the standard…
The Fibonacci numbers are the prototypical example of a recursive sequence, but grow too quickly to enumerate sets of integer partitions. The same is true for the other classical sequences $a(n)$ defined by Fibonacci-like recursions: the…
In a paper entitled Binary lambda calculus and combinatory logic, John Tromp presents a simple way of encoding lambda calculus terms as binary sequences. In what follows, we study the numbers of binary strings of a given size that represent…
In this paper, we derive several formulas of counting families of non-intersecting paths for two-sided ladder-shaped regions. As an application, we give a new proof to a combinatorial interpretation of Fibonacci numbers obtained by G.…
In certain fields where compositional data are studied, the compositional components, called parts, can be combined into certain subsets, called amalgamations, that are based on domain knowledge. Furthermore, these subsets can form a…
We enumerate derangements with descents in prescribed positions. A generating function was given by Guo-Niu Han and Guoce Xin in 2007. We give a combinatorial proof of this result, and derive several explicit formulas. To this end, we…
We investigate binary voting systems with two types of voters and a hierarchy among the members in each type, so that members in one class have more influence or importance than members in the other class. The purpose of this paper is to…
In the context of Higman embeddings of recursive groups into finitely presented groups we suggest an algorithm which uses Higman operations to explicitly constructs the specific recursively enumerable sets of integer sequences arising…