Related papers: On Discovering Interesting Combinatorial Integer S…
In this paper, we analyze properties of prime number sequences produced by the alternating sum of higher-order subsequences of the primes. We also introduce a new sieve which will generate these prime number sequences via the systematic…
Standard sequential generation methods assume a pre-specified generation order, such as text generation methods which generate words from left to right. In this work, we propose a framework for training models of text generation that…
This article gives a brief introduction to the On-Line Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms…
Tewodros Amdeberhan and Armin Straub initiated the study of enumerating subfamilies of the set of (s,t)-core partitions. While the enumeration of (n+1,n+2)-core partitions into distinct parts is relatively easy (in fact it equals the…
Recent developments in termination analysis for declarative programs emphasize the use of appropriate models for the logical theory representing the program at stake as a generic approach to prove termination of declarative programs. In…
In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many…
Conditional neural text generation models generate high-quality outputs, but often concentrate around a mode when what we really want is a diverse set of options. We present a search algorithm to construct lattices encoding a massive number…
The purpose of this paper is to show that some combinatorial sequences, such as second-order Eulerian numbers and Eulerian numbers of type $B$, can be generated by context-free grammars.
This article investigates integer sequences that partition the sequence into blocks of various lengths - irregular arrays. The main result of the article is explicit formulas for numbering of irregular arrays. A generalization of Cantor…
We show how to enumerate words in $1^{m_1} \dots n^{m_n}$ that avoid the increasing consecutive pattern $12 \dots r$ for any $r \geq 2$. Our approach yields an $O(n^{s+1})$ algorithm to enumerate words in $1^s \dots n^s$, avoiding the…
Non-linear recurrences which generate integers in a surprising way have been studied by many people. Typically people study recurrences that are linear in the highest order term. In this paper I consider what happens when the recurrence is…
The first author recently introduced an integer sequence now numbered A355519 in OEIS. This sequence arose from counting bracket tournaments; its study evokes the analysis of the Catalan triangle (sequence A009766 in OEIS) and the related…
This paper presents a distinctive prime detection approach. This method use GM-(n+1) sequences to effectively eliminate complex numbers. The sequences, which consist of odd a number of (n+1), exclude all components except for the initial…
In this article, we discuss a novel approach to solving number sequence problems, in which sequences of numbers following unstated rules are given, and missing terms are to be inferred. We develop a methodology of decomposing test sequences…
We introduce a class of stochastic integer sequences. In these sequences, every element is a sum of two previous elements, at least one of which is chosen randomly. The interplay between randomness and memory underlying these sequences…
We find generating functions the number of strings (words) containing a specified number of occurrences of certain types of order-isomorphic classes of substrings called subword patterns. In particular, we find generating functions for the…
Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences $(a_n)_{n\ge 0}$ of primitive eta-products that satisfy the generic…
In this study, several interesting iterative sequences were investigated. First, we define the iterative sequences. We fix function f(n). An iterative sequence starts with a natural number n, and calculates the sequence f(n),f(f(n)),…
A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of…
For any integer s >= 0, we derive a combinatorial interpretation for the family of sequences generated by the recursion (parameterized by s) h_s(n) = h_s(n - s - h_s(n - 1)) + h_s(n - 2 - s - h_s(n - 3)), n > s + 3, with the initial…