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Signs behaviour of sums of weighted numbers of compositions

Filip Gawron, Maciej Ulas

Abstract

Let $A$ be a subset of positive integers. For a given positive integer $n$ and $0\leq i\leq n$ let $c_{A}(i,n)$ denotes the number of $A$-compositions of $n$ with exactly $i$ parts. In this note we investigate the sign behaviour of the sequence $(S_{A,k}(n))_{n\in\N}$, where $S_{A,k}(n)=\sum_{i=0}^{n}(-1)^{k}i^{k}c_{A}(i,n)$. We prove that for a broad class of subsets $A$, the number $(-1)^{n}S_{A,k}(n)$ is non-negative for all sufficiently large $n$. Moreover, we show that there is $A\subset \N_{+}$ such that the sign behaviour of $S_{A,k}(n)$ is not periodic.

Signs behaviour of sums of weighted numbers of compositions

Abstract

Let be a subset of positive integers. For a given positive integer and let denotes the number of -compositions of with exactly parts. In this note we investigate the sign behaviour of the sequence , where . We prove that for a broad class of subsets , the number is non-negative for all sufficiently large . Moreover, we show that there is such that the sign behaviour of is not periodic.
Paper Structure (4 sections, 10 theorems, 49 equations)

This paper contains 4 sections, 10 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Basic observations
  3. Results
  4. Some problems and a question

Key Result

Lemma 2.1

Theorems & Definitions (25)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Corollary 3.1
  • ...and 15 more