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Partitions in which every term but the smallest one is consecutive

Rajat Gupta, Noah Lebowitz-Lockard

Abstract

In this article, we introduce the notion of almost consecutive partitions. A partition is almost consecutive if every term is consecutive, with the possible exception of the smallest one. We find formulas relating to the smallest parts of consecutive and almost consecutive partitions. We also find an alternate combinatorial interpretation of the number of almost consecutive partitions of a given integer $n$ and an asymptotic formula for this quantity.

Partitions in which every term but the smallest one is consecutive

Abstract

In this article, we introduce the notion of almost consecutive partitions. A partition is almost consecutive if every term is consecutive, with the possible exception of the smallest one. We find formulas relating to the smallest parts of consecutive and almost consecutive partitions. We also find an alternate combinatorial interpretation of the number of almost consecutive partitions of a given integer and an asymptotic formula for this quantity.
Paper Structure (7 sections, 17 theorems, 49 equations)

This paper contains 7 sections, 17 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Consecutive partitions
  3. A bijection between partitions
  4. Applying the bijection
  5. Special cases
  6. Another partition equivalence
  7. Competing Interests and Data Statements

Key Result

Theorem 1

Let $p_o (n)$ (resp., $p_e (n)$) be the number of partitions of $n$ into an odd (resp., even) number of distinct parts. Then,

Theorems & Definitions (26)

  • Theorem 1: And1
  • Theorem 2: And1
  • Theorem 3: Uch
  • Theorem 4
  • Theorem 5
  • Definition
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • ...and 16 more