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Unique path partitions: Characterization and Congruences

Christine Bessenrodt, Jørn B. Olsson, James A. Sellers

Abstract

We give a complete classification of the unique path partitions and study congruence properties of the function which enumerates such partitions.

Unique path partitions: Characterization and Congruences

Abstract

We give a complete classification of the unique path partitions and study congruence properties of the function which enumerates such partitions.

Paper Structure

This paper contains 4 sections, 20 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. The classification of up-partitions
  3. On up-partitions and restricted binary partitions
  4. Congruences for the number of up-partitions

Key Result

Lemma \oldthetheorem

(1) If $\mu=(a_1,a_2,...,a_k)$ is a up-partition with $a_k=2$, then $\mu'=(a_1,a_2,...,a_{k-1}, 1^2)$ is also a up-partition. (2) If $\mu=(a_1,a_2,...,a_k)$ is a up-partition with $k \ge 2$, then $\mu_2=(a_2,...,a_k)$ is also a up-partition.

Theorems & Definitions (26)

  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 16 more