Front representation of set partitions
Jang Soo Kim

TL;DR
This paper introduces the front representation of set partitions, providing recurrence relations for specific pattern-avoiding partitions and analyzing symmetric distributions of crossings and nestings.
Contribution
It defines the front representation of set partitions and derives new recurrence relations for pattern-avoiding and noncrossing partitions, extending understanding of their combinatorial properties.
Findings
Derived recurrence relations for 12...k12-avoiding partitions.
Established recurrence relations for k-distant noncrossing partitions.
Proved symmetric distribution properties of crossings and nestings in front representation.
Abstract
Let be a set partition of . The standard representation of is the graph on the vertex set whose edges are the pairs of integers with in the same block which does not contain any integer between and . The front representation of is the graph on the vertex set whose edges are the pairs of integers with in the same block whose smallest integer is . Using the front representation, we find a recurrence relation for the number of -avoiding partitions for . Similarly, we find a recurrence relation for the number of -distant noncrossing partitions for . We also prove that the front representation has several joint symmetric distributions for crossings and nestings as the standard representation does.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Bayesian Methods and Mixture Models
