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On distribution of the depth index on perfect matchings

Yonah Cherniavsky, Yuval Khachatryan-Raziel

Abstract

We study the restriction of depth index statistic on the set of perfect matchings. In particular, we provide additional combinatorial description of the statistic for perfect matchings and calculate the generating polynomial. The main result of the present short paper is that the depth index on perfect matchings is equidistributed with the rank function of the Bruhat order.

On distribution of the depth index on perfect matchings

Abstract

We study the restriction of depth index statistic on the set of perfect matchings. In particular, we provide additional combinatorial description of the statistic for perfect matchings and calculate the generating polynomial. The main result of the present short paper is that the depth index on perfect matchings is equidistributed with the rank function of the Bruhat order.
Paper Structure (3 sections, 13 theorems, 28 equations, 3 figures)

This paper contains 3 sections, 13 theorems, 28 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main result

Key Result

Theorem 2.10

GeometricInterpretationIntertwiningNumber For any set partition $A\in\Pi_n$, we have

Figures (3)

  • Figure 2.1: The arc diagram of the set partition $\pi = 1378|26|45$.
  • Figure 2.2: The extended arc diagram of the set partition $\pi = 1378|26|45$.
  • Figure 3.1: Crossing, nesting and alignment of two edges

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 27 more