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Enumerating Flat Fubini Rankings

Kenny Barrese, Jennifer Elder, Pamela E. Harris, Anthony Simpson

TL;DR

This work introduces and systematically studies flattened and weakly flattened Fubini rankings, bridging the combinatorics of runs of ascents with content/reduced-content encodings. It develops explicit enumeration formulas for weakly flattened rankings with runs of weak ascents (including fixed content) and proves structural bounds on the number of runs; it then treats flattened rankings with runs of ascents, giving a sharp content-based condition for existence and a constructive assembly. The paper also presents mixed-settings results, including nested-sum enumeration formulas and bounds for cases with both strict and weak constraints, and it closes with conjectures linking to second-order Eulerian numbers and open problems about associated matrices and generating functions. Together, these results advance the enumeration and structural understanding of flattened Fubini rankings and point to rich connections with parking-function-type objects and Eulerian-like statistics.

Abstract

Recall that the set of Fubini rankings on $n$ competitors consists of the $n$-tuples that encode the possible rankings of $n$ competitors in a competition allowing ties. Moreover, recall that a run (weak run) in a tuple is a subsequence of consecutive ascents (weak ascents). If the leading terms of the set of maximally long runs (weak runs) of a tuple are in increasing (weakly increasing) order, then the tuple is said to be flattened (weakly flattened). We define the set of strictly flattened Fubini rankings, which is the subset of Fubini rankings with runs of strict ascents whose leading term are strictly increasing. Analogously, we define the set of weakly flattened Fubini rankings, which is the subset of Fubini rankings with runs of weak ascents whose leading terms are in weakly increasing order. Our main results give formulas for the enumeration of strictly flattened Fubini rankings and weakly flattened Fubini rankings. We also provide some conjectures for further study.

Enumerating Flat Fubini Rankings

TL;DR

This work introduces and systematically studies flattened and weakly flattened Fubini rankings, bridging the combinatorics of runs of ascents with content/reduced-content encodings. It develops explicit enumeration formulas for weakly flattened rankings with runs of weak ascents (including fixed content) and proves structural bounds on the number of runs; it then treats flattened rankings with runs of ascents, giving a sharp content-based condition for existence and a constructive assembly. The paper also presents mixed-settings results, including nested-sum enumeration formulas and bounds for cases with both strict and weak constraints, and it closes with conjectures linking to second-order Eulerian numbers and open problems about associated matrices and generating functions. Together, these results advance the enumeration and structural understanding of flattened Fubini rankings and point to rich connections with parking-function-type objects and Eulerian-like statistics.

Abstract

Recall that the set of Fubini rankings on competitors consists of the -tuples that encode the possible rankings of competitors in a competition allowing ties. Moreover, recall that a run (weak run) in a tuple is a subsequence of consecutive ascents (weak ascents). If the leading terms of the set of maximally long runs (weak runs) of a tuple are in increasing (weakly increasing) order, then the tuple is said to be flattened (weakly flattened). We define the set of strictly flattened Fubini rankings, which is the subset of Fubini rankings with runs of strict ascents whose leading term are strictly increasing. Analogously, we define the set of weakly flattened Fubini rankings, which is the subset of Fubini rankings with runs of weak ascents whose leading terms are in weakly increasing order. Our main results give formulas for the enumeration of strictly flattened Fubini rankings and weakly flattened Fubini rankings. We also provide some conjectures for further study.

Paper Structure

This paper contains 11 sections, 16 theorems, 17 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weakly flattened Fubini rankings with runs of weak ascents
  4. Weakly flattened Fubini rankings with k runs of weak ascents
  5. Weakly flattened Fubini rankings with runs of weak ascents and a fixed content
  6. Flattened Fubini rankings with runs of ascents
  7. Flattened Fubini rankings with k runs of ascents
  8. Mixed flattened Fubini rankings
  9. Weakly flattened with runs of ascents
  10. Flattened with runs of weak ascents
  11. Future work

Key Result

Lemma 2.1

Let $\mathrm{\mathrm{FR}_n^\uparrow}$ denote the set of weakly increasing Fubini rankings and $\mathrm{\mathrm{FR}_n^\downarrow}$ denote the set of weakly decreasing Fubini rankings. If $n\geq 1$, then

Theorems & Definitions (38)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 28 more