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Approval Ballot Triangles and Strict-Sense Ballots

Andrew Beveridge, Ian Calaway

TL;DR

The paper addresses how approval-ballot structures can be connected to classic combinatorial objects, by introducing approval ballot triangles (ABTs) and approval ballot hypertriangles (ABHs). It develops explicit bijections ABT$\leftrightarrow$TSSCPP and ABH$\leftrightarrow$SSB, with SSBs corresponding to shifted standard Young tableaux of staircase shape $(n,n-1,\ldots,1)$, thereby decomposing strict-sense ballots into sequential ABTs. The core method uses non-crossing lattice-path frameworks (NILPs and NCLPs) to realize these correspondences and to translate 2D plane-partition structures into triangular ABTs. The results unify ballot-problem theory, plane partitions, and lattice-path combinatorics, yielding both exact enumerations (e.g., $\#\mathrm{TSSCPP}(n)=\prod_{k=0}^{n-1} \frac{(3k+1)!}{(n+k)!}$) and constructive decompositions that illuminate how simple combinatorial pieces assemble into richer families.

Abstract

We consider a family of binary triangular arrays, called approval ballot triangles (ABTs), that are in bijection with totally symmetric self-complementary plane partitions (TSSCPPs). These triangles correspond to a ballot process in which voters select their collection of approved candidates rather than voting for a single person. We situate ABTs within the ballot problem literature and then show that a strict-sense ballot can be decomposed into a list of sequentially compatible ABTs.

Approval Ballot Triangles and Strict-Sense Ballots

TL;DR

The paper addresses how approval-ballot structures can be connected to classic combinatorial objects, by introducing approval ballot triangles (ABTs) and approval ballot hypertriangles (ABHs). It develops explicit bijections ABTTSSCPP and ABHSSB, with SSBs corresponding to shifted standard Young tableaux of staircase shape , thereby decomposing strict-sense ballots into sequential ABTs. The core method uses non-crossing lattice-path frameworks (NILPs and NCLPs) to realize these correspondences and to translate 2D plane-partition structures into triangular ABTs. The results unify ballot-problem theory, plane partitions, and lattice-path combinatorics, yielding both exact enumerations (e.g., ) and constructive decompositions that illuminate how simple combinatorial pieces assemble into richer families.

Abstract

We consider a family of binary triangular arrays, called approval ballot triangles (ABTs), that are in bijection with totally symmetric self-complementary plane partitions (TSSCPPs). These triangles correspond to a ballot process in which voters select their collection of approved candidates rather than voting for a single person. We situate ABTs within the ballot problem literature and then show that a strict-sense ballot can be decomposed into a list of sequentially compatible ABTs.
Paper Structure (7 sections, 7 theorems, 20 equations, 11 figures)

This paper contains 7 sections, 7 theorems, 20 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Ballot Problems
  3. Approval Ballot Triangles
  4. Totally Symmetric Self-Complementary Plane Partitions
  5. Non-crossing Lattice Paths
  6. Approval Ballot Hypertriangles
  7. Conclusion

Key Result

Theorem 1.2

The set $\mathcal{A}_{n-1}$ of approval ballot triangles of size $n-1$ are in bijection with TSSCPPs inside a $2n \times 2n \times 2n$ box.

Figures (11)

  • Figure 1: The 42 approval ballot triangles of size 3. The zero entries are rendered blank for visual clarity.
  • Figure 2: An approval ballot hypertriangle $(A_5,A_4,A_3,A_2,A_1)$ of size 5.
  • Figure 3: An example of the ABH triangle constraint of Definition \ref{['def:abt-hyper']}. We have an ABH $(A_{9}, A_{8}, A_7, A_6, \ldots)$. The three ones in $A_{9}(7,[2:7])$ are located at $(t,u) \in \{(7,3), (7,4), (7,7)\}$. Each of these ones induces two constraints on one of the next three triangles in the ABH. Each one in row 7 of $A_9$ has the same shading as the two rows affected by the corresponding constraint.
  • Figure 4: (a) The last two rows of the five ABTs corresponding to Dyck paths of length 6. (b) The last two rows of the nine ABTs corresponding to Motzkin paths of length 4.
  • Figure 5: (a) A TSSCPP of order 5 along with its (b) fundamental domain, and (c) magog triangle.
  • ...and 6 more figures

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • ...and 16 more