[Paper Review] A Note on Boolean Lattices and Farey Sequences

Andrey O. Matveev|ArXiv.org|Feb 5, 2007

Limits and Structures in Graph Theory4 citations

TL;DR

This paper establishes monotone bijections between the standard Farey sequence 𝒻ₘ and the left and right halfsequences of the Farey subsequence ℱ(ℬ(2m), m) associated with rank-m elements in the Boolean lattice of subsets of a 2m-set. It derives novel combinatorial identities involving binomial coefficients indexed by fractions in these sequences, revealing deep structural connections between Boolean lattices and Farey sequences through rank-based subset enumeration and modular arithmetic constraints.

ABSTRACT

We establish monotone bijections between the Farey sequences of order m and the halfsequences of Farey subsequences associated with the rank m elements of the Boolean lattice of subsets of a 2m-set. We also present a few related combinatorial identities.

Motivation & Objective

• To establish a structural correspondence between the standard Farey sequence 𝒻ₘ and the halfsequences of the Farey subsequence ℱ(ℬ(2m), m) in the Boolean lattice of subsets of a 2m-element set.
• To explore the combinatorial properties of ℱ(ℬ(2m), m), particularly its partitioning into rank-based subsets and the enumeration of such subsets via binomial coefficients.
• To derive new combinatorial identities by leveraging the bijections and symmetry properties of the Farey subsequence ℱ(ℬ(2m), m), especially under order-reversing maps.
• To extend these results to the standard Farey sequence 𝒻ₘ via the established bijections, yielding analogous identities for 𝒻ₘ.

Proposed method

• Constructs a monotone bijection between the Farey sequence 𝒻ₘ and the left and right halfsequences of ℱ(ℬ(2m), m), using modular inverses to define predecessor and successor fractions.
• Applies the order-reversing map 𝒻(ℬ(n), m) → 𝒻(ℬ(n), n−m) defined by h/k ↦ (k−h)/k, which preserves bijectivity and symmetry.
• Uses the poset structure of the Boolean lattice ℬ(n), particularly order ideals and filters generated by rank-m elements, to partition the subposet ℱ(𝕀(a′) ∩ ℬ(n)⁽¹⁾).
• Derives a key identity by equating two expressions for the cardinality 2ⁿ − 2ⁿ⁻ᵐ, one via subset enumeration and the other via summing binomial coefficients over fractions in ℱ(ℬ(n), m).
• Applies the symmetry between ℱ(ℬ(n), m) and ℱ(ℬ(n), n−m) to derive Proposition 2, which equates two sums of binomial coefficients to 2ⁿ − 2ᵐ − 2ⁿ⁻ᵐ + 1.
• Specializes the general case to n = 2m, leading to Proposition 7 and Corollary 8, which present symmetric identities for ℱ(ℬ(2m), m) and 𝒻ₘ involving sums over fractions in specific intervals and binomial coefficient products.

Experimental results

Research questions

• RQ1 How are the Farey sequence 𝒻ₘ and the halfsequences of the Farey subsequence ℱ(ℬ(2m), m) related via monotone bijections?
• RQ2 What combinatorial identities emerge from the partitioning of the subposet ℱ(𝕀(a′) ∩ ℬ(n)⁽¹⁾) in the Boolean lattice ℬ(n)?
• RQ3 How does the symmetry between ℱ(ℬ(n), m) and ℱ(ℬ(n), n−m) via the map h/k ↦ (k−h)/k lead to balanced binomial coefficient sums?
• RQ4 What identities for the standard Farey sequence 𝒻ₘ can be derived from the bijections established for ℱ(ℬ(2m), m)?
• RQ5 What role do modular inverses and floor functions play in determining the predecessor and successor of a fraction in ℱ(ℬ(2m), m)?

Key findings

• A monotone bijection exists between the Farey sequence 𝒻ₘ and the left and right halfsequences of ℱ(ℬ(2m), m), with predecessor and successor fractions defined via modular inverses of h modulo (k−h).
• The sum of binomial coefficients ∑_{h/k ∈ ℱ(ℬ(2m), m), 0 < h/k < 1} ∑_{s ≤ ⌊min{m/h, m/(k−h)}⌋} ₑ(m, sh)⋅₇(m, s(k−h)) equals 2²ᵐ − 2ᵐ⁺¹ + 1.
• The sum over fractions with h/k < 1/2 equals the sum over h/k > 1/2, and both equal 2²ᵐ⁻¹ − 2ᵐ − ½⋅₇(2m, m) + 1.
• The sum over h/k ∈ (0, 1/3) and (1/3, 1/2) of terms involving ₑ(m, s·h) + ₑ(m, s·(k−2h)) equals the sum over h/k ∈ (1/2, 2/3) and (2/3, 1), with the common value 2²ᵐ⁻¹ − 2ᵐ − ½⋅₇(2m, m) − ∑_{t≤⌊m/2⌋} ₑ(m, 2t)⋅₇(m, t) + 1.
• The standard Farey sequence 𝒻ₘ satisfies the identity ∑_{h/k ∈ 𝒻ₘ, 0 < h/k < 1} ∑_{s ≤ ⌊m/k⌋} ₑ(m, sh)⋅₇(m, sk) = 2²ᵐ⁻¹ − 2ᵐ − ½⋅₇(2m, m) + 1.
• For 𝒻ₘ, the sum over h/k < 1/2 of ₑ(m, sk)(ₑ(m, sh) + ₑ(m, s(k−h))) equals the sum over h/k > 1/2, and both equal 2²ᵐ⁻¹ − 2ᵐ − ½⋅₇(2m, m) − ∑_{t≤⌊m/2⌋} ₑ(m, 2t)⋅₇(m, t) + 1.