SwingingFactorial

A Comprehensive Study on Swinging Factorials

Introduction

The concept of swinging factorials presents a unique and fascinating aspect of combinatorial mathematics, extending the traditional understanding of factorials with intriguing properties and applications. This document aims to provide a thorough examination of swinging factorials, including their definitions, mathematical properties, generating functions, and significant relations to other combinatorial structures.

Definitions

The swinging factorial of a number $n$, denoted by $a(n)$, is defined in several equivalent ways. One prominent definition is given by:

$$ a(n) = 2^{n-(n \mod 2)} \prod_{k=1}^{n} k^{(-1)^{k+1}}, $$

where $n \mod m$ denotes the modulus operation. An alternative formulation involves factorials and the floor function:

$$ a(n) = \frac{n!}{\left(\lfloor \frac{n}{2} \rfloor!\right)^2}. $$

Properties

Swinging factorials exhibit several notable properties:

• They relate to the enumeration of swinging orbitals in combinatorial structures.
• The sequence connects with central binomial coefficients, especially evident in its relationship with least common multiples (LCM) of consecutive central binomial coefficients.
• For odd prime numbers $p$, the sequence shows a pattern where $p$ consecutive multiples of $p$ follow the least positive multiple of $p$.

Generating Functions and Recurrences

The swinging factorial is characterized by various generating functions and recurrence relations:

• Exponential generating function (E.g.f.): $(1+x) \cdot \text{BesselI}(0, 2x)$.
• Ordinary generating function (O.g.f.): $a(n) = \text{SeriesCoeff}_{n}\left(\frac{1+z/(1-4z^2)}{\sqrt{1-4z^2}}\right)$.
• Polynomial generating function (P.g.f.): $a(n) = \text{PolyCoeff}_{n}\left((1+z^2)^n+nz(1+z^2)^{n-1}\right)$.
• D-finite recurrence: $n \cdot a(n) + (n-2) \cdot a(n-1) + 4(-2n+3) \cdot a(n-2) + 4(-n+1) \cdot a(n-3) + 16(n-3) \cdot a(n-4) = 0$.

Applications and Interpretations

Swinging factorials play a significant role in various mathematical contexts:

• They are related to the enumeration of geometric structures, particularly in the context of hypercubes intersecting with hyperplanes.
• The sequence contributes to the study of number theory and combinatorial geometry, including implications for the lonely runner conjecture.

Examples

To illustrate the concept, consider the following examples:

• $a(10) = 252$, calculated as $10! / 5!^2$.
• $a(11) = 2772$, calculated as $11! / 5!^2$.

Conclusion

The study of swinging factorials unveils a rich mathematical structure interlinking factorials, binomial coefficients, and geometric interpretations. It underscores the depth and breadth of combinatorial mathematics, revealing intricate relationships and properties that extend beyond the conventional factorial function.