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Decomposing Dedekind Numbers: A Polynomial Representation with Powers of 2

YongQing Liu

Abstract

In this paper, we reveal an internal structure within Dedekind numbers, demonstrating that they can be expressed as polynomials of powers of 2. This discovery is based on innovative concepts and methods, offering a new perspective on the nature of these numbers.

Decomposing Dedekind Numbers: A Polynomial Representation with Powers of 2

Abstract

In this paper, we reveal an internal structure within Dedekind numbers, demonstrating that they can be expressed as polynomials of powers of 2. This discovery is based on innovative concepts and methods, offering a new perspective on the nature of these numbers.
Paper Structure (3 sections, 8 theorems, 4 equations)

This paper contains 3 sections, 8 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Recursive Partition Formula of the Dedekind Number

Key Result

Theorem 1

For any subset $A=\{a_1,\dots,a_k\}$ of the poset $S\subseteq E^n$, the $n$-dimensional local Dedekind number $D(S)$ on $S$ can be decomposed into the sum of $D(A)$ parts: We call the set of all terms of the above polynomial the partition of $D(S)$ induced by $A$.

Theorems & Definitions (23)

  • Definition 1: Cube
  • Definition 2: Point Weight
  • Definition 3: Comparable, Incomparable
  • Definition 4: Cover
  • Definition 5: Antichain
  • Definition 6: Poset Dual
  • Definition 7: Monotone Boolean Function
  • Definition 8: Dedekind Number
  • Definition 9: Function Local Restriction
  • Definition 10: Upper set, Lower set
  • ...and 13 more