[Paper Review] On Cobweb Admissible Sequences - The Production Theorem

Maciej Dziemiańczuk|arXiv (Cornell University)|Jan 30, 2008

Advanced Mathematical Theories1 references9 citations

TL;DR

This paper establishes a complete characterization of cobweb admissible sequences by proving they are pointwise products of primary cobweb admissible sequences—each valued in powers of a single prime p. Using a novel tree-based algorithm, the authors construct all such sequences via recursive path generation on weighted trees, solving Problem 1 from prior work and providing an effective algorithmic production method for the entire family of cobweb admissible sequences.

ABSTRACT

In this note further clue decisive observations on cobweb admissible sequences are shared with the audience. In particular an announced proof of the Theorem 1 (by Dziemia\'nczuk) from [1] announced in India -Kolkata- December 2007 is delivered here. Namely here and there we claim that any cobweb admissible sequence F is at the point product of primary cobweb admissible sequences taking values one and/or certain power of an appropriate primary number p. Here also an algorithm to produce the family of all cobweb-admissible sequences i.e. the Problem 1 from [1] i.e. one of several problems posed in source papers [2,3] is solved using the idea and methods implicitly present already in [4]

Motivation & Objective

• To solve Problem 1 from prior work: effectively characterize and produce all cobweb admissible sequences.
• To establish a structural decomposition of any cobweb admissible sequence into primary components based on prime powers.
• To develop an algorithmic framework for generating all primary cobweb admissible sequences using a recursive tree structure.
• To formalize the necessary and sufficient conditions for a sequence to be cobweb admissible via exponent sum inequalities.
• To define and characterize the family of all primary cobweb admissible sequences through a labeled infinite tree graph.

Proposed method

• Define a primary cobweb admissible sequence as one whose terms are 1 or powers of a fixed prime p.
• Introduce a weighted rooted tree G(p) where vertex weights δ(v) represent exponents in the prime power decomposition of sequence terms.
• Construct the tree recursively: a path (v₀,…,vₙ) can be extended to vₙ₊₁ iff the sum of weights in the last k terms of the path is at least the sum in the first k terms.
• Define the successor set ∆ₙ for each path as all vertices v(m) with m ≥ max{Kₖ − Nₙ₋₁,ₖ : k=1,…,⌊n/2⌋}, ensuring admissibility.
• Use the tree structure to encode all finite initial segments of primary cobweb admissible sequences via paths from the root.
• Prove that infinite paths in G(p) correspond exactly to primary cobweb admissible sequences, with nF = p^{δ(vₙ)}.

Experimental results

Research questions

• RQ1 What is the complete algebraic structure of cobweb admissible sequences?
• RQ2 Can all cobweb admissible sequences be generated algorithmically from simpler components?
• RQ3 What conditions on the exponent sequence {nB(F)} ensure that the binomial coefficient (n/k)_F remains an integer for all n,k?
• RQ4 How can the family of all primary cobweb admissible sequences be systematically enumerated?
• RQ5 Is there a recursive tree-based construction that generates all such sequences while preserving admissibility?

Key findings

• Any cobweb admissible sequence F is the pointwise product of primary cobweb admissible sequences P(p), where each P(p) takes values in {1, p, p², ...} for a fixed prime p.
• The family of all cobweb admissible sequences is isomorphic to the Cartesian product ×ₛ A(pₛ), where A(p) is the family of all primary cobweb admissible sequences for prime p.
• A primary cobweb admissible sequence F is characterized by the condition that for all n,k with 1≤k≤⌊n/2⌋, the sum of exponents in the last k terms of the exponent sequence B(F) is at least the sum in the first k terms.
• An infinite path in the tree G(p) corresponds exactly to a primary cobweb admissible sequence, with vertex weight δ(vₙ) equal to the exponent of p in nF.
• The algorithm to generate successors at each node ensures that all generated paths maintain the required inequality condition for admissibility.
• The tree construction guarantees that every valid sequence is generated, and every generated path yields a valid cobweb admissible sequence.