On Characteristic Polynomials of the Family of Cobweb Posets

This note is a response to one of problems posed by A.K. Kwasniewski in one of his recent papers. Namely for the sequence of finite cobweb subposets, the looked for explicit formulas for corresponding sequence of characteristic polynomials are discovered and delivered here. The recurrence relation defining arbitrary family of charactristic polynomials of finite cobweb posets is also derived.

💡 Research Summary

The paper addresses a problem posed by A.K. Kwasniewski concerning the explicit description of characteristic polynomials for the family of finite cobweb sub‑posets {Pₙ} generated by an arbitrary natural‑valued sequence {Fₙ} with F₀=1. A cobweb poset Π is defined by levels Φₛ containing Fₛ elements h_{j,s} (1≤j≤Fₛ); the finite sub‑poset Pₙ consists of the union of levels 0 through n and inherits the partial order from Π. Because Π satisfies the Jordan‑chain condition, each Pₙ has rank n. The authors exploit the fact that the Möbius function μ(0,x) depends only on the rank r(x) and obtain a closed form μ(0,x)=(-1)^{r(x)}∏{i=1}^{r(x)-1}(F_i−1). Consequently the Whitney numbers of the first kind are w_k(Π)=F_k·(-1)^k·∏{i=1}^{k-1}(F_i−1) for k>0, with w₀(Π)=1. Using the standard definition χ_P(t)=∑_{x∈P} μ(0,x) t^{n−r(x)} the characteristic polynomial of Pₙ becomes

χₙ(t)=tⁿ+∑{k=1}^{n}(-1)^k F_k · ∏{i=1}^{k-1}(F_i−1) t^{n−k}.

This formula is completely explicit and works for any admissible sequence {Fₙ}. In the special case F₁=1 (equivalently |Φ₁|=1) the product ∏_{i=1}^{k-1}(F_i−1) collapses to 1 for all k, yielding the remarkably simple χₙ(t)=tⁿ−t^{n−1} for n≥1, which recovers the known result for the Fibonacci cobweb poset.

Beyond the closed form, the authors derive a recurrence relation that allows χₙ(t) to be built iteratively from lower‑degree polynomials:

χ₀(t)=1, χ₁(t)=t−F₁,

χₙ(t)=t·χ_{n−1}(t)+(-1)^{n} F_n · ∏_{i=1}^{n-1}(F_i−1) for n≥2.

This recurrence eliminates the need to compute the full product for each term and makes the computation of higher‑degree characteristic polynomials tractable.

The paper supplies several illustrative families of sequences. For Fₙ=n+1 (the natural numbers) the characteristic polynomials exhibit coefficients 2,4,18,120,… and are listed up to n=6. For F₁=1 and Fₙ=2n+1 (odd numbers) the coefficients become 3,10,56,432,…, again displayed for several n. When F₁=1 and Fₙ=k (constant k>1) the polynomials simplify to

χₙ(t)=∑_{j=0}^{n}(-1)^{j} k (k−1)^{j-1} t^{n−j},

which clearly shows the geometric progression in the coefficients.

The authors also discuss the relationship of their results to earlier work on the Fibonacci cobweb poset, confirming that the general formulas specialize correctly. They emphasize that the dependence of μ(0,x) solely on rank is a key structural property of cobweb posets, enabling the uniform treatment of all admissible sequences.

Finally, the paper points out potential applications: the explicit characteristic polynomials can be used to study eigenvalue distributions of incidence matrices, factorization properties (as in Sagan’s work), and connections to algebraic combinatorics, graph theory, and non‑commutative prefabs. By providing both a closed‑form expression and an efficient recurrence, the work equips researchers with practical tools for further exploration of cobweb posets and their algebraic invariants.