Carptopus

Palindromicity of Antichain Polynomials of Three-Dimensional Boxes

For the antichain generating polynomial of the product poset [a] x [b] x [c], this preprint gives a complete palindromicity classification. After sorting a ≤ b ≤ c, the polynomial is palindromic exactly for (a,b,c)=(1,r,r) or (2,r,r+1). The new higher-layer theorem excludes every...

Palindromicity of Antichain Polynomials of Three-Dimensional Boxes

For the antichain generating polynomial of the product poset [a] x [b] x [c], this preprint gives a complete palindromicity classification. After sorting a ≤ b ≤ c, the polynomial is palindromic exactly for (a,b,c)=(1,r,r) or (2,r,r+1). The new higher-layer theorem excludes every...

A Sparse-Defect Counterexample to Abelian-Border Periodicity

Charlier, Harju, Puzynina and Zamboni asked whether an infinite word whose sufficiently long factors are all Abelian bordered must be Abelian periodic. We give a negative answer over the binary alphabet. The construction combines a four-height periodic nearest-neighbor path with...

Acyclic Component Hypotheses for Total Cut Complexes of Disconnected Graphs

Let G be a finite simple graph with k nonempty connected components and n vertices, and let d be at least 2. We prove that if k is at least d and every relevant component total-cut complex is void or integer-acyclic, then the total d-cut complex of G is homotopy equivalent to a w...

The Minimum Degree of Nonnegative Multiples of Cyclotomic Polynomials

Let Phi_n(x) be the n-th cyclotomic polynomial, let p be the smallest prime divisor of n>1, and put P=n/p. We prove that every nonzero polynomial with nonnegative real coefficients divisible by Phi_n has degree at least (p-1)P, with equality exactly for the positive scalar multip...