Sequencings in Semidirect Products via the Polynomial Method

Abstract

The partial sums of a sequence $\mathbf{x}=x_1,x_2,\dots ,x_k$ of distinct non-identity elements of a group $(G,\cdot )$ are $s_0=i{d}_G$ and $s_j={\prod }_{i=1}^j{x}_i$ for $0<j\le k$. If the partial sums are all different then $\mathbf{x}$ is a linear sequencing and if the partial sums are all different when $|i-j|\le t$ then $\mathbf{x}$ is a $t$-weak sequencing. We investigate these notions of sequenceability in semidirect products using the polynomial method. We show that every subset of order $k$ of the non-identity elements of the dihedral group of order $2m$ has a linear sequencing when $k\le 12$ and either $m>3$ is prime or every prime factor of $m$ is larger than $k!$, unless $s_k$ is unavoidably the identity; that every subset of order $k$ of a non-abelian group of order three times a prime has a linear sequencing when $5<k\le 10$, unless $s_k$ is unavoidably the identity; and that if the order of a group is $pe$ then all sufficiently large subsets of the non-identity elements are $t$-weakly sequenceable when $p>3$ is prime, $e\le 3$ and $t\le 6$.

arXiv admin note: text overlap with arXiv:2203.16658