Few subsequence sums in Z_n x Z_n

Definitions and mathematical statement

Let $n\ge2$ and write $G=(\mathbb Z/n\mathbb Z)^2$ additively. A sequence $S=(s_1,\ldots,s_\ell)$ in $G$ may have repeated terms. Its set of nonempty subsequence sums is \[ \Sigma(S)=\left\{\sum_{i\in I}s_i: \varnothing\ne I\subseteq\{1,\ldots,\ell\}\right\}. \] It is zero-sum-free if $0\notin\Sigma(S)$. Distinctness here concerns the resulting group elements, not the index subsets producing them.

For $0\le t\le n-1$, the proposed extremal bound is \[ |S|=n-1+t,\quad 0\notin\Sigma(S) \quad\Longrightarrow\quad |\Sigma(S)|\ge n(t+1)-1. \] This is precisely the source's comparison with the sequence having $n-1$ copies of $e_1=(1,0)$ and $t$ copies of $e_2=(0,1)$. For that sequence, the sums are all pairs $(a,b)$ with $0\le a\le n-1$, $0\le b\le t$, except $(0,0)$; hence it is zero-sum-free and has exactly the displayed number of sums. The empty subsequence is excluded consistently on both sides. Prove this bound for every $n,t$ and every sequence, or produce a zero-sum-free sequence with fewer sums. Classification of every equality case would be additional information, not part of the stated minimum-value claim.

Short English statement

Among zero-sum-free sequences of the specified length in a two-dimensional modular group, do repeated coordinate vectors minimize the number of distinct nonempty sums?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3377 (OPG-156), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Few subsequence sums in Z_n x Z_n. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/limiting_subsequence_sums_in_z_n_x_z_n.