arXiv subjects

Nicholas B. Jones

Publications and source records attributed to Nicholas B. Jones.

2 recordsLinked to original sources

arXiv2026

A lattice path model for the volume of the Monge polytope

Monge matrices arise throughout combinatorial optimization and algorithm design; the Monge polytope $\mathcal{M}_{pq}$ is the set of $p \times q$ Monge matrices lying inside the standard simplex on the set of matrix coordinates. We find a Stanley decomposition of the associated affine semigroup, and use it to obtain a volume formula for $\mathcal{M}_{pq}$ expressed as a sum over "Z-avoiding" Delannoy paths in a $p \times q$ grid. An efficient dynamic-programming implementation of this formula computes the volume in dimensions far beyond the reach of general-purpose exact-volume algorithms (e.g., the volume of $\mathcal{M}_{20,20}$, which has dimension 399, is computed in a fraction of a second). As a corollary of our Stanley decomposition, we also obtain a combinatorial closed form for the Ehrhart series of $\mathcal{M}_{pq}$.

math.CO↗

arXiv2024

Homometric subsets of $\mathbb{Z}_n$ with cardinality 5: classification and enumeration

Two subsets of $\mathbb{Z}_n$ are said to be homometric if they have the same multiset of pairwise cyclic (i.e., Lee) distances. Homometric subsets necessarily have the same cardinality, say $k$. In this paper, for all positive integers $n$, we classify the homometric subsets of $\mathbb{Z}_n$ with cardinality $k=5$ (modulo cyclic shifts and reflections). Our classification consists of six families of homometric pairs, and one family of homometric triples. We also give a closed-form generating function that counts these homometric pairs and triples for all $n$. The same problem for $k \leq 4$ was partially solved by Erd\H{o}s and ultimately settled by Rosenblatt-Berman (1984). As an immediate application of our result, one obtains an explicit criterion for the solvability of the crystallographic phase retrieval problem, in the setting of binary signals supported on $k=5$ many atoms.

math.CO↗