Alejandro B. Galván | Searcharxiv

arXiv2026

Enumerating separable derangements

We give a polynomial-time algorithm to compute the number $b_n$ of separable derangements of $[n]$. This algorithm is based on a generating function technique which tracks permutations along with their occupied diagonals, where each permutation is counted once for every such diagonal. We provide bounds for the proportion of separable permutations which are derangements, show that $b_n$ and the large Schröder numbers have the same exponential growth constant $(3 + 2 \sqrt{2})$, and use the first 3000 terms to conjecture more explicit asymptotic behavior. This partially answers several questions about separable derangements recently posed by Vatter.

math.CO↗

arXiv2023

Triangular partitions: enumeration, structure, and generation

A triangular partition is a partition whose Ferrers diagram can be separated from its complement (as a subset of $\mathbb{N}^2$) by a straight line. Having their origins in combinatorial number theory and computer vision, triangular partitions have been studied from a combinatorial perspective by Onn and Sturmfels, and by Corteel et al. under the name plane corner cuts, and more recently by Bergeron and Mazin. In this paper we derive new enumerative, geometric and algorithmic properties of such partitions. We give a new characterization of triangular partitions and the cells that can be added or removed while preserving the triangular condition, and use it to describe the Möbius function of the restriction of Young's lattice to triangular partitions. We obtain a formula for the number of triangular partitions whose Young diagram fits inside a square, deriving, as a byproduct, a new proof of Lipatov's enumeration theorem for balanced words. Finally, we present an algorithm that generates all the triangular partitions of a given size, which is significantly more efficient than previous ones and allows us to compute the number of triangular partitions of size up to $10^5$.

math.CO↗