arXiv subjects

Samuel Wilson

Publications and source records attributed to Samuel Wilson.

8 recordsLinked to original sources

arXiv2026

Families of Congruences for Partitions with $k$-colored odd parts

The study of integer partitions and their congruences dates back to 1919 when Ramanujan discovered his famous congruences for the partition function, $p(n)$. Since then, many other kinds of partition functions have been discovered, as well as their respective congruences. Recently, Hirschorn and Sellers have consider partitions in which the odd parts may appear in $k$ colors and the even parts are restricted to at most one color. It turns out that these partitions exhibit fascinating families of congruences. In this paper, we look at a set of congruences that give rise to infinite families modulo 3. We also give some questions at the end that could aid further research into these partitions.

math.NT↗

arXiv2026

An Exploration of Crank Generating functions for $t$-core partitions

In 1919, Ramanujan discovered his famous congruences for the partition function. Not too long after, Freeman Dyson conjectured a combinatorial statistic existed that explained the three congruences, which he dubbed the \textit{crank}. A crank generating function for the partition function was discovered in 1988 by George Andrews and Frank Garvan. Since then other crank generating functions have been found for many other kinds of partitions. In this paper, we give a family of crank generating functions which explain some partition congruences for $t$-core partitions.

math.CO↗

arXiv2024

A proposed crank for $(k+j)$-colored partitions, with $j$ colors having distinct parts

In 1988, George Andrews and Frank Garvan discovered a crank for $p(n)$. In 2020, Larry Rolen, Zack Tripp, and Ian Wagner generalized the crank for p(n) in order to accommodate Ramanujan-like congruences for $k$-colored partitions. In this paper, we utilize the techniques used by Rolen, Tripp, and Wagner for crank generating functions in order to define a crank generating function for $(k + j)$-colored partitions where $j$ colors have distinct parts. We provide three infinite families of crank generating functions and conjecture a general crank generating function for such partitions.

math.CO↗

arXiv2023

d-Fold Partition Diamonds

In this work we introduce new combinatorial objects called $d$--fold partition diamonds, which generalize both the classical partition function and the partition diamonds of Andrews, Paule and Riese, and we set $r_d(n)$ to be their counting function. We also consider the Schmidt type $d$--fold partition diamonds, which have counting function $s_d(n).$ Using partition analysis, we then find the generating function for both, and connect the generating functions $\sum_{n= 0}^\infty s_d(n)q^n$ to Eulerian polynomials. This allows us to develop elementary proofs of infinitely many Ramanujan--like congruences satisfied by $s_d(n)$ for various values of $d$, including the following family: for all $d\geq 1$ and all $n\geq 0,$ $s_d(2n+1) \equiv 0 \pmod{2^d}.$

math.NT↗

arXiv2022

SAFE: Sensitivity-Aware Features for Out-of-Distribution Object Detection

We address the problem of out-of-distribution (OOD) detection for the task of object detection. We show that residual convolutional layers with batch normalisation produce Sensitivity-Aware FEatures (SAFE) that are consistently powerful for distinguishing in-distribution from out-of-distribution detections. We extract SAFE vectors for every detected object, and train a multilayer perceptron on the surrogate task of distinguishing adversarially perturbed from clean in-distribution examples. This circumvents the need for realistic OOD training data, computationally expensive generative models, or retraining of the base object detector. SAFE outperforms the state-of-the-art OOD object detectors on multiple benchmarks by large margins, e.g. reducing the FPR95 by an absolute 30.6% from 48.3% to 17.7% on the OpenImages dataset.

cs.CV↗

arXiv2022

On symmetric representations of $\text{SL}_2(\mathbb{Z})$

We introduce the notions of symmetric and symmetrizable representations of $\text{SL}_2(\mathbb{Z})$. The linear representations of $\text{SL}_2(\mathbb{Z})$ arising from modular tensor categories are symmetric and have congruence kernel. Conversely, one may also reconstruct modular data from finite-dimensional symmetric, congruence representations of $\text{SL}_2(\mathbb{Z})$. By investigating a $\mathbb{Z}/2\mathbb{Z}$-symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations of $\text{SL}_2(\mathbb{Z})$ are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of $\text{SL}_2(\mathbb{Z})$ that are subrepresentations of a symmetric one.

math.QA↗

arXiv2021

Hyperdimensional Feature Fusion for Out-Of-Distribution Detection

We introduce powerful ideas from Hyperdimensional Computing into the challenging field of Out-of-Distribution (OOD) detection. In contrast to most existing work that performs OOD detection based on only a single layer of a neural network, we use similarity-preserving semi-orthogonal projection matrices to project the feature maps from multiple layers into a common vector space. By repeatedly applying the bundling operation $\oplus$, we create expressive class-specific descriptor vectors for all in-distribution classes. At test time, a simple and efficient cosine similarity calculation between descriptor vectors consistently identifies OOD samples with better performance than the current state-of-the-art. We show that the hyperdimensional fusion of multiple network layers is critical to achieve best general performance.

cs.CV↗

arXiv2020

Cranks for Ramanujan-type congruences of $k$-colored partitions

Dyson famously provided combinatorial explanations for Ramanujan's partition congruences modulo $5$ and $7$ via his rank function, and postulated that an invariant explaining all of Ramanujan's congruences modulo $5$, $7$, and $11$ should exist. Garvan and Andrews-Garvan later discovered such an invariant called the crank, fulfilling Dyson's goal. Many further examples of congruences of partition functions are known in the literature. Here, we provide a framework for discovering and proving such invariants for families of congruences. As a first example, we find a family of crank functions that simultaneously explains most known congruences for colored partition functions. The method used, which utilizes Gritsenko, Skoruppa, and Zagier's powerful recent theory of theta blocks, should also be useful for studying other combinatorial functions.

math.NT↗