Shannon Starr

collaborators

6 papers

math.CO2026

Pal's permanent conjecture: proof for block uniform matrices

Andrea Ottolini, Shannon Starr

Consider a symmetric function $C (x, y)$ on $[0, 1] \times [0, 1]$ which is twice continuously differentiable up to the boundary, and which satisfies $ \mathcal{C}(x,y)=\mathca…

math.NT2026

Asymptotics of the d'Arcais Numbers at Small $k$

Shannon Starr

The d'Arcais numbers are the triangular array ${A (2, n, k) : n = 0, 1, \dots, k = 0, \dots, n}$ , such that $\sum_{n=0}^{\infty} \sum_{k=0}^{n} A(2,n,k) x^k z^n/n! = ((z;z)_{\infty})^…

math.PR2026

Large Deviations for the d'Arcais Numbers

Shannon Starr

The d'Arcais polynomials $P_{n} (z)$ for $n \in {0, 1, \dots}$ are defined as $\sum_{n = 0}^{\infty} P_{n} (z) q^{n} = exp (- z ln ((q; q)_{\infty}))$ where the $q$ -Pochhammer symbol is $(q;q)_{\…

math.CO2025

About the Moments of the Generalized Ulam Problem

Samen Hossein, Shannon Starr

Given $I ¨ \in S_{n}$ , let $Z_{n, k} (I ¨ ) = \sum_{1 \leq i_{1} < \dots < i_{k} \leq n} 1 ({I ¨ _{i_{1}} < \dots < I ¨ _{i_{k}}}$ denote the number of increasing subsequences of length $k$ . Consider t…

math.CO2025

Some Observations about the "Generalized Abundancy Index"

Shannon Starr

Let $A (ℓ, n) \subset S_{n}^{ℓ}$ denote the set of all $ℓ$ -tuples $(I ¨ _{1}, \dots, I ¨ _{ℓ})$ , for $I ¨ _{1}, \dots, I ¨ _{ℓ} \in S_{n}$ satisfying: $\forall i < j$ we have $Ï_…

math.CO2025

Multifold Convolutions, Generating Functions and 1d Random Walks

Timothy Li, Shannon Starr

We consider multifold convolutions of a combinatorial sequence $(a_{n})_{n = 0}^{\infty}$ : namely, for each $k \in N$ the $k$ -fold convolution is $\mathcal{M}^{(k)}_n(\boldsymbol{a})…

Pal's permanent conjecture: proof for block uniform matrices

Asymptotics of the d'Arcais Numbers at Small k

Large Deviations for the d'Arcais Numbers

About the Moments of the Generalized Ulam Problem

Some Observations about the "Generalized Abundancy Index"

Multifold Convolutions, Generating Functions and 1d Random Walks

Asymptotics of the d'Arcais Numbers at Small $k$