arXiv subjects

Zack Tripp

Publications and source records attributed to Zack Tripp.

5 recordsLinked to original sources

arXiv2021

Infinite families of crank functions, Stanton-type conjectures, and unimodality

Dyson's rank function and the Andrews--Garvan crank function famously give combinatorial witnesses for Ramanujan's partition function congruences modulo 5, 7, and 11. While these functions can be used to show that the corresponding sets of partitions split into 5, 7, or 11 equally sized sets, one may ask how to make the resulting bijections between partitions organized by rank or crank combinatorially explicit. Stanton recently made conjectures which aim to uncover a deeper combinatorial structure along these lines, where it turns out that minor modifications of the rank and crank are required. Here, we prove two of these conjectures. We also provide abstract criteria for quotients of polynomials by certain cyclotomic polynomials to have non-negative coefficients based on unimodality and symmetry. Furthermore, we extend Stanton's conjecture to an infinite family of cranks. This suggests further applications to other combinatorial objects. We also discuss numerical evidence for our conjectures, connections with other analytic conjectures such as the distribution of partition ranks.

math.NT↗

arXiv2020

Combinatorial results on $t$-cores and sums of squares

We classify the connection between $t$-cores and self-conjugate $t$-cores to sums of squares. To do so, we provide explicit maps between $t$-core partitions and self-conjugate $t$-core partitions of a positive integer $n$ to representations of certain numbers as sums of squares. For example, the self-conjugate $4$-core partition $\lambda=(4,1,1,1)$ corresponds uniquely to the solution $61=6^2+5^2$. As a corollary, we completely classify the relationship between $t$-cores and Hurwitz class numbers. Using these tools, we see how certain sets of representations as sums of squares naturally decompose into families of $t$-cores. Finally, we construct an explicit map on partitions to explain the equality $2\operatorname{sc}_7(8n+1) = \operatorname{c}_4(7n+2)$ previously studied by Bringmann, Kane, and the first author.

math.CO↗

arXiv2020

Fractional partitions and conjectures of Chern-Fu-Tang and Heim-Neuhauser

Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern-Fu-Tang and Heim-Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov-Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern-Fu-Tang conjecture and to show the Heim-Neuhauser conjecture in a certain range. The explicit error terms provided will also be useful in the future study of partition inequalities. These are laid out in a user-friendly way for the researcher in combinatorics interested in such analytic questions.

math.NT↗

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↗

arXiv2019

Pair correlation for Dedekind zeta functions of abelian extensions

Here we study problems related to the proportions of zeros, especially simple and distinct zeros on the critical line, of Dedekind zeta functions. We obtain new bounds on a counting function that measures the discrepancy of the zeta functions from having all zeros simple. In particular, for quadratic number fields, we deduce that more than 45% of the zeros are distinct. This extends work based on Montgomery's pair correlation approach for the Riemann zeta function. Our optimization problems can be interpreted as interpolants between the pair correlation bound for the Riemann zeta function and the Cohn-Elkies sphere packing bound in dimension 1. We compute the bounds through optimization over Schwartz functions using semidefinite programming and also show how semidefinite programming can be used to optimize over functions with bounded support.

math.NT↗