Bi-banded Paths, a Bijection and the Narayana Numbers

We find a bijection between bi-banded paths and peak-counting paths, applying to two classes of lattice paths including Dyck paths. Thus we find a new interpretation of Narayana numbers as coefficients of weight polynomials enumerating bi-banded Dyck paths, which class of paths has arisen naturally in previous literature in a solution of the stationary state of the `TASEP' stochastic process.Comment: 10 pages, 5 figure

On minors of maximal determinant matrices

By an old result of Cohn (1965), a Hadamard matrix of order n has no proper Hadamard submatrices of order m > n/2. We generalise this result to maximal determinant submatrices of Hadamard matrices, and show that an interval of length asymptotically equal to n/2 is excluded from the allowable orders. We make a conjecture regarding a lower bound for sums of squares of minors of maximal determinant matrices, and give evidence in support of the conjecture. We give tables of the values taken by the minors of all maximal determinant matrices of orders up to and including 21 and make some observations on the data. Finally, we describe the algorithms that were used to compute the tables.Comment: 35 pages, 43 tables, added reference to Cohn in v

Maximal determinants and saturated D-optimal designs of orders 19 and 37

A saturated D-optimal design is a {+1,-1} square matrix of given order with maximal determinant. We search for saturated D-optimal designs of orders 19 and 37, and find that known matrices due to Smith, Cohn, Orrick and Solomon are optimal. For order 19 we find all inequivalent saturated D-optimal designs with maximal determinant, 2^30 x 7^2 x 17, and confirm that the three known designs comprise a complete set. For order 37 we prove that the maximal determinant is 2^39 x 3^36, and find a sample of inequivalent saturated D-optimal designs. Our method is an extension of that used by Orrick to resolve the previously smallest unknown order of 15; and by Chadjipantelis, Kounias and Moyssiadis to resolve orders 17 and 21. The method is a two-step computation which first searches for candidate Gram matrices and then attempts to decompose them. Using a similar method, we also find the complete spectrum of determinant values for {+1,-1} matrices of order 13.Comment: 28 pages, 4 figure

Probabilistic lower bounds on maximal determinants of binary matrices

Let ${\mathcal D}(n)$ be the maximal determinant for $n \times n$ $\{\pm 1\}$-matrices, and $\mathcal R(n) = {\mathcal D}(n)/n^{n/2}$ be the ratio of ${\mathcal D}(n)$ to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on ${\mathcal D}(n)$ and $\mathcal R(n)$ in terms of $d = n-h$, where $h$ is the order of a Hadamard matrix and $h$ is maximal subject to $h \le n$. For example, $\mathcal R(n) > (\pi e/2)^{-d/2}$ if $1 \le d \le 3$, and $\mathcal R(n) > (\pi e/2)^{-d/2}(1 - d^2(\pi/(2h))^{1/2})$ if $d > 3$. By a recent result of Livinskyi, $d^2/h^{1/2} \to 0$ as $n \to \infty$, so the second bound is close to $(\pi e/2)^{-d/2}$ for large $n$. Previous lower bounds tended to zero as $n \to \infty$ with $d$ fixed, except in the cases $d \in \{0,1\}$. For $d \ge 2$, our bounds are better for all sufficiently large $n$. If the Hadamard conjecture is true, then $d \le 3$, so the first bound above shows that $\mathcal R(n)$ is bounded below by a positive constant $(\pi e/2)^{-3/2} > 0.1133$.Comment: 17 pages, 2 tables, 24 references. Shorter version of arXiv:1402.6817v4. Typos corrected in v2 and v3, new Lemma 7 in v4, updated references in v5, added Remark 2.8 and a reference in v6, updated references in v

Hunger on the Richest Hill: A case study of linked historical and contemporary food insecurity in Butte, Montana

On the “Richest Hill on Earth” or Butte, Montana, a dilemma of food insecurity persists. This dilemma broadly consists of limited access to affordable, healthy, and appropriate nutrition for the city’s urban population. This thesis constructs a historically and contextually-informed understanding of food insecurity in Butte since the city’s establishment in 1864. Food insecurity research largely lacks place-specific consideration of historical events and processes which contribute to and reinforce the root factors contributing to food insecurity. Butte was once a thriving copper-mining boom town yet saw the decline of its primary industry in the late 20th century. In 2020, nearly one fifth of the population was living below the federal poverty line, and 19.5% of all residents were determined to be food insecure. In light of its industrial history and current socio-economic vulnerabilities, Butte represents a prime location for a case study of food insecurity in the post-industrial Rocky Mountain West. The first part of the study presents an investigation of the dynamic factors contributing to food insecurity through three main phases of Butte’s history: pre- and early settlement, industrial boom times, and post-industrial decline. The analysis reveals the persistence of food insecurity that influenced a portion of the lived experiences of Butte’s population, even before the loss of its primary industry of copper mining in the late-1900s. The process and outcomes of post-industrial decline which impacted the local food system and transformed the drivers of food insecurity are also examined. The second part of the study examines the contemporary situation, including the environmental and socio-economic factors contributing to food insecurity in Butte. Current efforts to address the challenges as well as a path forward into a more food-secure future are explored. Through a mixed-methods approach that combines qualitative key informant and oral history interviews and participant observation with quantitative GIS analysis, this research seeks to analyze and contextualize the historical chains of explanation that help to decode contemporary patterns and experiences of food insecurity in Butte, Montana. The findings suggest that efforts in local food infrastructure and community planning could support urban residents’ capacities in addressing their food needs

Some binomial sums involving absolute values

We consider several families of binomial sum identities whose definition involves the absolute value function. In particular, we consider centered double sums of the form $$S_{\alpha,\beta}(n) := \sum_{k,\;\ell}\binom{2n}{n+k}\binom{2n}{n+\ell} |k^\alpha-\ell^\alpha|^\beta,$$ obtaining new results in the cases $\alpha = 1, 2$. We show that there is a close connection between these double sums in the case $\alpha=1$ and the single centered binomial sums considered by Tuenter.Comment: 15 pages, 19 reference

Bounds on minors of binary matrices

We prove an upper bound on sums of squares of minors of {+1, -1} matrices. The bound is sharp for Hadamard matrices, a result due to de Launey and Levin (2009), but our proof is simpler. We give several corollaries relevant to minors of Hadamard matrices, and generalise a result of Turan on determinants of random {+1,-1} matrices.Comment: 9 pages, 1 table. Typo corrected in v2. Two references and Theorem 2 added in v