Combinatorial Properties of Rogers-Ramanujan-Type Identities Arising from Hall-Littlewood Polynomials

Here we consider the $q$-series coming from the Hall-Littlewood polynomials, \begin{equation*} R_\nu(a,b;q)=\sum_{\substack{\lambda \\[1pt] \lambda_1\leq a}} q^{c|\lambda|} P_{2\lambda}\big(1,q,q^2,\dots;q^{2b+d}\big). \end{equation*} These series were defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congruence \begin{equation*} p(5n+4)\equiv0\pmod{5}. \end{equation*}Comment: 16 pages v2: Minor changes included, to appear in Annals of Combinatoric

Large Sums of Fourier Coefficients of Cusp Forms

Let $N$ be a fixed positive integer, and let $f\in S_k(N)$ be a primitive cusp form given by the Fourier expansion $f(z)=\sum_{n=1}^{\infty} \lambda_f(n)n^{\frac{k-1}{2}}e(nz)$. We consider the partial sum $S(x,f)=\sum_{n\leq x}\lambda_f(x)$. It is conjectured that $S(x,f)=o(x\log x)$ in the range $x\geq k^{\epsilon}$. Lamzouri proved in arXiv:1703.10582 [math.NT] that this is true under the assumption of the Generalized Riemann Hypothesis (GRH) for $L(s,f)$. In this paper, we prove that this conjecture holds under a weaker assumption than GRH. In particular, we prove that given $\epsilon>(\log k)^{-\frac{1}{8}}$ and $1\leq T\leq (\log k)^{\frac{1}{200}}$, we have $S(x,f)\ll \frac{x\log x}{T}$ in the range $x\geq k^{\epsilon}$ provided that $L(s,f)$ has no more than $\epsilon^2\log k/5000$ zeros in the region $\left\{s\,:\, \Re(s)\geq \frac34, \, |\Im(s)-\phi| \leq \frac14\right\}$ for every real number $\phi$ with $|\phi|\leq T$.Comment: 14 page

Pangolins in global camera trap data: Implications for ecological monitoring

Despite being heavily exploited, pangolins (Pholidota: Manidae) have been subject to limited research, resulting in a lack of reliable population estimates and standardised survey methods for the eight extant species. Camera trapping represents a unique opportunity for broad-scale collaborative species monitoring due to its largely non-discriminatory nature, which creates considerable volumes of data on a relatively wide range of species. This has the potential to shed light on the ecology of rare, cryptic and understudied taxa, with implications for conservation decision-making. We undertook a global analysis of available pangolin data from camera trapping studies across their range in Africa and Asia. Our aims were (1) to assess the utility of existing camera trapping efforts as a method for monitoring pangolin populations, and (2) to gain insights into the distribution and ecology of pangolins. We analysed data collated from 103 camera trap surveys undertaken across 22 countries that fell within the range of seven of the eight pangolin species, which yielded more than half a million trap nights and 888 pangolin encounters. We ran occupancy analyses on three species (Sunda pangolin Manis javanica, white-bellied pangolin Phataginus tricuspis and giant pangolin Smutsia gigantea). Detection probabilities varied with forest cover and levels of human influence for P. tricuspis, but were low (<0.05) for all species. Occupancy was associated with distance from rivers for M. javanica and S. gigantea, elevation for P. tricuspis and S. gigantea, forest cover for P. tricuspis and protected area status for M. javanica and P. tricuspis. We conclude that camera traps are suitable for the detection of pangolins and large-scale assessment of their distributions. However, the trapping effort required to monitor populations at any given study site using existing methods appears prohibitively high. This may change in the future should anticipated technological and methodological advances in camera trapping facilitate greater sampling efforts and/or higher probabilities of detection. In particular, targeted camera placement for pangolins is likely to make pangolin monitoring more feasible with moderate sampling efforts

Pangolins in Global Camera Trap Data: Implications for Ecological Monitoring

Despite being heavily exploited, pangolins (Pholidota: Manidae) have been subject to limited research, resulting in a lack of reliable population estimates and standardised survey methods for the eight extant species. Camera trapping represents a unique opportunity for broad-scale collaborative species monitoring due to its largely non-discriminatory nature, which creates considerable volumes of data on a relatively wide range of species. This has the potential to shed light on the ecology of rare, cryptic and understudied taxa, with implications for conservation decision-making. We undertook a global analysis of available pangolin data from camera trapping studies across their range in Africa and Asia. Our aims were (1) to assess the utility of existing camera trapping efforts as a method for monitoring pangolin populations, and (2) to gain insights into the distribution and ecology of pangolins. We analysed data collated from 103 camera trap surveys undertaken across 22 countries that fell within the range of seven of the eight pangolin species, which yielded more than half a million trap nights and 888 pangolin encounters. We ran occupancy analyses on three species (Sunda pangolin Manis javanica, white-bellied pangolin Phataginus tricuspis and giant pangolin Smutsia gigantea). Detection probabilities varied with forest cover and levels of human influence for P. tricuspis, but were low (M. javanica and S. gigantea, elevation for P. tricuspis and S. gigantea, forest cover for P. tricuspis and protected area status for M. javanica and P. tricuspis. We conclude that camera traps are suitable for the detection of pangolins and large-scale assessment of their distributions. However, the trapping effort required to monitor populations at any given study site using existing methods appears prohibitively high. This may change in the future should anticipated technological and methodological advances in camera trapping facilitate greater sampling efforts and/or higher probabilities of detection. In particular, targeted camera placement for pangolins is likely to make pangolin monitoring more feasible with moderate sampling efforts

A lattice model for super LLT polynomials

We introduce a solvable lattice model for supersymmetric LLT polynomials, also known as super LLT polynomials, based upon particle interactions in super $n$-ribbon tableaux. Using related Heisenberg operators on a Fock space, we prove Cauchy and Pieri identities for super LLT polynomials, simultaneously generalizing the Cauchy, dual Cauchy, and Pieri identities for LLT polynomials. Lastly, we construct a solvable semi-infinite Cauchy lattice model with a surprising Yang-Baxter equation and examine its connections to the Pieri and Cauchy identities.Mathematics Subject Classifications: 05E05, 82B20, 05E10Keywords: Lattice models, super LLT polynomial

A lattice model for super LLT polynomials

We introduce a solvable lattice model for supersymmetric LLT polynomials, also known as super LLT polynomials, based upon particle interactions in super $n$-ribbon tableaux. Using related Heisenberg operators on a Fock space, we prove Cauchy and Pieri identities for super LLT polynomials, simultaneously generalizing the Cauchy, dual Cauchy, and Pieri identities for LLT polynomials. Lastly, we construct a solvable semi-infinite Cauchy lattice model with a surprising Yang-Baxter equation and examine its connections to the Pieri and Cauchy identities.Mathematics Subject Classifications: 05E05, 82B20, 05E10Keywords: Lattice models, super LLT polynomial