Higher Order SPT-Functions

Andrews' spt-function can be written as the difference between the second symmetrized crank and rank moment functions. Using the machinery of Bailey pairs a combinatorial interpretation is given for the difference between higher order symmetrized crank and rank moment functions. This implies an inequality between crank and rank moments that was only know previously for sufficiently large n and fixed order. This combinatorial interpretation is in terms of a weighted sum of partitions. A number of congruences for higher order spt-functions are derived.Comment: 21 pages (previous version was 19 pages), added reference to Andrews and Rose's recent paper, MacMahon's paper and OEIS, changed some wordin

New symmetries for Dyson's rank function

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of $R(\zeta_p,q)$, where $R(z,q)$ is the two-variable generating function of Dyson's rank function and $\zeta_p$ is a primitive $p$-th root of unity. In his lost notebook Ramanujan gives the $5$-dissection of $R(\zeta_5,q)$. This result is related to Dyson's famous rank conjecture which was proved by Atkin and Swinnerton-Dyer. In 2016 the first author showed that there is an analogous result for the $p$-dissection of $R(\zeta_p,q)$ when $p$ is any prime greater than $3$, by extending work of Bringmann and Ono, and Ahlgren and Treneer. It was also shown how the group $\Gamma_1(p)$ acts on the elements of the $p$-dissection of $R(\zeta_p,q)$. We extend this to the group $\Gamma_0(p)$, thus revealing new and surprising symmetries for Dyson's rank function.Comment: 51 page

Anthropology and GIS: Temporal and Spatial Distribution of the Philippine Negrito Groups

The Philippine negrito groups comprise a diverse group of populations speaking over 30 different languages, who are spread all over the archipelago, mostly in marginal areas of Luzon Island in the north, the central Visayas islands, and Mindanao in the south. They exhibit physical characteristics that are different from more than 100 Philippine ethnolinguistic groups that are categorized as non-negritos. Given their numbers, it is not surprising that Philippine negritos make up a major category in a number of general ethnographic maps produced since the nineteenth century. Reports from various ethnological surveys during this period, however, have further enriched our understanding regarding the extent and distribution of negrito populations. Using the data contained in these reports, it is possible to plot and create a map showing the historical locations and distribution of negrito groups. Using geographic information systems (GIS), the location and distribution of negrito groups at any given time can be overlaid on historical or current maps. In the present study, a GIS layer was compiled and extracted from the 2000 Philippine Census of population at the village level and overlaid on existing maps of the Philippines. The maps that were generated from this project will complement ongoing anthropological and genetic studies of negrito groups that inhabit different locations within the Philippine archipelago