74 research outputs found
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 , where is the two-variable
generating function of Dyson's rank function and is a primitive
-th root of unity. In his lost notebook Ramanujan gives the -dissection
of . 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 -dissection of when
is any prime greater than , by extending work of Bringmann and Ono, and
Ahlgren and Treneer. It was also shown how the group acts on the
elements of the -dissection of . We extend this to the group
, 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
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