Let p be an odd prime. The bar partitions with sign and p-bar-core partitions with sign respectively label the spin characters and p-defect zero spin characters of the double cover of the symmetric group, and by restriction, those of the alternating group. The generating functions for these objects have been determined by J. Olsson. We study these functions from an arithmetic perspective, using classical analytic tools and elementary generating function manipulation to obtain many Ramanujan-like congruences.
Bibliographical noteFunding Information:
The first author thanks George Andrews for supporting his visit to Pennsylvania State University in November 2014 where this research began. He also thanks Henry Cohn and the Massachusetts Institute of Technology where he was a visitor in March and April 2015. The first author was supported by grant PSC-CUNY TRADA-46-493 and a 2014–2015 Faculty Fellowship Leave from York College, City University of New York. Both authors would like to thank Shi-Chao Chen, Jorn Olsson, and the anonymous referee for their helpful comments and suggestions.
- Generating function