Atul.DixitAffiliation: Indian Institute of Technology Gandhinagar Title Of Talk: Non-Rascoe partitions and a rank parity function associated with the Rogers-Ramanujan partitions
URLS: Abstract: We study an interesting analogue of Ramanujan’s celebrated quantum modular form $\sigma(q)$. It is the generating function of the excess number of Rogers-Ramanujan partitions with odd rank over those with even rank. Using combinatorial and analytical techniques, we show that this generating function is closely connected with an interesting class of restricted partitions termed here as $\textbf{non-Rascoe partitions}$. These are the partitions into distinct parts where the number of parts is not a part. We derive several arithmetic properties of the number of such partitions via this connection and conjecture an interesting mod 4 congruence. Generalizations of most of these results in a parameter $\ell$ are obtained in conjunction with the generalized Rogers–Ramanujan partitions associated with some results of Garrett, Ismail and Stanton. Using a generalized modular relation occurring on page 27 of Ramanujan’s Lost Notebook, we obtain a congruence involving the number of non-Rascoe partitions and coefficients of certain tenth order mock theta functions. This is joint work with Gaurav Kumar and Aviral Srivastava.
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Last update made Tue Mar 10 21:28:03 CDT 2026.
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