On the rank and the crank modulo 4 and 8

Lewis, Richard; Santa-Gadea, Nicolas

doi:10.1090/S0002-9947-1994-1136545-0

On the rank and the crank modulo $4$ and $8$
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by Richard Lewis and Nicolas Santa-Gadea PDF

Trans. Amer. Math. Soc. 341 (1994), 449-465 Request permission

Abstract:

In this paper we prove some identities, conjectured by Lewis, for the rank and crank of partitions concerning the modulo $4$ and $8$. These identities are similar to Dyson’s identities for the rank modulo $5$ and $7$ which give a combinatorial interpretation to Ramanujan’s partition congruences. For this, we use multisection of series and some of the results that Watson established for the third order mock theta functions.

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