TY - JOUR

T1 - On ranks and cranks of partitions modulo 4 and 8

AU - Mortenson, Eric T.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Denote by p(n) the number of partitions of n and by N(a,M;n) the number of partitions of n with rank congruent to a modulo M. By considering the deviation D(a,M):=∑n=0∞(N(a,M;n)−[Formula presented])qn, we give new proofs of recent results of Andrews, Berndt, Chan, Kim and Malik on mock theta functions and ranks of partitions. By considering deviations of cranks, we give new proofs of Lewis and Santa-Gadea's rank-crank identities. We revisit ranks and cranks modulus M=5 and 7, with our results on cranks appearing to be new. We also demonstrate how deviations of ranks and cranks resolve Lewis’ long-standing conjectures on identities and inequalities for rank-crank differences of modulus M=8.

AB - Denote by p(n) the number of partitions of n and by N(a,M;n) the number of partitions of n with rank congruent to a modulo M. By considering the deviation D(a,M):=∑n=0∞(N(a,M;n)−[Formula presented])qn, we give new proofs of recent results of Andrews, Berndt, Chan, Kim and Malik on mock theta functions and ranks of partitions. By considering deviations of cranks, we give new proofs of Lewis and Santa-Gadea's rank-crank identities. We revisit ranks and cranks modulus M=5 and 7, with our results on cranks appearing to be new. We also demonstrate how deviations of ranks and cranks resolve Lewis’ long-standing conjectures on identities and inequalities for rank-crank differences of modulus M=8.

KW - Crank

KW - Mock theta functions

KW - Partitions

KW - Rank

UR - http://www.scopus.com/inward/record.url?scp=85050666416&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2018.07.009

DO - 10.1016/j.jcta.2018.07.009

M3 - Article

AN - SCOPUS:85050666416

VL - 161

SP - 51

EP - 80

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

ER -