# On ranks and cranks of partitions modulo 4 and 8

@article{Mortenson2019OnRA,
title={On ranks and cranks of partitions modulo 4 and 8},
author={Eric T. Mortenson},
journal={J. Comb. Theory, Ser. A},
year={2019},
volume={161},
pages={51-80}
}
A mock theta function identity related to the partition rank modulo 3 and 9
• Mathematics, Computer Science
• 2020
A new mock theta function identity related to the partition rank modulo 3 and 9 is proved and the dissection of the rank generating function modulo [Formula: see text] is obtained.
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The definitions of the rank and crank for overpartitions were given by Bringmann, Lovejoy and Osburn. Let $$\overline{N}(s,l;n)$$ N ¯ ( s , l ; n ) (resp. $$\overline{M}(s,l;n)$$ M ¯ ( s , l ; n ) ,
FOUR IDENTITIES FOR THIRD ORDER MOCK THETA FUNCTIONS
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Nagoya Mathematical Journal
• 2018
In 2005, using a famous lemma of Atkin and Swinnerton-Dyer (Some properties of partitions, Proc. Lond. Math. Soc. (3) 4 (1954), 84–106), Yesilyurt (Four identities related to third order mock theta
Four identities related to third-order mock theta functions
• Mathematics
The Ramanujan Journal
• 2020
Ramanujan presented four identities for third order mock theta functions in his Lost Notebook. In 2005, with the aid of complex analysis, Yesilyurt first proved these four identities. Recently,
Recent Work on Mock Theta Functions
• Mathematics
• 2018
The work of Ramanujan has had a wide ranging impact in many branches of mathematics. Among many fields of research influenced by Ramanujan, few are as currently vibratingly active as the area of mock
On three identities of Ramanujan
In this paper, we represent the generating function of the rank function as a summation of four parts—a constant, two Lambert series and a product. Applying it to three of Ramanujan’s identities we
Identities and inequalities for the $$M_2$$-rank of partitions without repeated odd parts modulo 8
• Mathematics
The Ramanujan Journal
• 2021
In 2002, Berkovich and Garvan introduced the $$M_2$$ -rank of partitions without repeated odd parts. Let $$N_2(a, M, n)$$ denote the number of partitions of n without repeated odd parts in which
Tenth Order Mock Theta Functions: Part III, Identities for χ 10 ( q ), X 10 ( q )
• Mathematics
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The previous chapter provided an account of S. Zwegers’ ingenious proofs of the first four identities that appear on page 9 of Ramanujan’s Lost Notebook [232]. Identities (5) and (6) have not yielded

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Abstract Let N(i,m;n) be the number of partitions of n with rank (Dyson) congruent to i (mod m) and let M(j,m;n) be the number of partitions of n with crank (Andrews, Garvan) congruent to j (mod m).
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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. We find and prove a general formula for Dyson’s ranks by considering
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holds. He was thus led to conjecture the existence of some other partition statistic (which he called the crank); this unknown statistic should provide a combinatorial interpretation of ^-p(lln + 6)
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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
FOUR IDENTITIES FOR THIRD ORDER MOCK THETA FUNCTIONS
• Mathematics
Nagoya Mathematical Journal
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In 2005, using a famous lemma of Atkin and Swinnerton-Dyer (Some properties of partitions, Proc. Lond. Math. Soc. (3) 4 (1954), 84–106), Yesilyurt (Four identities related to third order mock theta
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In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge, to examine the papers of the late G.N. Watson. Among these papers, Andrews
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By introducing a dual notion between partial theta functions and Appell–Lerch sums, we find and prove a formula which expresses Hecke‐type double sums in terms of Appell–Lerch sums. Not only does our
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