Ranks of overpartitions modulo 4 and 8

Cui, Su-Ping; Gu, Nancy S. S.; Su, Chen-Yang

doi:10.1142/s1793042120501171

Cited by 4 publications

(6 citation statements)

References 17 publications

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“…The inequalities from Theorems 1, 2 and 4 also shed light on the signs of the coefficients of the rank differences found for ∈ {3, 4, 5, 7, 8} in [14], [17] and [21]. A study of rank inequalities based on q-series expansions was done in [18] for ∈ {6, 10}, but this was only possible in the case of some fairly simple expressions for which it is not difficult to conclude, say, that the coefficients are all positive (see, e.g., the proof of [18,Theorem 1.4]).…”

Section: Remarkmentioning

confidence: 78%

“…By computing rank differences for = 4 and = 8, Cui, Gu and Su [14] established very recently 1 a few identities and inequalities, see Theorems 1.2-1.5 in [14], most of which also follow from Theorem 1 and the next result. However, while Theorem 1.5 in [14] gives several inequalities that hold modulo 8 for n in certain residue classes, it does not capture the full behavior of the inequalities. This is answered by Theorem 1 of the current paper, applied to the case c = 8.…”

Section: Resultsmentioning

confidence: 97%

“…Lovejoy and Osburn [21] found formulas for the rank differences N (s, , n + d) − N (t, , n +d) in terms of modular functions and generalized Lambert series for = 3 and = 5, whereas Jennings-Shaffer [17] computed the rank differences for = 7 using the result of Bringmann and Lovejoy [9] that the overpartition rank function is the holomorphic part of a harmonic Maass form. More recently, Cui, Gu and Su [14] computed the rank differences for = 4 and = 8, obtaining a few identities and inequalities that were also independently formulated and proven in the current paper, while Ji, Zhang and Zhao [18], and Wei and Zhang [23] computed the rank differences for = 6 and = 10 by relating them to Ramanujan's third and tenth order mock theta functions. They also established a few inequalities and left several others as conjectures, all of which were subsequently proven, with different methods, by the author [10].…”

Section: Motivationmentioning

confidence: 87%

“…However, this is also the reason for which other inequalities cannot be proven with that approach. For ∈ {3, 4, 5, 7, 8}, the rank differences are written as sums of various quotients of infinite products, and the sign of the coefficients of these rather complicated q-series expansions (see Theorems 1.3-1.4 of [14], Theorem 1.1 of [17], or Theorems 1.1-1.2 in [21]) can generally not be guessed a priori.…”

Section: Remarkmentioning

confidence: 99%

“…The author only became aware of[14] after the completion of the present paper, and shortly before its submission. Any overlapping results are therefore to be seen as independent of one another.…”

mentioning

confidence: 99%

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Inequalities between overpartition ranks for all moduli

Ciolan

2021

Ramanujan J

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In this paper we give a full description of the inequalities that can occur between overpartition ranks modulo $$ c\ge 2. $$ c ≥ 2 . If $$ \overline{N}(a,c,n) $$ N ¯ ( a , c , n ) denotes the number of overpartitions of n with rank congruent to a modulo c, we prove that for any $$ c\ge 7 $$ c ≥ 7 and $$ 0\le a<b\le \left\lfloor \frac{c}{2}\right\rfloor $$ 0 ≤ a < b ≤ c 2 we have $$ \overline{N}(a,c,n)>\overline{N}(b,c,n) $$ N ¯ ( a , c , n ) > N ¯ ( b , c , n ) for n large enough. That the sign of the rank differences $$ \overline{N}(a,c,n)-\overline{N}(b,c,n) $$ N ¯ ( a , c , n ) - N ¯ ( b , c , n ) depends on the residue class of n modulo c in the case of small moduli, such as $$ c=6, $$ c = 6 , is known due to the work of Ji et al. (J Number Theory 184:235–269, 2018) and Ciolan (Int J Number Theory 16(1):121–143, 2020). We show that the same behavior holds for $$ c\in \{2,3, 4,5\}. $$ c ∈ { 2 , 3 , 4 , 5 } .

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