ℓ-Adic properties of the partition function

Folsom, Amanda; Kent, Zachary A.; Ono, Ken

doi:10.1016/j.aim.2011.11.013

Cited by 27 publications

(31 citation statements)

References 22 publications

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“…Finally, as a direct corollary of Theorem 7.1 in [8], in the case r (r) = 1 we have that the forms P (r, b; 24z) (mod m ) converge to Hecke eigenforms as b, m → ∞. This is the analogue of Theorem 1.3 in [8].…”

Section: Remarkssupporting

confidence: 53%

“…Ramanujan congruences such as p (5 m n + δ 5 (m)) ≡ 0 (mod 5 m ), where 24δ 5 (m) ≡ 1 (mod 5 m ), have been known for quite some time. Using the theory of -adic modular forms, Folsom, Kent, and Ono established a general framework for partition congruences in [8], which not only explains this phenomenon but also allows them to show additional congruences such as p(13 4 n + 27371) ≡ 45p(13 2 n + 162) (mod 13 2 ).…”

Section: Introductionmentioning

confidence: 99%

“…To state our results, we define a sequence of generating functions related to p r (n) which are analogous to those that appear in [8] and [6],…”

Section: Introductionmentioning

confidence: 99%

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$$\ell $$ ℓ -Adic properties of partition functions

et al. 2013

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Folsom, Kent, and Ono used the theory of modular forms modulo to establish remarkable "self-similarity" properties of the partition function and give an overarching explanation of many partition congruences. We generalize their work to analyze powers p r of the partition function as well as Andrews's spt-function. By showing that certain generating functions reside in a small space made up of reductions of modular forms, we set up a general framework for congruences for p r and spt on arithmetic progressions of the form m n + δ modulo powers of . Our work gives a conceptual explanation of the exceptional congruences of p r observed by Boylan, as well as striking congruences of spt modulo 5, 7, and 13 recently discovered by Andrews and Garvan.

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Section: Remarkssupporting

confidence: 53%

Section: Introductionmentioning

confidence: 99%

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$$\ell $$ ℓ -Adic properties of partition functions

et al. 2013

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“…A number of congruences satisfied by k-colored partitions have been found (see [2,3,6,10,14,16], to name a few). For example, Atkin [3] proved the following infinite families of congruences modulo powers of prime.…”

Section: Final Remarksmentioning

confidence: 99%

Congruences modulo powers of 5 for k-colored partitions

Tang

2018

Journal of Number Theory

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Abstract. Let p −k (n) enumerate the number of k-colored partitions of n. In this paper, we establish some infinite families of congruences modulo 25 for k-colored partitions. Furthermore, we prove some infinite families of Ramanujan-type congruences modulo powers of 5 for p −k (n) with k = 2, 6, and 7. For example, for all integers n ≥ 0 and α ≥ 1, we prove thatand p −2 5 2α n + 11 × 5 2α + 1 12 ≡ 0 (mod 5 α+1 ).

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“…More recently in 2012, Folsom, Kent, and Ono [6] were studying the -adic behavior of p(n) and discovered that it displays fractal-like behavior, similar to the Mandelbrot set's behavior of self-similarity. This means that there exist infinitely many Ramanujan congruences with a prime modulus ≥ 5.…”

Section: Applications Of Generating Functions To Partitionsmentioning

confidence: 99%

Integer Partitions and Why Counting Them is Hard

Ortíz¹

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ℓ-Adic properties of the partition function

Cited by 27 publications

References 22 publications

$$\ell $$ ℓ -Adic properties of partition functions

$$\ell $$ ℓ -Adic properties of partition functions

Congruences modulo powers of 5 for k-colored partitions

Integer Partitions and Why Counting Them is Hard

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