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

Belmont, Eva; Lee, Holden; Musat, Alexandra; Trebat-Leder, Sarah

doi:10.1007/s00605-013-0586-y

Cited by 9 publications

(9 citation statements)

References 19 publications

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“…Remark 3. Recent work of Belmont, Lee, Musat, and Trebat-Leder [9] adapts Theorems 1.1, 1.2, and 1.4, to Andrews' smallest parts partition function, spt(n), and to the rth power partition function, p r (n).…”

Section: Reformulation Of Main Resultsmentioning

confidence: 99%

The partition function modulo prime powers

Boylan

Webb

2012

Trans. Amer. Math. Soc.

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Let ≥ 5 be prime, let m ≥ 1 be an integer, and let p(n) denote the partition function. Folsom, Kent, and Ono recently proved that there exists a positive integer b (m) of size roughly m 2 such that the module formed from the Z/ m Z-span of generating functions for p b n+1 24 with odd b ≥ b (m) has finite rank. The same result holds with "odd" b replaced by "even" b. Furthermore, they proved an upper bound on the ranks of these modules. This upper bound is independent of m; it is +12 24. In this paper, we prove, with a mild condition on , that b (m) ≤ 2m − 1. Our bound is sharp in all computed cases with ≥ 29. To deduce it, we prove structure theorems for the relevant Z/ m Z-modules of modular forms. This work sheds further light on a question of Mazur posed to Folsom, Kent, and Ono.

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Section: Reformulation Of Main Resultsmentioning

confidence: 99%

The partition function modulo prime powers

Boylan

Webb

2012

Trans. Amer. Math. Soc.

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“…If we want to obtain the accurate value of h(n), we can use the recursion formula (13) and write a program based on it, while sometimes we need to know the estimation value in a program for technique reason especially when we use a general programming language.…”

Section: Discussionmentioning

confidence: 99%

“…We can easily obtain the solutions of equation ( 2) by hand when n < 13. By equation (13), we can obtain the number h(n) of solutions of equation ( 2) by writing a small program in some Computer Algebra System (CAS) software such as "maple," "maxima," and "axiom" or some other software likewise (be aware of that 0 is not a valid index value in some software such like maple).…”

Section: Some Formulae For H(n)mentioning

confidence: 99%

“…e recursion formula equation (13) for h(n) is not convenient in practical for a lot of people who do not want to write programs. Sometimes we need the approximation value, such as the cases mentioned in [1], so an estimation formula is necessary.…”

Section: The Estimation Of H(n)mentioning

confidence: 99%

“…ere are also a lot of literatures on the number of some types of restricted partitions of n (such as [6][7][8][9][10][11][12][13]) or on the congruence properties of (restricted) partition function (such as [14][15][16][17][18][19][20][21]).…”

Section: Introductionmentioning

confidence: 99%

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On the Number of Conjugate Classes of Derangements

Cheng²,

Liu

2021

Journal of Mathematics

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The number of conjugate classes of derangements of order n is the same as the number h n of the restricted partitions with every portion greater than 1. It is also equal to the number of isotopy classes of 2 × n Latin rectangles. Sometimes the exact value is necessary, while sometimes we need the approximation value. In this paper, a recursion formula of h n will be obtained and also will some elementary approximation formulae with high accuracy for h n be presented. Although we may obtain the value of h n in some computer algebra system, it is still meaningful to find an efficient way to calculate the approximate value, especially in engineering, since most people are familiar with neither programming nor CAS software. This paper is mainly for the readers who need a simple and practical formula to obtain the approximate value (without writing a program) with more accuracy, such as to compute the value in a pocket science calculator without programming function. Some methods used here can also be applied to find the fitting functions for some types of data obtained in experiments.

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Congruences of multipartition functions modulo powers of primes

Chen

Hou

et al. 2013

Ramanujan J

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Let p r (n) denote the number of r-component multipartitions of n, and let S γ,λ be the space spanned by η(24z) γ φ(24z), where η(z) is the Dedekind's eta function and φ(z) is a holomorphic modular form in M λ (SL 2 (Z)). In this paper, we show that the generating function of p r ( m k n+r 24 ) with respect to n is congruent to a function in the space S γ,λ modulo m k . As special cases, this relation leads to many well known congruences including the Ramanujan congruences of p(n) modulo 5, 7, 11 and Gandhi's congruences of p 2 (n) modulo 5 and p 8 (n) modulo 11. Furthermore, using the invariance property of S γ,λ under the Hecke operator T ℓ 2 , we obtain two classes of congruences pertaining to the m k -adic property of p r (n).

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

Cited by 9 publications

References 19 publications

The partition function modulo prime powers

The partition function modulo prime powers

On the Number of Conjugate Classes of Derangements

Congruences of multipartition functions modulo powers of primes

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