The spt-crank for overpartitions

Frank G. Garvan; Chris Jennings-Shaffer

Displaying similar documents to “The spt-crank for overpartitions”

Some new infinite families of congruences modulo 3 for overpartitions into odd parts

Ernest X. W. Xia (2016)

Colloquium Mathematicae

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Let $p ̅_{o} (n)$ denote the number of overpartitions of n in which only odd parts are used. Some congruences modulo 3 and powers of 2 for the function $p ̅_{o} (n)$ have been derived by Hirschhorn and Sellers, and Lovejoy and Osburn. In this paper, employing 2-dissections of certain quotients of theta functions due to Ramanujan, we prove some new infinite families of Ramanujan-type congruences for $p ̅_{o} (n)$ modulo 3. For example, we prove that for n, α ≥ 0, $p ̅_{o} (4^{α} (24 n + 17)) \equiv p ̅_{o} (4^{α} (24 n + 23)) \equiv 0 (m o d 3)$ .

New infinite families of Ramanujan-type congruences modulo 9 for overpartition pairs

Ernest X. W. Xia (2015)

Colloquium Mathematicae

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Let $\bar{p p} (n)$ denote the number of overpartition pairs of n. Bringmann and Lovejoy (2008) proved that for n ≥ 0, $\bar{p p} (3 n + 2) \equiv 0 (m o d 3)$ . They also proved that there are infinitely many Ramanujan-type congruences modulo every power of odd primes for $\bar{p p} (n)$ . Recently, Chen and Lin (2012) established some Ramanujan-type identities and explicit congruences for $\bar{p p} (n)$ . Furthermore, they also constructed infinite families of congruences for $\bar{p p} (n)$ modulo 3 and 5, and two congruence relations modulo 9. In this paper, we prove several...

Polynomial analogues of Ramanujan congruences for Han's hooklength formula

William J. Keith (2013)

Acta Arithmetica

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This article considers the eta power $\prod_{(1 - q^{k})}^{b - 1}$ . It is proved that the coefficients of $q^{n} / n!$ in this expression, as polynomials in b, exhibit equidistribution of the coefficients in the nonzero residue classes mod 5 when n = 5j+4. Other symmetries, as well as symmetries for other primes and prime powers, are proved, and some open questions are raised.

Integers not of the form $c (2^{a} + 2^{b}) + p^{α}$

Pingzhi Yuan (2004)

Acta Arithmetica

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Congruences for certain families of Apéry-like sequences

Zhi-Hong Sun (2022)

Czechoslovak Mathematical Journal

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We systematically investigate the expressions and congruences for both a one-parameter family ${G_{n} (x)}$ as well as a two-parameter family ${G_{n} (r, m)}$ of sequences.

Non-abelian $p$ -adic $L$ -functions and Eisenstein series of unitary groups – The CM method

Thanasis Bouganis (2014)

Annales de l’institut Fourier

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In this work we prove various cases of the so-called “torsion congruences” between abelian $p$ -adic $L$ -functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of $n$ variables and we obtain more explicit results...

On the integers not of the form $p + 2^{a} + 2^{b}$

Hao Pan (2011)

Acta Arithmetica

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Some congruences involving binomial coefficients

Hui-Qin Cao, Zhi-Wei Sun (2015)

Colloquium Mathematicae

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Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that $T_{p - 1} \equiv (p / 3) 3^{p - 1} (m o d p ²)$ , where the central trinomial coefficient Tₙ is the constant term in the expansion of $(1 + x + x^{- 1}) ⁿ$ . We also prove three congruences modulo p³ conjectured by Sun, one of which is $\sum_{k = 0}^{p - 1} (\binom{p - 1}{k}) (\binom{2 k}{k}) ({(- 1)}^{k} - {(- 3)}^{- k}) \equiv (p / 3) (3^{p - 1} - 1) (m o d p ³)$ . In addition, we get some new combinatorial identities.

On Alternatives of Polynomial Congruences

Mariusz Skałba (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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What should be assumed about the integral polynomials $f ₁ (x), . . ., f_{k} (x)$ in order that the solvability of the congruence $f ₁ (x) f ₂ (x) \dots f_{k} (x) \equiv 0 (m o d p)$ for sufficiently large primes p implies the solvability of the equation $f ₁ (x) f ₂ (x) \dots f_{k} (x) = 0$ in integers x? We provide some explicit characterizations for the cases when $f_{j} (x)$ are binomials or have cyclic splitting fields.

Linear congruences and a conjecture of Bibak

Chinnakonda Gnanamoorthy Karthick Babu, Ranjan Bera, Balasubramanian Sury (2024)

Czechoslovak Mathematical Journal

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We address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences $\sum_{i = 1}^{k} a_{i} x_{i} \equiv b (mod n)$ . In particular, we obtain explicit expressions for the number of solutions, where $x_{i}$ ’s are squares modulo $n$ . In addition, we obtain expressions for the number of solutions with order restrictions $x_{1} \geq \dots \geq x_{k}$ or with strict order restrictions $x_{1} > \dots > x_{k}$ in some special cases. In these results, the expressions for the number of solutions involve...

On the congruences $σ (n) \equiv a (m o d n)$ and $n \equiv a (m o d φ (n))$

Carl Pomerance (1975)

Acta Arithmetica

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Congruences for ${(A + \sqrt (A ² + m B ²))}^{(p - 1) / 2)}$ and ${(b + \sqrt (a ² + b ²))}^{(p - 1) / 4} (m o d p)$

Zhi-Hong Sun (2011)

Acta Arithmetica

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Some Parity Statistics in Integer Partitions

Aubrey Blecher, Toufik Mansour, Augustine O. Munagi (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study integer partitions with respect to the classical word statistics of levels and descents subject to prescribed parity conditions. For instance, a partition with summands $λ ₁ \geq \dots \geq λ_{k}$ may be enumerated according to descents $λ_{i} > λ_{i + 1}$ while tracking the individual parities of $λ_{i}$ and $λ_{i + 1}$ . There are two types of parity levels, E = E and O = O, and four types of parity-descents, E > E, E > O, O > E and O > O, where E and O represent arbitrary even and odd summands. We obtain functional equations...

Congruences and homomorphisms on $Ω$ -algebras

Elijah Eghosa Edeghagba, Branimir Šešelja, Andreja Tepavčević (2017)

Kybernetika

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The topic of the paper are $Ω$ -algebras, where $Ω$ is a complete lattice. In this research we deal with congruences and homomorphisms. An $Ω$ -algebra is a classical algebra which is not assumed to satisfy particular identities and it is equipped with an $Ω$ -valued equality instead of the ordinary one. Identities are satisfied as lattice theoretic formulas. We introduce $Ω$ -valued congruences, corresponding quotient $Ω$ -algebras and $Ω$ -homomorphisms and we investigate connections among these notions....

Arithmetic theory of harmonic numbers (II)

Zhi-Wei Sun, Li-Lu Zhao (2013)

Colloquium Mathematicae

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For k = 1,2,... let $H_{k}$ denote the harmonic number $\sum_{j = 1}^{k} 1 / j$ . In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have $\sum_{k = 1}^{p - 1} (H_{k}) / (k 2^{k}) \equiv 7 / 24 p B_{p - 3} (m o d p ²)$ , $\sum_{k = 1}^{p - 1} (H_{k, 2}) / (k 2^{k}) \equiv - 3 / 8 B_{p - 3} (m o d p)$ , and $\sum_{k = 1}^{p - 1} (H ²_{k, 2 n}) / (k^{2 n}) \equiv ((\binom{6 n + 1}{2 n - 1}) + n) / (6 n + 1) p B_{p - 1 - 6 n} (m o d p ²)$ for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_{k, m} : = \sum_{j = 1}^{k} 1 / (j^{m})$ .

Subspaces of $L_{p}$ , p > 2, determined by partitions and weights

Dale E. Alspach, Simei Tong (2003)

Studia Mathematica

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Many of the known complemented subspaces of $L_{p}$ have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well known complemented subspaces of $L_{p}$ . It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given...

A note on representing dowling geometries by partitions

František Matúš, Aner Ben-Efraim (2020)

Kybernetika

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We prove that a rank $\geq 3$ Dowling geometry of a group $H$ is partition representable if and only if $H$ is a Frobenius complement. This implies that Dowling group geometries are secret-sharing if and only if they are multilinearly representable.

$q$ -analogues of two supercongruences of Z.-W. Sun

Cheng-Yang Gu, Victor J. W. Guo (2020)

Czechoslovak Mathematical Journal

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We give several different $q$ -analogues of the following two congruences of Z.-W. Sun: $\sum_{k = 0}^{(p^{r} - 1) / 2} \frac{1}{8^{k}} (\binom{2 k}{k}) \equiv (\frac{2}{p^{r}}) (mod p^{2}) and \sum_{k = 0}^{(p^{r} - 1) / 2} \frac{1}{16^{k}} (\binom{2 k}{k}) \equiv (\frac{3}{p^{r}}) (mod p^{2}),$ where $p$ is an odd prime, $r$ is a positive integer, and $(\frac{m}{n})$ is the Jacobi symbol. The proofs of them require the use of some curious $q$ -series identities, two of which are related to Franklin’s involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012.

On the $q$ -Pell sequences and sums of tails

Alexander E. Patkowski (2017)

Czechoslovak Mathematical Journal

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We examine the $q$ -Pell sequences and their applications to weighted partition theorems and values of $L$ -functions. We also put them into perspective with sums of tails. It is shown that there is a deeper structure between two-variable generalizations of Rogers-Ramanujan identities and sums of tails, by offering examples of an operator equation considered in a paper published by the present author. The paper starts with the classical example offered by Ramanujan and studied by previous...