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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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.

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...

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...

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...

On a magnetic characterization of spectral minimal partitions

Bernard Helffer, Thomas Hoffmann-Ostenhof (2013)

Journal of the European Mathematical Society

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Given a bounded open set $Ω$ in $ℝ^{n}$ (or in a Riemannian manifold) and a partition of $Ω$ by $k$ open sets $D_{j}$ , we consider the quantity ${𝚖𝚊𝚡}_{j} λ (D_{j})$ where $λ (D_{j})$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_{j}$ . If we denote by $ℒ_{k} (Ω)$ the infimum over all the $k$ -partitions of ${𝚖𝚊𝚡}_{j} λ (D_{j})$ , a minimal $k$ -partition is then a partition which realizes the infimum. When $k = 2$ , we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$ ’s is non trivial and quite interesting. In this...

On square classes in generalized Fibonacci sequences

Zafer Şiar, Refik Keskin (2016)

Acta Arithmetica

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Let P and Q be nonzero integers. The generalized Fibonacci and Lucas sequences are defined respectively as follows: U₀ = 0, U₁ = 1, V₀ = 2, V₁ = P and $U_{n + 1} = P U ₙ + Q U_{n - 1}$ , $V_{n + 1} = P V ₙ + Q V_{n - 1}$ for n ≥ 1. In this paper, when w ∈ 1,2,3,6, for all odd relatively prime values of P and Q such that P ≥ 1 and P² + 4Q > 0, we determine all n and m satisfying the equation Uₙ = wUₘx². In particular, when k|P and k > 1, we solve the equations Uₙ = kx² and Uₙ = 2kx². As a result, we determine all n such that Uₙ = 6x². ...

Linear combinations of partitions of unity with restricted supports

Christian Richter (2002)

Studia Mathematica

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Given a locally finite open covering of a normal space X and a Hausdorff topological vector space E, we characterize all continuous functions f: X → E which admit a representation $f = \sum_{C \in} a_{C} φ_{C}$ with $a_{C} \in E$ and a partition of unity $φ_{C} : C \in$ subordinate to . As an application, we determine the class of all functions f ∈ C(||) on the underlying space || of a Euclidean complex such that, for each polytope P ∈ , the restriction ${f |}_{P}$ attains its extrema at vertices of P. Finally, a class of extremal functions on the...

Ramsey partitions and proximity data structures

Manor Mendel, Assaf Naor (2007)

Journal of the European Mathematical Society

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This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion.We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (also known as the non-linear version of the isomorphic Dvoretzky...

On the lattice of congruences on inverse semirings

Anwesha Bhuniya, Anjan Kumar Bhuniya (2008)

Discussiones Mathematicae - General Algebra and Applications

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Let S be a semiring whose additive reduct (S,+) is an inverse semigroup. The relations θ and k, induced by tr and ker (resp.), are congruences on the lattice C(S) of all congruences on S. For ρ ∈ C(S), we have introduced four congruences $ρ_{m i n}, ρ_{m a x}, ρ^{m i n}$ and $ρ^{m a x}$ on S and showed that $ρ θ = [ρ_{m i n}, ρ_{m a x}]$ and $ρ κ = [ρ^{m i n}, ρ^{m a x}]$ . Different properties of ρθ and ρκ have been considered here. A congruence ρ on S is a Clifford congruence if and only if $ρ_{m a x}$ is a distributive lattice congruence and $ρ^{m a x}$ is a skew-ring congruence on S. If η (σ) is the...

Extension of point-finite partitions of unity

Haruto Ohta, Kaori Yamazaki (2006)

Fundamenta Mathematicae

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A subspace A of a topological space X is said to be $P^{γ}$ -embedded ( $P^{γ}$ (point-finite)-embedded) in X if every (point-finite) partition of unity α on A with |α| ≤ γ extends to a (point-finite) partition of unity on X. The main results are: (Theorem A) A subspace A of X is $P^{γ}$ (point-finite)-embedded in X iff it is $P^{γ}$ -embedded and every countable intersection B of cozero-sets in X with B ∩ A = ∅ can be separated from A by a cozero-set in X. (Theorem B) The product A × [0,1] is $P^{γ}$ (point-finite)-embedded...

On simple partitions of ${[κ]}^{κ}$

David Asperó (2003)

Fundamenta Mathematicae

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For every uncountable regular cardinal κ, every κ-Borel partition of the space of all members of ${[κ]}^{κ}$ whose enumerating function does not have fixed points has a homogeneous club.

Pressure and recurrence

Véronique Maume-Deschamps, Bernard Schmitt, Mariusz Urbański, Anna Zdunik (2003)

Fundamenta Mathematicae

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We deal with a subshift of finite type and an equilibrium state μ for a Hölder continuous function. Let αⁿ be the partition into cylinders of length n. We compute (in particular we show the existence of the limit) $l i m_{n \to \infty} n^{- 1} l o g \sum_{j = 0}^{τ ₙ (x)} μ (α ⁿ (T^{j} (x)))$ , where $α ⁿ (T^{j} (x))$ is the element of the partition containing $T^{j} (x)$ and τₙ(x) is the return time of the trajectory of x to the cylinder αⁿ(x).

Some q-supercongruences for truncated basic hypergeometric series

Victor J. W. Guo, Jiang Zeng (2015)

Acta Arithmetica

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For any odd prime p we obtain q-analogues of van Hamme’s and Rodriguez-Villegas’ supercongruences involving products of three binomial coefficients such as $\sum_{k = 0}^{(p - 1) / 2} {[\binom{2 k}{k}]}_{q ²}^{3} (q^{2 k}) / ((- q ²; q ²) ²_{k} (- q; q) ²_{2 k} ²) \equiv 0 (m o d [p] ²)$ for p≡ 3 (mod 4), $\sum_{k = 0}^{(p - 1) / 2} {[\binom{2 k}{k}]}_{q ³} ({(q; q ³)}_{k} {(q ²; q ³)}_{k} q^{3 k}) ({(q ⁶; q ⁶)}_{k} ²) \equiv 0 (m o d [p] ²)$ for p≡ 2 (mod 3), where $[p] = 1 + q + \dots + q^{p - 1}$ and $(a; q) ₙ = (1 - a) (1 - a q) \dots (1 - a q^{n - 1})$ . We also prove q-analogues of the Sun brothers’ generalizations of the above supercongruences. Our proofs are elementary in nature and use the theory of basic hypergeometric series and combinatorial q-binomial identities including a new q-Clausen type summation formula. ...

Modular symbols, Eisenstein series, and congruences

Jay Heumann, Vinayak Vatsal (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $E$ and $f$ be an Eisenstein series and a cusp form, respectively, of the same weight $k \geq 2$ and of the same level $N$ , both eigenfunctions of the Hecke operators, and both normalized so that $a_{1} (f) = a_{1} (E) = 1$ . The main result we prove is that when $E$ and $f$ are congruent mod a prime $𝔭$ (which we take in this paper to be a prime of $\bar{ℚ}$ lying over a rational prime $p > 2$ ), the algebraic parts of the special values $L (E, χ, j)$ and $L (f, χ, j)$ satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions, ...