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Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$. A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$, $i\not =c$, the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$. Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.
Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least
$$|S| - m\ln |G|$$
elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least
$$|G{|^{1 - c{ \in ^l}}}$$
for certain c=c(m) and
$$ \in = \in (m) < 1$$
; we use the probabilistic method to give sharper values of c(m) and
$$ \in (m)$$
in the case when G is a vector space; and we give new proofs of related known results.
We find a new refinement of Fine’s partition theorem on partitions into distinct parts with the minimum part odd. As a consequence, we obtain two companion partition identities. Both analytic and combinatorial proofs are provided.
The aim of this article is to provoke discussion concerning arithmetic properties of the function $p_{d}(n)$ counting partitions of a positive integer n into dth powers, where $d\geq 2$. Apart from results concerning the asymptotic behaviour of $p_{d}(n)$, little is known. In the first part of the paper, we prove certain congruences involving functions counting various types of partitions into dth powers. The second part of the paper is experimental and contains questions and conjectures concerning the arithmetic behaviour of the sequence $(p_{d}(n))_{n\in \mathbb {N}}$, based on computations of $p_{d}(n)$ for $n\leq 10^5$ for $d=2$ and $n\leq 10^{6}$ for $d=3, 4, 5$.
We show that there are biases in the number of appearances of the parts in two residue classes in the set of ordinary partitions. More precisely, let
$p_{j,k,m} (n)$
be the number of partitions of n such that there are more parts congruent to j modulo m than parts congruent to k modulo m for
$m \geq 2$
. We prove that
$p_{1,0,m} (n)$
is in general larger than
$p_{0,1,m} (n)$
. We also obtain asymptotic formulas for
$p_{1,0,m}(n)$
and
$p_{0,1,m}(n)$
for
$m \geq 2$
.
Andrews introduced the partition function $\overline {C}_{k, i}(n)$, called the singular overpartition function, which counts the number of overpartitions of n in which no part is divisible by k and only parts $\equiv \pm i\pmod {k}$ may be overlined. We prove that $\overline {C}_{6, 2}(n)$ is almost always divisible by $2^k$ for any positive integer k. We also prove that $\overline {C}_{6, 2}(n)$ and $\overline {C}_{12, 4}(n)$ are almost always divisible by $3^k$. Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of $2$ satisfied by $\overline {C}_{6, 2}(n)$.
We investigate the sum of the parts in all the partitions of n into distinct parts and give two infinite families of linear inequalities involving this sum. The results can be seen as new connections between partitions and divisors.
We construct eta-quotient representations of two families of q-series involving the Rogers–Ramanujan continued fraction by establishing related recurrence relations. We also display how these eta-quotient representations can be utilised to dissect certain q-series identities.
In this paper, we decompose
$\overline {D}(a,M)$
into modular and mock modular parts, so that it gives as a straightforward consequencethe celebrated results of Bringmann and Lovejoy on Maass forms. Let
$\overline {p}(n)$
be the number of partitions of n and
$\overline {N}(a,M,n)$
be the number of overpartitions of n with rank congruent to a modulo M. Motivated by Hickerson and Mortenson, we find and prove a general formula for Dyson’s ranks by considering the deviation of the ranks from the average:
Let r be an integer with 2 ≤ r ≤ 24 and let pr(n) be defined by $\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$. In this paper, we provide uniform methods for discovering infinite families of congruences and strange congruences for pr(n) by using some identities on pr(n) due to Newman. As applications, we establish many infinite families of congruences and strange congruences for certain partition functions, such as Andrews's smallest parts function, the coefficients of Ramanujan's ϕ function and p-regular partition functions. For example, we prove that for n ≥ 0,
In this note, we evaluate sums of partial theta functions. Our main tool is an application of an extended version of the Bailey transform to an identity of Gasper and Rahman on $q$-hypergeometric series.
For positive integers $n$ and $k$, let $r_{k}(n)$ denote the number of representations of $n$ as a sum of $k$ squares, where representations with different orders and different signs are counted as distinct. For a given positive integer $m$, by means of some properties of binomial coefficients, we derive some infinite families of congruences for $r_{k}(n)$ modulo $2^{m}$. Furthermore, in view of these arithmetic properties of $r_{k}(n)$, we establish many infinite families of congruences for the overpartition function and the overpartition pair function.
Andrews [‘Binary and semi-Fibonacci partitions’, J. Ramanujan Soc. Math. Math. Sci.7(1) (2019), 1–6] recently proved a new identity between the cardinalities of the set of semi-Fibonacci partitions and the set of partitions into powers of 2 with all parts appearing an odd number of times. We extend the identity to the set of semi-
$m$
-Fibonacci partitions of
$n$
and the set of partitions of
$n$
into powers of
$m$
in which all parts appear with multiplicity not divisible by
$m$
. We also give a new characterisation of semi-
$m$
-Fibonacci partitions and some congruences satisfied by the associated number sequence.
In this paper, we investigate
$\unicode[STIX]{x1D70B}(m,n)$
, the number of partitions of the bipartite number
$(m,n)$
into steadily decreasing parts, introduced by Carlitz [‘A problem in partitions’, Duke Math. J.30 (1963), 203–213]. We give a relation between
$\unicode[STIX]{x1D70B}(m,n)$
and the crank statistic
$M(m,n)$
for integer partitions. Using this relation, we establish some uniform asymptotic formulas for
$\unicode[STIX]{x1D70B}(m,n)$
.
Let
${\mathcal{A}}$
be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate
$t{\mathcal{A}}$
is asymptotically
$\frac{6}{\unicode[STIX]{x1D70B}^{2}}\text{Area}(t{\mathcal{A}})$
as
$t\rightarrow \infty$
. We show that the error term is both
$\unicode[STIX]{x1D6FA}_{\pm }(t\sqrt{\log \log t})$
and
$O(t(\log t)^{2/3}(\log \log t)^{4/3})$
. Both bounds extend (to the above class of polygons) known results for the isosceles right triangle, which appear in the literature as bounds for the error term in the summatory function for Euler’s
$\unicode[STIX]{x1D719}(n)$
.
Recently, Brietzke, Silva and Sellers [‘Congruences related to an eighth order mock theta function of Gordon and McIntosh’, J. Math. Anal. Appl.479 (2019), 62–89] studied the number
$v_{0}(n)$
of overpartitions of
$n$
into odd parts without gaps between the nonoverlined parts, whose generating function is related to the mock theta function
$V_{0}(q)$
of order 8. In this paper we first present a short proof of the 3-dissection for the generating function of
$v_{0}(2n)$
. Then we establish three congruences for
$v_{0}(n)$
along certain progressions which are subsequences of the integers
$4n+3$
.
Let
$\overline{t}(n)$
be the number of overpartitions in which (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd then it is overlined. Ramanujan-type congruences for
$\overline{t}(n)$
modulo small powers of
$2$
and
$3$
have been established. We present two infinite families of congruences modulo
$5$
and
$27$
for
$\overline{t}(n)$
, the first of which generalises a recent result of Chern and Hao [‘Congruences for two restricted overpartitions’, Proc. Math. Sci.129 (2019), Article 31].
We prove that every sufficiently large even integer can be represented as the sum of two squares of primes, four cubes of primes and 28 powers of two. This improves the result obtained by Liu and Lü [‘Two results on powers of 2 in Waring–Goldbach problem’, J. Number Theory131(4) (2011), 716–736].