618 research outputs found
The number of cubic partitions modulo powers of 5
The notion of cubic partitions is introduced by Hei-Chi Chan and named by
Byungchan Kim in connection with Ramanujan's cubic continued fractions. Chan
proved that cubic partition function has Ramanujan Type congruences modulo
powers of . In a recent paper, William Y.C. Chen and Bernard L.S. Lin
studied the congruent property of the cubic partition function modulo . In
this note, we give Ramanujan type congruences for cubic partition function
modulo powers of .Comment: 17 pages,Submitte
Relations among Ramanujan-type congruences I
We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific congruences for Hecke eigenvalues. In particular, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra. More generally, we show for weakly holomorphic modular forms of integral weight, that Ramanujan-type congruences naturally occur for shifts in the union of two square-classes as opposed to single square-classes that appear in the literature on the partition function. We also rule out the possibility of square-free periods, whose scarcity in the case of the partition function was investigated recently. We complement our obstructions on maximal Ramanujan-type congruences with several existence statements. Our results are based on a framework that leverages classical results on integral models of modular curves via modular representation theory, and applies to congruences of all weakly holomorphic modular forms. Steinberg representations govern all maximal Ramanujan-type congruences for integral weights. We discern the scope of our framework in the case of half-integral weights through example calculations
Non-existence of Ramanujan congruences in modular forms of level four
Ramanujan famously found congruences for the partition function like p(5n+4)
= 0 modulo 5. We provide a method to find all simple congruences of this type
in the coefficients of the inverse of a modular form on Gamma_{1}(4) which is
non-vanishing on the upper half plane. This is applied to answer open questions
about the (non)-existence of congruences in the generating functions for
overpartitions, crank differences, and 2-colored F-partitions.Comment: 19 page
d-Fold Partition Diamonds
In this work we introduce new combinatorial objects called --fold
partition diamonds, which generalize both the classical partition function and
the partition diamonds of Andrews, Paule and Riese, and we set to be
their counting function. We also consider the Schmidt type --fold partition
diamonds, which have counting function Using partition analysis, we
then find the generating function for both, and connect the generating
functions to Eulerian polynomials. This allows
us to develop elementary proofs of infinitely many Ramanujan--like congruences
satisfied by for various values of , including the following
family: for all and all Comment: 16 pages, 3 figures; v2: added a new result concerning Eulerian
polynomials and several subsequent congruences for , and corrected
a mistake in the proof of Proposition 1.
Computer-Assisted Proofs of Congruences for Multipartitions and Divisor Function Convolutions, based on Methods of Differential Algebra
This paper provides algebraic proofs for several types of congruences
involving the multipartition function and self-convolutions of the divisor
function. Our computations use methods of Differential Algebra in
, implemented in a couple of MAPLE programs available
as ancillary files on arXiv.
The first results of the paper are Ramanujan-type congruences of the form
and , where and
are the partition and divisor functions, is prime, and
denotes -order self-convolution. We prove all the valid
congruences of this form for , including the three
Ramanujan congruences, and a nontrivial one for . All such
multipartition congruences have already been settled in principle up to a
numerical verification due to D. Eichhorn and K. Ono via modular forms, but our
proofs are purely algebraic. On the other hand, the majority of the divisor
function congruences are new results.
We then proceed to search for more general congruences modulo small primes,
concerning linear combinations of for different values of
, as well as weighted convolutions of and with polynomial
weights. The paper ends with a few corollaries and extensions for the divisor
function congruences, including proofs for three conjectures of N. C.
Bonciocat.Comment: 25 pages, 1 figure, 2 tables; 6 ancillary files containing program
code in MAPLE and C+
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