Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA supernormal partition statistic
76
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study a bijective map from integer partitions to the prime factorizations of integers that we call the supernorm of a partition, in which the multiplicities of the parts of partitions are mapped to the multiplicities of prime factors of natural numbers. The supernorm is connected to a family of maps we define, which suggests the potential to apply techniques from partition theory to identify and prove multiplicative properties of integers. We make a brief study of pertinent analytic aspects of the supernorm. Then, as an application of supernorma mappings (i.e., pertaining to the supernorm statistic), we prove an analogue of a formula of Kural-McDonald-Sah to give arithmetic densities of subsets of $mathbb N$ instead of natural densities in $mathbb P$ like previous formulas of this type; this builds on works of Alladi, Ono, Wagner, and the first and third authors. Finally, using a table of supernormal additive-multiplicative correspondences, we conjecture Abelian-type formulas that specialize to our main theorem and other known results.
rate research
Read More
One of the most basic results concerning the number-theoretic properties of the partition function $p(n)$ is that $p(n)$ takes each value of parity infinitely often. This statement was first proved by Kolberg in 1959, and it was strengthened by Subbarao in 1966 to say that both $p(2n)$ and $p(2n+1)$ take each value of parity infinitely often. These results have received several other proofs, each relying to some extent on manipulating generating functions. We give a new, self-contained proof of Subbaraos result by constructing a series of bijections and involutions, along the way getting a more general theorem concerning the enumeration of a special subset of integer partitions.
Using a theorem of Frobenius filtered through partition generating function techniques, we prove partition-theoretic and $q$-series Abelian theorems, yielding analogues of Abels convergence theorem for complex power series, and related formulas. As an application we give a limiting formula for the $q$-bracket of Bloch and Okounkov, an operator from statistical physics connected to the theory of modular forms, as $qto 1$ from within the unit disk.
In this article we study the norm of an integer partition, which we define to be the product of the parts. This partition-theoretic statistic has appeared here and there in the literature of the last century or so, and is at the heart of current research by both authors. We survey known results and give new results related to this all-but-overlooked object, which, it turns out, plays a comparable role in partition theory to the size, length, and other standard partition statistics.
We examine partition zeta functions analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over partitions of fixed length -- which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahons partial fraction decomposition of the generating function for partitions of fixed length.
In 1917, Hardy and Ramanujan obtained the asymptotic formula for the classical partition function $p(n)$. The classical partition function $p(n)$ has been extensively studied. Recently, Luca and Ralaivaosaona obtained the asymptotic formula for the square-root function. Many mathematicians have paid much attention to congruences on some special colored partition functions. In this paper, we investigate the general colored partition functions. Given positive integers $1=s_1<s_2<dots <s_k$ and $ell_1, ell_2,dots , ell_k$. Let $g(mathbf{s}, mathbf{l}, n)$ be the number of $ell$-colored partitions of $n$ with $ell_i$ of the colors appearing only in multiplies of $s_i (1le ile k)$, where $ell = ell_1+cdots +ell_k$. By using the elementary method we obtain an asymptotic formula for the partition function $g(mathbf{s}, mathbf{l}, n)$ with an explicit error term.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
