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Register a new userArithmetic partition sums and orbits of Z_n^k under the symmetric group S_k
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We study M(n,k,r), the number of orbits of {(a_1,...,a_k)in Z_n^k | a_1+...+a_k = r (mod n)} under the action of S_k. Equivalently, M(n,k,r) sums the partition numbers of an arithmetic sequence: M(n,k,r) = sum_{t geq 0} p(n-1,k,r+nt), where p(a,b,t) denotes the number of partitions of t into at most b parts, each of which is at most a. We derive closed formulas and various identities for such arithmetic partition sums. These results have already appeared in Elashvili/Jibladze/Pataraia, Combinatorics of necklaces and Hermite reciprocity, J. Alg. Combin. 10 (1999) 173-188, and the main result was also published by Von Sterneck in Sitzber. Akad. Wiss. Wien. Math. Naturw. Class. 111 (1902), 1567-1601 (see Lemma 2 and references in math.NT/9909121). Thanks to Don Zagier and Robin Chapman for bringing these references to our attention.
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A $k$-sum of a set $Asubseteq mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $kwedge A$ for the set of $k$-sums of $A$ we prove that [ frac{|(k+1)wedge A|}{|kwedge A|}, le , frac{|A|-k}{k+1} ] whenever $|A|ge (k^{2}+7k)/2$. The inequality is tight -- the above ratio being attained when $A$ is a geometric progression. This answers a question of Ruzsa.
In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers $mathbb N$ as limiting values of $q$-series as $qto zeta$ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of $mathbb N$ by analogous structures in the integer partitions $mathcal P$. In recent work, Wang obtains a wide generalization of Alladis original theorem, in which arithmetic densities of subsets of prime numbers are computed as values of Dirichlet series arising from Dirichlet convolutions. Here the authors prove that Wangs extension has a partition-theoretic analogue as well, yielding new $q$-series density formulas for any subset of $mathbb N$. To do so, we outline a theory of $q$-series density calculations from first principles, based on a statistic we call the $q$-density of a given subset. This theory in turn yields infinite families of further formulas for arithmetic densities.
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.
We identify a class of semi-modular forms invariant on special subgroups of $GL_2(mathbb Z)$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an Eisenstein-like series summed over integer partitions, and use it to construct families of semi-modular forms.
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.
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