Related papers: On the complexity of computing Kronecker coefficie…

We obtain new lower bounds for the number of Fourier coefficients of a weakly holomorphic modular form of half-integral weight not divisible by some prime $\ell$. Among the applications of this we show that there are $\gg \sqrt{X}/\log \log…

In the $k$-cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into $k$ connected components. Algorithms of Karger \& Stein can solve this in roughly $O(n^{2k})$ time. On the other hand,…

We consider a type of divided symmetrization $\overrightarrow{D}_{\lambda,G}$ where $\lambda$ is a nonincreasing partition on $n$ and where $G$ is a graph. We discover that in the case where $\lambda$ is a hook shape partition with first…

In the late 1930's Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n sufficiently large, the values of the Kronecker coefficients appearing in the product of two Schur…

We present new proofs and generalizations of unimodality of the q-binomial coefficients \binom{n}{k}_q as polynomials in q. We use an algebraic approach by interpreting the differences between numbers of certain partitions as Kronecker…

Let $K\subset\mathbb{C}$ be a compact set in the plane whose logarithmic capacity $c(K)$ is strictly positive. Let $\mathscr{P}_n(K)$ be the space of monic polynomials of degree $n,$ \emph{all} of whose zeros lie in $K.$ For $p\in…

We define solvable quantum mechanical systems on a Hilbert space spanned by bipartite ribbon graphs with a fixed number of edges. The Hilbert space is also an associative algebra, where the product is derived from permutation group…

We construct bases of the simple modules for partition algebras which are indexed by paths in an alcove geometry. This allows us to give a concrete interpretation (and new proof) of the monotone convergence property for Kronecker…

For a fixed positive integer $k$, let $C(k,n)$ denote the number of two-color partitions of $n$ with odd smallest part and restrictions on even parts, and let $C_k(q)$ be its generating function. We show that $C(1,n)\equiv d(2n-1)\pmod{4}$…

In this paper, assuming the low-degree conjecture, we provide evidence of computational hardness for two problems: (1) the (partial) matching recovery problem in the sparse correlated Erd\H{o}s-R\'enyi graphs $\mathcal G(n,q;\rho)$ when the…

For each integer $k\ge 1$, we define an algorithm which associates to a partition whose maximal value is at most $k$ a certain subset of all partitions. In the case when we begin with a partition $\lambda$ which is square, i.e…

Two algorithms for computing $P(n,m)$, the number of integer partitions of $n$ into exactly $m$ parts, are described, and using a combination of these two algorithms, the resulting algorithm is $O(n^{3/2})$. The second algorithm uses a list…

Motivated in part by hook-content formulas for certain restricted partitions in representation theory, we consider the total number of hooks of fixed length in odd versus distinct partitions. We show that there are more hooks of length $2$,…

Let $r \geq 0$, and let $\lambda$ and $\mu$ be partitions such that $\lambda_1 \leq r + 1$. We present a combinatorial interpretation of the plethysm coefficient $\langle s_\lambda, s_\mu[s_r] \rangle$. As a consequence, we solve the…

In this paper, we consider the asymptotic properties of hook numbers of partitions in restricted classes. More specifically, we compare the frequency with which partitions into odd parts and partitions into distinct parts have hook numbers…

Partition functions of certain classes of "spin glass" models in statistical physics show strong connections to combinatorial graph invariants. Also known as homomorphism functions they allow for the representation of many such invariants,…

We study the topological version of the partition calculus in the setting of countable ordinals. Let $\alpha$ and $\beta$ be ordinals and let $k$ be a positive integer. We write $\beta\to_{top}(\alpha,k)^2$ to mean that, for every red-blue…

Let $b^{k}_{\ell,m}(n)$ denotes the number of $k-$colored partitions of $n$ into parts that are not multiples of $\ell$ or $m$. We establish several congruence relations for $b_{\ell,m}(n)$. For instance, for any nonnegative integer $n$…

An positive even number is said to be a Kronecker number if it can be written in infinitely many ways as the difference between two primes, and it is believed that all even numbers are Kronecker numbers. We will study the division and…

Let A and M be nonempty sets of positive integers. A partition of the positive integer n with parts in A and multiplicities in M is a representation of n in the form n = \sum_{a\in A} m_a a, where m_a is in M U {0} for all a in A, and m_a…