Related papers: Hook lengths and 3-cores
Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern-Fu-Tang and Heim-Neuhauser gave…
We demonstrate a method for listing all ordinary partitions of n as binary words of length (n-1). The resulting family imbued with the hamming distance yields subgraphs of the Hamming Graphs. The existence of a 2-Gray Code for ordinary…
An asymptotic formula for the number of partitions into p-cores is derived. As a byproduct some integer valued trigonometric sums are found
Let $A(p,n,k)$ be the number of $p$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We formulate the conjecture that, for every fixed $p$ and $n$, the…
Mills proved that there exists a real constant $A>1$ such that for all $n\in \mathbb{N}$ the values $\lfloor A^{3^n}\rfloor$ are prime numbers. No explicit value of $A$ is known, but assuming the Riemann hypothesis one can choose $A=…
The Akbari-Cameron-Khosrovshahi (ACK) conjecture, which appears to be unresolved, states that for any simple graph $G$ with at least one edge, there exists a nonzero {$\{0,1\}$}-vector in the row space of its adjacency matrix that is not a…
Integer partitions have long been of interest to number theorists, perhaps most notably Ramanujan, and are related to many areas of mathematics including combinatorics, modular forms, representation theory, analysis, and mathematical…
We are concerned with counting self-conjugate $(s,s+1,s+2)$-core partitions. A Motzkin path of length $n$ is a path from $(0,0)$ to $(n,0)$ which stays above the $x$-axis and consists of the up $U=(1,1)$, down $D=(1,-1)$, and flat $F=(1,0)$…
In 2003, Maroti showed that one could use the machinery of l-cores and l-quotients of partitions to establish lower bounds for p(n), the number of partitions of n. In this paper we explore these ideas in the case l=2, using them to give a…
We study $\nu_k(n)$, the number of partitions of $n$ into $k$ part sizes, and find numerous arithmetic progressions where $\nu_2$ and $\nu_3$ take on values divisible by 2 and 4. Expanding earlier work, we show $\nu_2(An+B) \equiv 0…
In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck.…
This note is concerned with the set of integral solutions of the equation $x^2+3y^2=12n+4$, where $n$ is a positive integer. We will describe a parametrization of this set using the 3-core partitions of n. In particular we construct a crank…
A sequence x=x_1 x_2...x_n $ is said to be an ascent sequence of length $n$ if it satisfies x_1=0 and $0\leq x_i\leq asc(x_1x_2...x_{i-1})+1$ for all $2\leq i\leq n$, where $asc(x_1x_2... x_{i-1})$ is the number of ascents in the sequence…
Recently, a simple proof of the hook length formula was given via the branching rule. In this paper, we extend the results to shifted tableaux. We give a bijective proof of the branching rule for the hook lengths for shifted tableaux;…
Let $L$ be a set of $n$ axis-parallel lines in $\mathbb{R}^3$. We are are interested in partitions of $\mathbb{R}^3$ by a set $H$ of three planes such that each open cell in the arrangement $\mathcal{A}(H)$ is intersected by as few lines…
Rota's basis conjecture, open since 1989, states that if B_1, B_2, ..., B_n are n bases of a vector space of rank n, then there is an nxn grid of vectors such that the vectors in the ith row are precisely the elements of B_i and such that…
Let $M_0(n)$ (resp. $M_1(n)$) denote the number of partitions of $n$ with even (reps. odd) crank. Choi, Kang and Lovejoy established an asymptotic formula for $M_0(n)-M_1(n)$. By utilizing this formula with the explicit bound, we show that…
A famous conjecture about group algebras of torsion-free groups states that there is no zero divisor in such group algebras. A recent approach to settle the conjecture is to show the non-existence of zero divisors with respect to the length…
Generalizing the concept of a perfect number, Sloane's sequences of integers A083207 lists the sequence of integers $n$ with the property: the positive factors of $n$ can be partitioned into two disjoint parts so that the sums of the two…
The well-known $abc$-conjecture concerns triples $(a,b,c)$ of non-zero integers that are coprime and satisfy ${a+b+c=0}$. The strong $n$-conjecture is a generalisation to $n$ summands where integer solutions of the equation ${a_1 + \ldots +…