Let $\mathbb{N}$ denote the set of positive integers and $\mathbb{N}_0 = \mathbb{N} \cup \lbrace 0\rbrace$. The positive diagonal integral ternary quadratic form $ax^2+by^2+cz^2 \, \, (a, b, c \in \mathbb{N})$ is said to be $(k,l)$-universal if it represents every integer in the arithmetic progression $\lbrace kn+l \, \, | \, \, n \in \mathbb{N}_0 \rbrace$, where $k, l \in \mathbb{N}$ are such that $l \leq k.$ We show that there are only finitely many $(k,l)$-universal positive integral ternary quadratic forms $ax^2+by^2+cz^2$ for a fixed pair $(k,l) \in \mathbb{N}^2$ with $l \leq k$. We also prove the existence of a finite set $S=S(k,l)$ of positive integers $\equiv l \pmod{k}$ such that if $ax^2+by^2+cz^2$ represents every integer in $S$ then $ax^2+by^2+cz^2$ is $(k,l)$-universal. Assuming that certain ternaries are $(k,l)$-universal we determine all the $(k,l)$-universal ternaries $ax^2+by^2+cz^2$ $(a, b, c \in \mathbb{N}),$ as well as the sets $S(k,l),$ for all $k,l \in \mathbb{N} $ with $1 \leq l \leq k \leq 11$.
Lerna Pehlivan, Kenneth S. Williams
In INTEGERS,
2018
For a given permutation $\tau$, let $P_N^{\tau}$ be the uniform probability distribution on the set of $N$-element permutations $\sigma$ that avoid the pattern $\tau$. For $\tau=\mu_k:=123\ldots k$, we consider $P_N^{\mu_k}(\sigma_I=J)$ where $I\sim \gamma N$ and $J\sim \delta N$ for $\gamma,\delta\in (0,1)$. If $\gamma+\delta\neq 1$, then we are in the large deviations regime with the probability decaying exponentially, and we calculate the limiting value of $P_N^{\mu_k}(\sigma_I=J)^{1/N}$.
Neal Madras, Lerna Pehlivan
In The Electronic Journal of Combinatorics,
2016
We evaluate the probabilities of various events under the uniform distribution on the set of $312$-avoiding permutations of $1,\ldots,N$. We derive exact formulas for the probability that the $i^{th}$ element of a random permutation is a specific value less than $i$, and for joint probabilities of two such events. In addition, we obtain asymptotic approximations to these probabilities for large $N$ when the elements are not close to the boundaries or to each other. We also evaluate the probability that the graph of a random $312$-avoiding permutation has $k$ specified decreasing points, and we show that for large $N$ the points below the diagonal look like trajectories of a random walk.
Neal Madras, Lerna Pehlivan
In Random Structures & Algorithms,
2015
We give some new evaluations of the Legendre symbol $\left(\frac{a + b\sqrt{q}}{p} \right)$ for certain integers $a$ and $b$ and certain primes $q$, where $p$ is an odd prime such that $\left(\frac{q}{p}\right) = 1,$ and $\sqrt{q}$ denotes an integer whose square is $q \pmod p$. For example it is shown that if $p$ is a prime $\equiv 1, 19, 25, 31, 37, 55 \pmod{84}$ then $$ \left(\frac{-5-2\sqrt{7}}{p} \right) = \left(\frac{x-2y}{7}\right) = \begin{cases} +1 & \text{if } x-2y \equiv 1,2,4 \pmod 7, \cr -1 & \text{if } x-2y \equiv 3,5,6 \pmod 7, \end{cases} $$ where $x$ and $y$ are the unique integers satisfying $p=x^2+xy+y^2$, $x \equiv 1 \pmod 4$, $y \equiv 3(p-1) \pmod 8$ and $\left(1-(-1)^{(p-1)/2}\right)x+y>0.$
Lerna Pehlivan, Kenneth S. Williams
In Acta Arithmetica,
2015
Feryal Alayont, Gizem Karaali, Lerna Pehlivan
In MAA FOCUS,
2013