Lerna Pehlivan
I am a visiting scholar in mathematics at Dalhousie University. My research lies in the areas of probability theory, enumerative combinatorics, representation theory on symmetric groups, and analytic number theory. In probability theory and enumerative combinatorics my reseach has evolved around problems related to pattern avoiding permutations, card shuffling problems as well as random labellings on graphs. In analytic number theory my work involves representations of integers by quadratic forms, Theta function identities, determining the number of solutions of quadratic equations, and reciprocity laws for the Legendre symbol.
Probability Theory
Markov Chain Theory
Representation Theory of Symmetric Groups
Analytic Number Theory
MS in Mathematics
Bogazici University
BS in Mathematics
$(k,l)$-universality of ternary quadratic forms $ax^2 + by^2+ cz^2$
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
Large deviations for permutations avoiding monotone patterns
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
PDF Project
Structure of random 312-avoiding permutations
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.
In Random Structures & Algorithms, 2015
Some new evaluations of the Legendre symbol $\left( \frac{a+b\sqrt{q}}{p}\right)$
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 $\left(1-(-1)^{(p-1)/2}\right)x+y>0.$
In Acta Arithmetica, 2015
What Does It Take to Teach Non-Majors Effectively?
Feryal Alayont, Gizem Karaali, Lerna Pehlivan
In MAA FOCUS, 2013
Lerna Pehlivan, Kenneth S. Williams. Positive-definite ternary quadratic forms which are (4,1)-universal and (4,3)-universal. Accepted for publication INTEGERS, 2018.
Lerna Pehlivan, Kenneth S. Williams. $(k,l)$-universality of ternary quadratic forms $ax^2 + by^2+ cz^2$. In INTEGERS, 2018.
Neal Madras, Lerna Pehlivan. Large deviations for permutations avoiding monotone patterns. In The Electronic Journal of Combinatorics, 2016.
Neal Madras, Lerna Pehlivan. Structure of random 312-avoiding permutations. In Random Structures & Algorithms, 2015.
Lerna Pehlivan, Kenneth S. Williams. Some new evaluations of the Legendre symbol $\left( \frac{a+b\sqrt{q}}{p}\right)$. In Acta Arithmetica, 2015.
Saban Alaca, Lerna Pehlivan, Kenneth S. Williams. On the number of representations of a positive integer as a sum of two binary quadratic forms. In International Journal of Number Theory, 2014.
Lerna Pehlivan, Kenneth S. Williams. The power series expansion of certain infinite products $q^r \prod_{n=1}^\infty(1-q^n)^{a_1}(1-q^{2n})^{a_2}\cdots (1-q^{mn})^{a_m}$. In Ramanujan Journal, 2014.
Feryal Alayont, Gizem Karaali, Lerna Pehlivan. What Does It Take to Teach Non-Majors Effectively?. In MAA FOCUS, 2013.
Saban Alaca, Lerna Pehlivan, Kenneth S. Williams. Some product-to-sum identities. In Journal of Combinatorics and Number Theory, 2012.
Lerna Pehlivan. No feedback card guessing for top to random shuffle. 2010.
I previously taught
Abstract Algebra I & II
Statistics I & II
Matrix Algebra
Applied Multivariate and Vector Calculus
Calculus I & II
Precalculus I & II
lr608779@dal.ca
Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada
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