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Gökhan Yıldırım

Publications and source records attributed to Gökhan Yıldırım.

8 recordsLinked to original sources
arXiv2023

An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences

We introduce an algorithmic approach based on generating tree method for enumerating the inversion sequences with various pattern-avoidance restrictions. For a given set of patterns, we propose an algorithm that outputs either an accurate description of the succession rules of the corresponding generating tree or an ansatz. By using this approach, we determine the generating trees for the pattern-classes $I_n(000, 021), I_n(100, 021)$, $I_n(110, 021), I_n(102, 021)$, $I_n(100,012)$, $I_n(011,201)$, $I_n(011,210)$ and $I_n(120,210)$. Then we use the kernel method, obtain generating functions of each class, and find enumerating formulas. Lin and Yan studied the classification of the Wilf-equivalences for inversion sequences avoiding pairs of length-three patterns and showed that there are 48 Wilf classes among 78 pairs. In this paper, we solve six open cases for such pattern classes.

math.CO↗
arXiv2021

Variations on Hammersley's interacting particle process

The longest increasing subsequence problem for permutations has been studied extensively in the last fifty years. The interpretation of the longest increasing subsequence as the longest 21-avoiding subsequence in the context of permutation patterns leads to many interesting research directions. We introduce and study the statistical properties of Hammersleytype interacting particle processes related to these generalizations and explore the finer structures of their distributions. We also propose three different interacting particle systems in the plane analogous to the Hammersley process in one dimension and obtain estimates for the asymptotic orders of the mean and variance of the number of particles in the systems.

math.PR↗
arXiv2021

Enumerations of bargraphs with respect to corner statistics

We study the enumeration of bargraphs with respect to some corner statistics. We find generating functions for the number of bargraphs that tracks the corner statistics of interest, the number of cells, and the number of columns. The bargraph representation of set partitions is also considered and some explicit formulas are obtained for the number of some specific types of corners in such representations.

math.CO↗
arXiv2020

Permutations avoiding 312 and another pattern, Chebyshev polynomials and longest increasing subsequences

We study the longest increasing subsequence problem for random permutations avoiding the pattern $312$ and another pattern $τ$ under the uniform probability distribution. We determine the exact and asymptotic formulas for the average length of the longest increasing subsequences for such permutation classes specifically when the pattern $τ$ is monotone increasing or decreasing, or any pattern of length four.

math.CO↗
arXiv2018

Directed polymers on a disordered tree with a defect subtree

We study the question of how the competition between $\textit{bulk disorder}$ and a $\textit{localized microscopic defect}$ affects the macroscopic behavior of a system in the directed polymer context at the free energy level. We consider the directed polymer model on a disordered $d$-ary tree and represent the localized microscopic defect by modifying the disorder distribution at each vertex in a single path (branch), or in a subtree, of the tree. The polymer must choose between following the microscopic defect and finding the best branches through the bulk disorder. We describe three possible phases, called the $\textit{fully pinned, partially pinned}$ and $\textit{depinned}$ phases. When the microscopic defect is associated only with a single branch, we compute the free energy and the critical curve of the model, and show that the partially pinned phase does not occur. When the localized microscopic defect is associated with a non-disordered regular subtree of the disordered tree, the picture is more complicated. We prove that all three phases are non-empty below a critical temperature, and that the partially pinned phase disappears above the critical temperature.

math.PR↗
arXiv2017

Longest monotone subsequences and rare regions of pattern-avoiding permutations

We consider the distributions of the lengths of the longest monotone and alternating subsequences in classes of permutations of size $n$ that avoid a specific pattern or set of patterns, with respect to the uniform distribution on each such class. We obtain exact results for any class that avoids two patterns of length 3, as well as results for some classes that avoid one pattern of length 4 or more. In our results, the longest monotone subsequences have expected length proportional to $n$ for pattern-avoiding classes, in contrast with the $\sqrt n$ behaviour that holds for unrestricted permutations. In addition, for a pattern $τ$ of length $k$, we scale the plot of a random $τ$-avoiding permutation down to the unit square and study the "rare region," which is the part of the square that is exponentially unlikely to contain any points. We prove that when $τ_1>τ_k$, the complement of the rare region is a closed set that contains the main diagonal of the unit square. For the case $τ_1=k,$ we also show that the lower boundary of the part of the rare region above the main diagonal is a curve that is Lipschitz continuous and strictly increasing on $[0,1]$.

math.CO↗
arXiv2015

Directed polymers in a random environment with a defect line

We study the depinning transition of the $1+1$ dimensional directed polymer in a random environment with a defect line. The random environment consists of i.i.d. potential values assigned to each site of $\mathbb{Z}^2$; sites on the positive axis have the potential enhanced by a deterministic value $u$. We show that for small inverse temperature $β$ the quenched and annealed free energies differ significantly at most in a small neighborhood (of size of order $β$) of the annealed critical point $u_c^a=0$. For the case $u=0$, we show that the difference between quenched and annealed free energies is of order $β^4$ as $β\to 0$, assuming only finiteness of exponential moments of the potential values, improving existing results which required stronger assumptions.

math.PR↗