arXiv subjects

Stoyan Dimitrov

Publications and source records attributed to Stoyan Dimitrov.

10 recordsLinked to original sources

arXiv2026

Counting Colored Trees

We consider the enumeration of plane trees (rooted ordered trees) whose vertices are colored according to a specific coloring rule that prescribes which possible pairs of colors can occur as the colors of a parent vertex and its child. This general construction covers many different examples that have been studied in the literature. Some general necessary and sufficient conditions for two different coloring rules to result in the same counting sequence are established. We also provide exhaustive lists of counting sequences arising from coloring rules with two or three colors, and we find formulas and closed form expressions for many of these sequences. The famous Fibonacci, Catalan, Narayana, and Schr\"oder sequences appear in several cases. Some of these coloring rules are extended to families of coloring rules with arbitrarily many colors.

math.CO↗

arXiv2024

BFS versus DFS for fixed-level targets in ordered trees

We find the average time complexity of the breadth-first search (BFS) and the depth-first search (DFS) algorithms, when one searches for a target node selected uniformly at random among all nodes at level $\ell$ in the set of ordered trees with $n$ edges. Intuition suggests that on average BFS must be asymptotically faster than DFS if and only if $\ell$, as a function of $n$, is below a certain threshold. We confirm this intuition by showing that there exists a unique constant $\lambda\approx 0.789004$, such that in expectation BFS is asymptotically faster than DFS if and only if $\ell\leq \lambda\sqrt{n}$. This gives us a practical rule to select between the two algorithms, even when we do not know the exact value of $\ell$, but only an estimate of it. Furthermore, we find the asymptotic average time complexity of BFS in the given setting for an arbitrary class of Galton--Watson trees, which includes ordered trees, binary trees, and other popular classes. We use results on the occupation measure of Brownian excursions, as well as combinatorial identities related to lattice paths. Finally, we consider the simple \textit{truncated DFS} algorithm, which can be shown easily to be asymptotically faster than both BFS and DFS when $\ell$ is known in advance. We show that in fact its asymptotic time complexity is $1/2$ of the asymptotic complexity of BFS, when $\ell = s\sqrt{n}$ for any constant $s$. Several further questions are also raised.

cs.DS↗

arXiv2023

Chess tableaux, powers of two and affine Lie algebras

Chess tableaux are a special kind of standard Young tableaux where, in the chessboard coloring of the Young diagram, even numbers always appear in white cells and odd numbers in black cells. If, for $λ$ a partition of $n$, $\text{Chess}(λ)$ denotes the number of chess tableaux of shape $λ$, then Chow, Eriksson and Fan observed that $\displaystyle\sum_{λ\vdash n} \text{Chess}(λ)^2$ is divisible by unusually large powers of $2$. In this paper, we give an explanation for this phenomenon, proving a lower bound of $n-O(\sqrt{n})$ for the $2$-adic valuation of this sum and a generalization of it. We do this by exploiting a connection with a certain representation of the affine Lie algebra $\widehat{\mathfrak{sl}_2}$ on the vector space with basis indexed by partitions. Our result about chess tableaux then follows from a study of the basic representation of $\widehat{\mathfrak{sl}_2}$ with coefficients taken from the ring of rational numbers with odd denominators.

math.CO↗

arXiv2022

A combinatorial proof of a symmetry for a refinement of the Narayana numbers

We establish a tantalizing symmetry of certain numbers refining the Narayana numbers. In terms of Dyck paths, this symmetry is interpreted in the following way: if $w_{n,k,m}$ is the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$, then $w_{2k+1,k,m}=w_{2k+1,k,k+1-m}$. We give a combinatorial proof of this fact, relying on the cycle lemma, and showing that the numbers $w_{2k+1,k,m}$ are multiples of the Narayana numbers. We prove a more general fact establishing a relationship between the numbers $w_{n,k,m}$ and a family of generalized Narayana numbers due to Callan. A closed-form expression for the even more general numbers $w_{n,k_{1},k_{2},\ldots , k_{r}}$ counting the semilength-$n$ Dyck paths with $k_{1}$ $UD$-factors, $k_{2}$ $UUD$-factors, $\ldots$ , and $k_{r}$ $U^{r}D$-factors is also obtained, as well as a more general form of the discussed symmetry for these numbers in the case when all rise runs are of certain minimal length. Finally, we investigate properties of the polynomials $W_{n,k}(t)= \sum_{m=0}^k w_{n,k,m} t^m$, including real-rootedness, $\gamma$-positivity, and a symmetric decomposition.

math.CO↗

arXiv2022

On Sums, Derivatives, and Flips of Riordan Arrays

We study three operations on Riordan arrays. First, we investigate when the sum of Riordan arrays yields another Riordan array. We characterize the $A$- and $Z$-sequences of these sums of Riordan arrays, and also identify an analog for $A$-sequences when the sum of Riordan arrays does not yield a Riordan array. In addition, we define the new operations `Der' and `Flip' on Riordan arrays. We fully characterize the Riordan arrays resulting from these operations applied to the Appell and Lagrange subgroups of the Riordan group. Finally, we study the application of these operations to various known Riordan arrays, generating many combinatorial identities in the process.

math.CO↗

arXiv2022

Sorting by shuffling methods and a queue

We study sorting by queues that can rearrange their content by applying permutations from a predefined set. These new sorting devices are called shuffle queues and we investigate those of them corresponding to sets of permutations defining some well-known shuffling methods. If $\mathbb{Q}_Σ$ is the shuffle queue corresponding to the shuffling method $Σ$, then we find a number of surprising results related to two natural variations of shuffle queues denoted by $\mathbb{Q}_Σ^{\prime}$ and $\mathbb{Q}_Σ^{\textsf{pop}}$. These require the entire content of the device to be unloaded after a permutation is applied or unloaded by each pop operation, respectively. First, we show that sorting by a deque is equivalent to sorting by a shuffle queue that can reverse its content. Next, we focus on sorting by cuts. We prove that the set of permutations that one can sort by using $\mathbb{Q}_{\text{cuts}}^{\prime}$ is the set of the $321$-avoiding separable permutations. We give lower and upper bounds to the maximum number of times the device must be used to sort a permutation. Furthermore, we give a formula for the number of $n$-permutations, $p_{n}(\mathbb{Q}_Σ^{\prime})$, that one can sort by using $\mathbb{Q}_Σ^{\prime}$, for any shuffling method $Σ$, corresponding to a set of irreducible permutations. We also show that $p_{n}(\mathbb{Q}_Σ^{\textsf{pop}})$ is given by the odd indexed Fibonacci numbers $F_{2n-1}$, for any shuffling method $Σ$ having a specific "back-front" property. The rest of the work is dedicated to a surprising conjecture inspired by Diaconis and Graham, which states that one can sort the same number of permutations of any given size by using the devices $\mathbb{Q}_{\text{In-sh}}^{\textsf{pop}}$ and $\mathbb{Q}_{\text{Monge}}^{\textsf{pop}}$, corresponding to the popular In-shuffle and Monge shuffling methods.

math.CO↗

arXiv2021

Moments of permutation statistics and central limit theorems

We show that if a permutation statistic can be written as a linear combination of bivincular patterns, then its moments can be expressed as a linear combination of factorials with constant coefficients. This generalizes a result of Zeilberger. We use an approach of Chern, Diaconis, Kane and Rhoades, previously applied on set partitions and matchings. In addition, we give a new proof of the central limit theorem (CLT) for the number of occurrences of classical patterns, which uses a lemma of Burstein and Hasto. We give a simple interpretation of this lemma and an analogous lemma that would imply the CLT for the number of occurrences of any vincular pattern. Furthermore, we obtain explicit formulas for the moments of the descents and the minimal descents statistics. The latter is used to give a new direct proof of the fact that we do not necessarily have asymptotic normality of the number of pattern occurrences in the case of bivincular patterns. Closed forms for some of the higher moments of several popular statistics on permutations are also obtained.

math.CO↗

arXiv2021

On permutation patterns with constrained gap sizes

We consider avoidance of permutation patterns with designated gap sizes between pairs of consecutive letters. We call the patterns having such constraints distant patterns (DPs) and we show their relation to other pattern notions investigated in the past. New results on DPs with 2 and 3 letters are obtained. Furthermore, we show how one can use DPs to prove two former conjectures of Kuszmaul without a computer. In addition, we deduce a surprising relation between the sets of permutations avoiding the classical patterns $123$ and $132$ by looking at a class of DPs with tight gap constraints. Some interesting analogues of the Stanley-Wilf former conjecture for DPs are also discussed.

math.CO↗

arXiv2021

Digraphs with exactly one Eulerian tour

We give two combinatorial proofs of the fact that the number of loopless digraphs on the vertex set $[n]$ with no isolated vertices and with exactly one Eulerian tour up to a cyclic shift is $\frac{1}{2}(n-1)!C_{n}$, where $C_{n}$ denotes the $n$-th Catalan number. We construct a bijection with a set of labeled rooted plane trees and with a set of valid parenthesis arrangements.

math.CO↗

arXiv2014

Reinforcement Learning Based Algorithm for the Maximization of EV Charging Station Revenue

This paper presents an online reinforcement learning based application which increases the revenue of one particular electric vehicles (EV) station, connected to a renewable source of energy. Moreover, the proposed application adapts to changes in the trends of the station's average number of customers and their types. Most of the parameters in the model are simulated stochastically and the algorithm used is a Q-learning algorithm. A computer simulation was implemented which demonstrates and confirms the utility of the model.

cs.CE↗