"The apex of mathematical achievement occurs when two or more fields which were thought to be entirely unrelated turn out to be closely intertwined!" *

-- Gian-Carlo Rota

Two main things are widely believed to be important in reasearch:

  1. What questions have been answered in a research work?
    How important, interesting and/or surprising are they?

    2. But how to ask good questions in the first place? Asking the right questions is determined by famous mathematicians (e.g. G.C.Rota in [1] and V.Arnold in [2]) to be the most important thing in research!

    Interestingly, one of the biggest names in Machine Learning, V.Vapnik, confirms that (in [3]) from the perspective of creating intelligence by saying:

    "How people ask questions is the most difficult thing to understand, in order to mirror intelligence!".

  2. What tools/techniques have been used?
    Are the tools novel? Can they be used to solve broader set of problems?

References:
[1] Kung, J. P., Rota, G. C., & Yan, C. H. (2009). Combinatorics: the Rota way.
[2] Arnold, V. I. (2004). Arnold's problems. See the "Author's Preface to the first edition".
[3] Vladimir Vapnik: Predicates, Invariants, and the Essence of Intelligence, AI Podcast.
[4] "Mathematics is the art of giving the same name to different things", an interview with H. Poincaré.

*H.Poincaré makes a statement in the same light: "Unexpected concurrences between different parts of science is what can make progress. Too much specialization would prohibit these concurrences." (see source [4] above)

List of publications:


  1. A combinatorial proof of a tantalizing symmetry on Catalan objects
    (with M. Bona, G. Labelle et al.)
    25 pages, arXiv:2212.10586
    [PDF file]
    We investigate a tantalizing symmetry on Catalan objects. 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 two proofs of this symmetry: an algebraic proof using generating functions, and a combinatorial proof which makes heavy use of the cycle lemma and an alternate interpretation of the numbers $w_{n,k,m}$ using plane trees. In particular, our combinatorial proof expresses the numbers w_{2k+1,k,m} in terms of Narayana numbers, and we generalize this to a relationship between the numbers $w_{n,k,m}$ and a family of generalized Narayana numbers due to Callan. Some further generalizations and applications of our combinatorial proofs are explored. Finally, we investigate properties of the polynomials $W_{n,k}(t)=\sum_{m=0}^{k}w_{n,k,m}t^{m}$, including real-rootedness, γ-positivity, and a symmetric decomposition.

  2. Chess tableaux, powers of two and affine Lie algebras
    (with A. Labelle)
    14 pages, Journal of Algebraic Combinatorics (to appear)
    [PDF file]
    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 $\lambda$ a partition of n, $Chess(\lambda)$ denotes the number of chess tableaux of shape $\lambda$, then Chow, Eriksson and Fan observed that $\sum_{\lambda |- n} Chess(\lambda)^{2}$ is divisible by unusually large powers of 2. In this paper, we give an explanation of this phenomenon, proving a lower bound of $n-\mathcal{O}(\sqrt{n})$ 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 $\hat{sl_{2}}$ on the vector space with basis indexed by partitions. Our result about chess tableaux then follows from a study of the basic representation of $\hat{sl_{2}}$ with coefficients taken from the ring of rational numbers with odd denominators.

  3. On Sums, Derivatives, and Flips Of Riordan Arrays
    (with S. Sundaram, M. von Bell, et al.)
    35 pages, Journal of Integer Sequences, Vol. 26, Issue 2 (2023)
    [PDF file]
    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.

  4. Moments of permutation statistics and central limit theorems
    (with N. Khare)
    27 pages, Advances in Applied Mathematics (to appear)
    [PDF file]
    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 H\"{a}st\"{o}. 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.

  5. Digraphs with exactly one Eulerian tour
    (with L. Grisales, A. Labelle and R. Posada)
    10 pages, Journal of Integer Sequences, Vol. 25, Issue 3 (2022)
    [PDF file]
    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 valid parenthesis arrangements.

  6. Sorting by shuffling methods and a queue
    29 pages, The Electronic Journal of Combinatorics, Vol. 29, Issue 3 (2022)
    [PDF file]
    We consider sorting by a queue that can apply a permutation from a given set over its content. This gives us a sorting device $\mathbb{Q}_{\Sigma}$ corresponding to any shuffling method $\Sigma$ since every such method is associated with a set of permutations. Two variations of these devices are considered - $\mathbb{Q}_{\Sigma}^{\prime}$ and $\mathbb{Q}_{\Sigma}^{\text{pop}}$. 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 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, $p_{n}(\mathbb{Q}_{\Sigma}^{\prime})$, that one can sort by using $\mathbb{Q}_{\Sigma}^{\prime}$, for any shuffling method $\Sigma$, such that the permutations associated with it are irreducible. Next, we prove a generalization of the fact that $\mathbb{Q}_{\text{cuts}}^{\text{pop}}$ can sort all permutations. We also show that $p_{n}(\mathbb{Q}_{\Sigma}^{\text{pop}})$ is given by the odd indexed Fibonacci numbers $F_{2n-1}$, for any shuffling method $\Sigma$ 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 $\mathbb{Q}_{\text{In-sh}}^{\text{pop}}$ and $\mathbb{Q}_{\text{Monge}}^{\text{pop}}$, corresponding to the popular In-shuffle and Monge shuffling methods.

  7. On permutation patterns with constrained gap sizes
    26 pages, submitted
    [PDF file]
    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 including a generating function found using the block-decomposition method in a non-trivial way. Furthermore, we prove two conjectures of Kuszmaul using a DP interpretation and we give that perspective to four of the other conjectures listed there. DPs with tight gap constraints are also considered in order to deduce a somewhat surprising relation between the sets of permutations avoiding the classical patterns 123 and 132. In addition, interesting Stanley-Wilf analogues for DPs are discussed, as well as some open questions.

  8. Adaptive Monte Carlo algorithm for Wigner kernel evaluation
    (with V. Todorov, I. Dimov and R. Georgieva)
    12 pages, Neural Computing and Applications, 2019: 1-12.
    [PDF file]
    In this paper, we study various numerical approaches for computation of multidimensional integrals, namely an adaptive Monte Carlo algorithm, a particular rank-1 lattice algorithm based on generalized Fibonacci numbers and a Monte Carlo algorithm based on Latin hypercube sampling. We compare the performance of the algorithms over three case studies— multidimensional integrals from Bayesian statistics, the so-called Genz test functions and the Wigner kernel—an important issue in quantum mechanics represented by multidimensional integrals. A comprehensive analysis of the computational complexity of the algorithms under consideration has been also presented.

  9. Branching processes in continuous time as models of mutations: Computational approaches and algorithms
    (with M. Bojkova and P. Trayanov)
    14 pages, Computational Statistics & Data Analysis 113, 111-124, 2017.: 1-12.
    [PDF file]
    The appearance of mutations in cancer development plays a crucial role in the disease control and its medical treatment. Motivated by the practical significance, it is of interest to model the event of occurrence of a mutant cell that will possibly lead to a path of indefinite survival. A two-type branching process model in continuous time is proposed for describing the relationship between the waiting time till the first escaping extinction mutant cell is born and the lifespan distribution of cells, which due to the applied treatment have small reproductive ratio. A numerical method and related algorithm for solving the integral equations are developed, in order to estimate the distribution of the waiting time to the escaping extinction mutant cell is born. Numerical studies demonstrate that the proposed approximation algorithm reveals the substantial difference of the results in discrete-time setting. In addition, to study the time needed for the mutant cell population to reach high levels a simulation algorithm for continuous two-type decomposable branching process is proposed. Two different computational approaches together with the theoretical studies might be applied to different kinds of cancer and their proper treatment.

  10. Reinforcement learning based algorithm for the maximization of EV charging station revenue
    (with R. Lguensat)
    5 pages, Proceedings of the Conference on ”Mathematics and Computers in Sciences and in Industry", 2014.
    [PDF file]
    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 the Q-learning algorithm. A computer simulation was implemented which demonstrates and confirms the utility of the model.

Articles in preparation:



Notes and unpublished work:


Here is a file I wrote with the MIT students Luz Grisales and Rodrigo Posada containing solutions to 150+ of the bijective proof problems of Richard Stanley.