**Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs**

arXiv:1711.09544, 2017

Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. This approach allows us to prove skew Cauchy identity which is our central result. We then derive various consequences and applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also explain the finite expansion property for products of Grothendieck polynomials.

**Schur operators and identities for skew stable Grothendieck polynomials**

Extended abstract, to appear in FPSAC 2018

**
Measuring the error in approximating the sub-level set topology of sampled scalar data**

with K. Beketayev, D. Morozov, G. Weber, B. Hamann

International Journal of Computational Geometry and Applications (IJCGA), to appear

This paper studies the influence of the definition of neighborhoods and methods used for creating point connectivity on topological analysis of scalar functions. It is assumed that a scalar function is known only at a finite set of points with associated function values. In order to utilize topological approaches to analyze the scalar-valued point set, it is necessary to choose point neighborhoods and, usually, point connectivity to meaningfully determine critical-point behavior for the point set. Two distances are used to measure the difference in topology when different point neighborhoods and means to define connectivity are used: (i) the bottleneck distance for persistence diagrams and (ii) the distance between merge trees. Usually, these distances define how different scalar functions are with respect to their topology. These measures, when properly adapted to point sets coupled with a definition of neighborhood and connectivity, make it possible to understand how topological characteristics depend on connectivity. Noise is another aspect considered. Five types of neighborhoods and connectivity are discussed: (i) the Delaunay triangulation; (ii) the relative neighborhood graph; (iii) the Gabriel graph; (iv) the k-nearest-neighbor (kNN) neighborhood; and (v) the Vietoris-Rips complex. It is discussed in detail how topological characterizations depend on the chosen connectivity.

**Duality and deformations of stable Grothendieck polynomials**

Journal of Algebraic Combinatorics, Vol. 45, 2017, 295-344

Stable Grothendieck polynomials can be viewed as a K-theory analog of Schur polynomials. We extend stable Grothendieck polynomials to a two-parameter version, which we call canonical stable Grothendieck functions. These functions have the same structure constants (with scaling) as stable Grothendieck polynomials, and (composing with parameter switching) are self-dual under the standard involutive ring automorphism. We study various properties of these functions, including combinatorial formulas, Schur expansions, Jacobi-Trudi type identities, and associated Fomin-Greene operators.

**Lattice path matroids: negative correlation and fast mixing**

with E. Cohen, P. Tetali

arXiv:1505.06710, 2015

We consider Markov chains on some of the realizations of the Catalan sequence. While our main result is in deriving an bound on the mixing time in (and hence total variation) distance for the random transposition chain on Dyck paths, we raise several open questions, including the optimality of the above bound. The novelty in our proof is in establishing a certain negative correlation property among random bases of lattice path matroids, including the so-called Catalan matroid which can be defined using Dyck paths.

**Walks, partitions, and normal ordering**

with A. Dzhumadil'daev

The Electronic Journal of Combinatorics, Vol. 22(4), 2015, #P4.10

We describe the relation between graph decompositions into walks and normal ordering of differential operators in the -th Weyl algebra. Under several specifications we study new types of restricted set partitions, and a generalization of Stirling numbers which we call -Stirling numbers.

**Path decompositions of digraphs and their applications to Weyl algebra**

with A. Dzhumadil'daev

Advances in Applied Mathematics, Vol. 67, 2015, 36-57

We consider decompositions of digraphs into edge-disjoint paths and describe their connection with the -th Weyl algebra of differential operators. This approach gives a graph-theoretic combinatorial view of the normal ordering problem and helps to study skew-symmetric polynomials on certain subspaces of Weyl algebra. For instance, path decompositions can be used to study minimal polynomial identities on Weyl algebra similarly as Eulerian tours applicable for Amitsur-Levitzki theorem.

**Stirling permutations on multisets**

with A. Dzhumadil'daev

European Journal of Combinatorics, Vol. 36, 2014, 377-392

A permutation of a multiset is called Stirling permutation if as soon as and . In other words, it is a 212-avoiding multiset permutation. We study Stirling polynomials that arise in the generating function for descent statistics on Stirling permutations of any multiset. We develop generalizations of classical Stirling numbers and present their combinatorial interpretations. In particular, we apply Stanley's theory of P-partitions. We also introduce Stirling numbers of odd type and generalizations of central factorial numbers.

**Measuring the distance between merge trees**

with K. Beketayev, D. Morozov, G. Weber, B. Hamann

in Topological Methods in Data Analysis and Visualization III, Springer-Verlag, 2014

Merge trees represent the topology of scalar functions. To assess the topological similarity of functions, one can compare their merge trees. To do so, one needs a notion of a distance between merge trees, which we define. We provide examples of using our merge tree distance and compare this new measure to other ways used to characterize topological similarity (bottleneck distance for persistence diagrams) and numerical difference.

**Power sums of binomial coefficients**

with A. Dzhumadil'daev

Journal of Integer Sequences, Vol. 16, 2013, Article 13.1.4

We establish an analog of Faulhaber's theorem for sums of powers of binomial coefficients. We study reciprocal power sums of binomial coefficients and Faulhaber coefficients for sums of powers of triangular numbers.

**Wolstenholme's theorem for binomial coefficients**

with A. Dzhumadil'daev

Siberian Electronic Math. Reports, Vol. 9, 2012, 460-463

**International Mathematical Olympiad (IMO) 2010. Shortlisted problems with solutions**

with Y. Baisalov, I. Bogdanov, G. Ko's, N. Sedrakyan, K. Yessenov, 2010

**Math Olympiads: Asian-Pacific and Silk Road** (in Russian, MCCME)

with A. Kungozhin, M. Kungozhin, Y. Baisalov, 2017

**Silk Road Mathematics Competition 2002-2010**

with A. Kerimov, Y. Baisalov, 2010

SRMC problems and Kazakhstan math olympiads related information can be found on matol.kz admined by Medeubek Kunogzhin.