**Grothendieck shenanigans: Permutons from pipe dreams via integrable probability**

with Alejandro H. Morales, Greta Panova, Leonid Petrov

arXiv:2407.21653

We study random permutations arising from reduced pipe dreams. Our main model is motivated by Grothendieck polynomials with parameter arising in K-theory of the flag variety. The probability weight of a permutation is proportional to the principal specialization (setting all variables to 1) of the corresponding Grothendieck polynomial. By mapping this random permutation to a version of TASEP (Totally Asymmetric Simple Exclusion Process), we describe the limiting permuton and fluctuations around it as the order of the permutation grows to infinity. The fluctuations are of order and have the Tracy-Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar-Parisi-Zhang universality class. We also investigate non-reduced pipe dreams and make progress on a recent open problem on the asymptotic number of inversions of the resulting permutation. Inspired by Stanley's question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for Grothendieck polynomials, and provide bounds for general .

-- See also the followinghighlightposted on IPAM's news.

**Some unimodal sequences of Kronecker coefficients**

with Alimzhan Amanov

arXiv:2312.17054

We conjecture unimodality for some sequences of generalized Kronecker coefficients and prove it for partitions with at most two columns. The proof is based on a hard Lefschetz property for corresponding highest weight spaces. We also study more general Lefschetz properties, show implications to a higher-dimensional analogue of the Alon--Tarsi conjecture on Latin squares and give related positivity results.

**Bounds on the number of higher-dimensional partitions**

Proceedings of the American Mathematical Society, Vol. 152, 2024, 955-965.

We establish some bounds on the number of higher-dimensional partitions by volume. In particular, we give bounds via vector partitions and MacMahon's numbers.

**Fundamental invariants of tensors, Latin hypercubes, and rectangular Kronecker coefficients**

with Alimzhan Amanov

International Mathematics Research Notices, Vol. 2023, Issue 20, 2023, 17552-17599.
(Published version)

We study polynomial SL-invariants of tensors, mainly focusing on fundamental invariants which are of smallest degrees. In particular, we prove that certain 3-dimensional analogue of the Alon--Tarsi conjecture on Latin cubes considered previously by Bürgisser and Ikenmeyer, implies positivity of (generalized) Kronecker coefficients at rectangular partitions and as a result provides values for degree sequences of fundamental invariants.

**MacMahon's statistics on higher-dimensional partitions**

with Alimzhan Amanov

Forum of Mathematics Sigma, Vol. 11, 2023, article e63.
Published version

We study some combinatorial properties of higher-dimensional partitions which generalize plane partitions. We present a natural bijection between d-dimensional partitions and d-dimensional arrays of nonnegative integers. This bijection has a number of important applications. We introduce a statistic on d-dimensional partitions, called the corner-hook volume, whose generating function has the formula of MacMahon's conjecture. We obtain multivariable formulas whose specializations give analogues of various formulas known for plane partitions. We also introduce higher-dimensional analogues of dual Grothendieck polynomials which are quasisymmetric functions and whose specializations enumerate higher-dimensional partitions of a given shape. Finally, we show probabilistic connections with a directed last passage percolation model in .

**Stability of the Levi-Civita tensors and an Alon–Tarsi type theorem**

Comptes Rendus Mathématique, Vol. 361, 2023, 1367-1373.

We show that the Levi-Civita tensors are semistable in the sense of Geometric Invariant Theory, which is equivalent to an analogue of the Alon–Tarsi conjecture on Latin squares. The proof uses the connection of Tao’s slice rank with semistable tensors. We also show an application to an asymptotic saturation-type version of Rota’s basis conjecture.

Note: an earlier version of this paper appeared as "Saturation of Rota's basis conjecture" at arXiv:2107.12926

**Tensor slice rank and Cayley's first hyperdeterminant**

with Alimzhan Amanov

Linear Algebra and Its Applications, Vol. 656, 2023, 224-246.

Cayley's first hyperdeterminant is a straightforward generalization of determinants for tensors. We prove that nonzero hyperdeterminants imply lower bounds on some types of tensor ranks. This result applies to the slice rank introduced by Tao and more generally to partition ranks introduced by Naslund. As an application, we show upper bounds on some generalizations of colored sum-free sets based on constraints related to order polytopes.

**Determinantal formulas for dual Grothendieck polynomials**

with Alimzhan Amanov

Proceedings of the American Mathematical Society, Vol. 150, 2022, 4113-4128.

We prove Jacobi-Trudi-type determinantal formulas for skew dual Grothendieck polynomials which are K-theoretic deformations of Schur polynomials. We also prove a bialternant-type formula analogous to the classical definition of Schur polynomials.

**Random plane partitions and corner distributions**

Algebraic Combinatorics, Vol. 4, 2021, 599-617.

We explore some probabilistic applications arising in connections with K-theoretic symmetric functions. For instance, we determine certain corner distributions of random lozenge tilings and plane partitions. We also introduce some distributions that are naturally related to the corner growth model. Our main tools are dual symmetric Grothendieck polynomials and normalized Schur functions.

**Enumeration of plane partitions by descents**

Journal of Combinatorial Theory Series A, Vol. 178, 2021, Article 105367, 18 p.

We study certain bijection between plane partitions and N-matrices. As applications, we prove a Cauchy-type identity for generalized dual Grothendieck polynomials. We introduce two statistics on plane partitions, whose generating functions are similar to classical MacMahon's formulas; one of these statistics is equidistributed with the usual volume. We also show natural connections with the longest increasing subsequences of words.

**Dual Grothendieck polynomials via last-passage percolation**

Comptes Rendus Mathématique, Vol. 358, 2020, 497-503.
Published version (open access)

The ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous K-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model.

**Positive specializations of symmetric Grothendieck polynomials**

Advances in Mathematics, Vol. 363, 2020, Article 107000, 35 p.

It is a classical fundamental result that Schur-positive specializations of the ring of symmetric functions are characterized via totally positive functions whose parametrization describes the Edrei-Thoma theorem. In this paper, we study positive specializations of symmetric Grothendieck polynomials, K-theoretic deformations of Schur polynomials.

**On the largest Kronecker and Littlewood-Richardson coefficients**

with Igor Pak and Greta Panova

Journal of Combinatorial Theory Series A, Vol. 165, 2019, 44-77

We give new bounds and asymptotic estimates for Kronecker and Littlewood-Richardson coefficients. Notably, we resolve Stanley's questions on the shape of partitions attaining the largest Kronecker and Littlewood-Richardson coefficients. We apply the results to asymptotics of the number of standard Young tableaux of skew shapes.My talk video at IPAM, Feb 2020

**Bounds on the largest Kronecker and induced multiplicities of finite groups**

with Igor Pak and Greta Panova

Communications in Algebra, Vol. 47:8, 2019, 3264-3279

We give new bounds and asymptotic estimates on the largest Kronecker and induced multiplicities of finite groups. The results apply to large simple groups of Lie type and other groups with few conjugacy classes.

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

Journal of Combinatorial Theory Series A, Vol. 161, 2019, 453-485

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. With this approach we prove skew Cauchy identity and then derive various applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also give a new explanation of the finite expansion property for products of Grothendieck polynomials.

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

with Kenes Beketayev, Dmitriy Morozov, Gunther Weber, Bernd Hamann

International Journal of Computational Geometry and Applications (IJCGA), Vol. 28, 2018, 57-77

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 Emma Cohen and Prasad 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 Askar Dzhumadil'daev

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 Askar 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 Askar 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 Kenes Beketayev, Dmitriy Morozov, Gunther Weber, Bernd 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 Askar 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 Askar Dzhumadil'daev

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

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

with Yerzhan Baisalov, Ilya Bogdanov, Geza Ko's, Nairi Sedrakyan, Kuat Yessenov, 2010

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

with Almas Kungozhin, Medeubek Kungozhin, Yerzhan Baisalov, 2017

**Silk Road Mathematics Competition 2002-2010**

with Azer Kerimov, Yerzhan Baisalov, 2010

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

May 2024, IPAM seminar series (Geometry, Statistical mechanics and Integrability program), IPAM UCLA

Everything we know about higher-dimensional integer partitions
slides

Apr 2024, SOCALDM'24, UCLA

Latin cubes, Kronecker coefficients, and tensor invariants
slides

Apr 2023, IMMM International mathematics conference, Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

Higher-dimensional partitions (Plenary talk)

Feb 2020,
Asymptotic Algebraic Combinatorics workshop,
IPAM UCLA

Inequalities and bounds for the Littlewood-Richardson coefficients
slides
video

Oct 2018, AICT'18, KBTU

Applications of spectral graph theory (tutorial)

May 2018, SOCALDM'18, USC

The largest Kronecker and Littlewood-Richardson coefficients
slides

Nov 2017, Combinatorics seminar, University of Minnesota

Identities for symmetric Grothendieck polynomials

Nov 2017,
AMS Fall Western Sectional Meeting,
UC Riverside

Schur operators and identities for skew stable Grothendieck polynomials
slides

May 2017, SOCALDM'17, UCLA

Ehrhart polynomial of some Schlafli simplices
slides

Mar 2017, Combinatorics seminar, Georgia Tech

Computing integer partitions

Oct 2016, Combinatorics seminar, USC

Duality and deformations of stable Grothendieck polynomials

Oct 2016, Combinatorics seminar, UCLA

Catalan shuffles

Oct 2014, Combinatorics seminar, University of Minnesota

Decomposing digraphs into increasing walks