## Damir Yeliussizov

### Research papers

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 $O(n^2 \log n)$ bound on the mixing time in $L_2$ (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
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 $n$-th Weyl algebra. Under several specifications we study new types of restricted set partitions, and a generalization of Stirling numbers which we call $\lambda$-Stirling numbers.

Path decompositions of digraphs and their applications to Weyl algebra
Advances in Applied Mathematics, Vol. 67, 2015, 36-57

We consider decompositions of digraphs into edge-disjoint paths and describe their connection with the $n$-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.

Measuring the error in approximating the sub-level set topology of sampled scalar data
with K. Beketayev, D. Morozov, G. Weber, B. Hamann
preprint, 2015

If the function is given only by samples, a ground truth function and its topology are usually attempted to be approximated using various methods. However, the amount of error in computed topology and how it correlates with a method selection is then an open question. We address this question by adapting two distances, the bottleneck distance between persistence diagrams and the distance between merge trees, as measures of error in approximating topological features of interest, in particular the persistence and evolution of sub-level sets of the scalar function. These measures are not biased by noise in the data, making them appealing for working with scientific data. We demonstrate the utility of such error measures by evaluating the quality of function approximation given by common mesh constructions.

Stirling permutations on multisets
European Journal of Combinatorics, Vol. 36, 2014, 377-392

A permutation $p$ of a multiset is called Stirling permutation if $p(s) \ge p(i)$ as soon as $p(i) = p(j)$ and $i < s < j$. In other words, it is 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
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
Siberian Electronic Math. Reports, Vol. 9, 2012, 460-463

### Books

International Mathematical Olympiad (IMO) 2010. Shortlisted problems with solutions
with Y. Baisalov, I. Bogdanov, G. Ko's, N. Sedrakyan, K. Yessenov, 2010