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Sergey Kirgizov

Publications and source records attributed to Sergey Kirgizov.

27 records · Page 2Linked to original sources
arXiv2021

Clique percolation method: memory efficient almost exact communities

Automatic detection of relevant groups of nodes in large real-world graphs, i.e. community detection, has applications in many fields and has received a lot of attention in the last twenty years. The most popular method designed to find overlapping communities (where a node can belong to several communities) is perhaps the clique percolation method (CPM). This method formalizes the notion of community as a maximal union of $k$-cliques that can be reached from each other through a series of adjacent $k$-cliques, where two cliques are adjacent if and only if they overlap on $k-1$ nodes. Despite much effort CPM has not been scalable to large graphs for medium values of $k$. Recent work has shown that it is possible to efficiently list all $k$-cliques in very large real-world graphs for medium values of $k$. We build on top of this work and scale up CPM. In cases where this first algorithm faces memory limitations, we propose another algorithm, CPMZ, that provides a solution close to the exact one, using more time but less memory.

cs.DS↗
arXiv2021

Pattern statistics in faro words and permutations

We study the distribution and the popularity of some patterns in $k$-ary faro words, i.e. words over the alphabet $\{1, 2, \ldots, k\}$ obtained by interlacing the letters of two nondecreasing words of lengths differing by at most one. We present a bijection between these words and dispersed Dyck paths (i.e. Motzkin paths with all level steps on the $x$-axis) with a given number of peaks. We show how the bijection maps statistics of consecutive patterns of faro words into linear combinations of other pattern statistics on paths. Then, we deduce enumerative results by providing multivariate generating functions for the distribution and the popularity of patterns of length at most three. Finally, we consider some interesting subclasses of faro words that are permutations, involutions, derangements, or subexcedent words.

math.CO↗
arXiv2021

Transformation à la Foata for special kinds of descents and excedances

A pure excedance in a permutation $π=π_1π_2\ldots π_n$ is a position $i<π_i$ such that there is no $j<i$ with $i\leq π_j<π_i$. We present a one-to-one correspondence on the symmetric group that transports pure excedances to descents of special kind. As a byproduct, we prove that the popularity of pure excedances equals those of pure descents on permutations, while their distributions are different.

math.CO↗
arXiv2019

Bijections between directed animals, multisets and Grand-Dyck paths

An $n$-multiset of $[k]=\{1,2,\ldots, k\}$ consists of a set of $n$ elements from $[k]$ where each element can be repeated. We present the bivariate generating function for $n$-multisets of $[k]$ with no consecutive elements. For $n=k$, these multisets have the same enumeration as directed animals in the square lattice. Then we give constructive bijections between directed animals, multisets with no consecutive elements and Grand-Dyck paths avoiding the pattern $DUD$, and we show how classical and novel statistics are transported by these bijections.

math.CO↗
arXiv2018

Enumeration of Łukasiewicz paths modulo some patterns

For any pattern $α$ of length at most two, we enumerate equivalence classes of Łukasiewicz paths of length $n\geq 0$ where two paths are equivalent whenever the occurrence positions of $α$ are identical on these paths. As a byproduct, we give a constructive bijection between Motzkin paths and some equivalence classes of Łukasiewicz paths.

math.CO↗
arXiv2018

Descent distribution on Catalan words avoiding a pattern of length at most three

Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern $p$ we provide a bivariate generating function where the coefficient of $x^ny^k$ in its series expansion is the number of length $n$ Catalan words with $k$ descents and avoiding $p$. As a byproduct, we enumerate the set of Catalan words avoiding $p$, and we provide the popularity of descents on this set. Some of the obtained enumerating sequences are not yet recorded in the On-line Encyclopedia of Integer Sequences.

math.CO↗
arXiv2016

Patterns in treeshelves

We study the distribution and the popularity of left children on sets of treeshelves avoiding a pattern of size three. (Treeshelves are ordered binary increasing trees where every child is connected to its parent by a left or a right link.) The considered patterns are sub-treeshelves, and for each such a pattern we provide exponential generating function for the corresponding distribution and popularity. Finally, we present constructive bijections between treeshelves avoiding a pattern of size three and some classes of simpler combinatorial objects.

cs.DM↗