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## Enumeration of Chord Diagrams without Loops and Parallel Chords

Electronic Journal of Combinatorics. 2017. Vol. 24. No. 3. P. 1–23.

Omelchenko A., Краско Е. С.

We enumerate chord diagrams without loops and without both loops and parallel chords. For labelled diagrams we obtain generating functions, for unlabelled ones we derive recurrence relations.

Краско Е. С., Лабутин И. Н., Omelchenko A., Записки научных семинаров ПОМИ РАН 2019 Т. 488 С. 119–142

We enumerate labelled and unlabelled Hamiltonian cycles in complete $n$-partite graphs $K_{d,d,\ldots,d}$ having exactly $d$ vertices in each part (in other words, Tur\'an graphs $T(nd, n))$. We obtain recurrence relations that allow us to find the exact values $b_{n}^{(d)}$ of such cycles for arbitrary $n$ and $d$. ...

Added: February 6, 2020

Kulakova E., Lando S., Mukhutdinova T. et al., European Journal of Combinatorics 2014 Vol. 41 P. 266–277

We introduce a new series Rk, k = 2, 3, 4, ..., of integer valued weight systems. The value of the weight system Rk on a chord diagram is a signed number of cycles of even length 2k in the intersection graph of the diagram.Weshow that this value depends on the intersection graph only. We ...

Added: October 26, 2014

Korpelainen N., Lozin V. V., Malyshev D. et al., Theoretical Computer Science 2011 No. 412 P. 3545–3554

The notion of a boundary graph property was recently introduced as a relaxation of that of a minimal property and was applied to several problems of both algorithmic and combinatorial nature. In the present paper, we first survey recent results related to this notion and then apply it to two algorithmic graph problems: Hamiltonian cycle ...

Added: September 11, 2012

Omelchenko A., Краско Е. С., European Journal of Combinatorics 2017 Vol. 62 P. 167–177

We obtain explicit formulas for enumerating rooted and unrooted 4-regular one-face maps on genus g surfaces. For rooted maps the result is combinatorially derived from Chapuy’s vertex cutting bijection and has a simple sum-free form similar to analogous formulas for general and cubic one-face maps. To enumerate unrooted maps we apply the approach of Liskovets, Mednykh and Nedela ...

Added: August 29, 2018

Smirnov E., Kleptsyn V., Journal of Knot Theory and Its Ramifications 2016 Vol. 26 P. 1642006

To each ribbon graph we assign a so-called L-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are ...

Added: January 15, 2016

Omelchenko A., Краско Е. С., Electronic Journal of Combinatorics 2015 Vol. 22 No. 1 P. 1–17

We present new functional equations connecting the counting series of plane and planar (in the sense of Harary and Palmer) dissections. Simple rigorous expressions for counting symmetric rr-dissections of polygons and planar SS-dissections are obtained. ...

Added: August 29, 2018

Yang Z., / Cornell University. Series arXiv "math". 2022.

The present paper has been motivated by an aspiration for understanding the weight system corresponding to the Lie algebra glN. The straightforward approach to computing the values of a Lie algebra weight system on a general chord diagram amounts to elaborating calculations in the noncommutative universal enveloping algebra, in spite of the fact that the result ...

Added: October 6, 2022

Omelchenko A., Краско Е. С., Discrete Mathematics 2019 Vol. 342 No. 2 P. 600–614

The second part of the paper is devoted to enumeration of r-regular maps on the torus up to all its homeomorphisms (unsensed maps). We describe in detail the periodic orientation reversing homeomorphisms of the torus which turn out to be representable as glide reflections. We show that considering quotients of the torus with respect to ...

Added: September 21, 2018

Yang Z., / Cornell University. Series arXiv "math". 2021.

Each knot invariant can be extended to singular knots according to the skein rule. A Vassiliev invariant of order at most n is defined as a knot invariant that vanishes identically on knots with more than n double points. A chord diagram encodes the order of double points along a singular knot. A Vassiliev invariant of order n gives rise to ...

Added: October 6, 2022

Alexander Dunaykin, Vyacheslav Zhukov, Moscow Mathematical Journal 2022 Vol. 22 No. 1 P. 69–81

To a singular knot K with n double points, one can associate a chord
diagram with n chords. A chord diagram can also be understood as a 4regular graph endowed with an oriented Euler circuit. L. Traldi introduced
a polynomial invariant for such graphs, called a transition polynomial.
We specialize this polynomial to a multiplicative weight system, that ...

Added: November 10, 2020

Kulakova E., Lando S., Mukhutdinova T. et al., / Cornell University. Series math "arxiv.org". 2013.

We introduce a new series~$R_k$, $k=2,3,4,\dots$, of integer valued weight systems. The value of the weight system~$R_k$ on a chord diagram is a signed number of cycles of even length~$2k$ in the intersection graph of the diagram. We show that this value depends on the intersection graph only. We check that for small orders of ...

Added: November 24, 2013

Zinova P., Функциональный анализ и его приложения 2020 Т. 54 № 3 С. 73–93

A weight system is a function on chord diagrams that satisfies the so-called four-term
relations. Vassiliev’s theory of finite-order knot invariants describes these invariants in terms of
weight systems. In particular, there is a weight system corresponding to the colored Jones polynomial.
This weight system can be easily defined in terms of the Lie algebra sl2, but this ...

Added: December 10, 2020

Lando S., Zhukov V., Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 741–755

Vassiliev (finite type) invariants of knots can be described in terms of weight systems. These are functions on chord diagrams satisfying so-called 4-term relations. The goal of the present paper is to show that one can define both the first and the second Vassiliev moves for binary delta-matroids and introduce a 4-term relation for them ...

Added: December 11, 2017

Omelchenko A., Краско Е. С., Discrete Mathematics 2019 Vol. 342 No. 2 P. 584–599

The work that consists of two parts is devoted to the problem of enumerating unrooted r-regular maps on the torus up to all its symmetries. We begin with enumerating near-r- regular rooted maps on the torus, the projective plane and the Klein bottle, as well as some special kinds of maps on the sphere: near-r-regular ...

Added: September 21, 2018

Kulakova E., Lando S., Mukhutdinova T. et al., / Cornell University. Series math "arxiv.org". 2013. No. 1307.4933.

We introduce a new series Rk, k=2,3,4,…, of integer valued weight systems. The value of the weight system Rk on a chord diagram is a signed number of cycles of even length 2k in the intersection graph of the diagram. We show that this value depends on the intersection graph only. We check that for ...

Added: December 18, 2014

Enatskaya N., Промышленные АСУ и контроллеры 2016 № 7 С. 32–36

All investigations of the scheme defined in its name are fulfiled by the direct enumeration of its outcomes, namely: the number of its outcomes of the scheme and its probability distribution are defined, the numbering problem for outcomes of the scheme is solved, this gives the possibility of a quick modeling of its possible values ...

Added: September 13, 2016

Краско Е. С., Omelchenko A., Acta Mathematica Universitatis Comenianae 2019 Vol. LXXXVIII No. 3 P. 885–890

The work is devoted to the problem of enumerating maps on an orientable or non-orientable surface of a given genus g up to all symmetries (so called unsensed maps). We obtain general formulas which reduce the problem of counting such maps to the problem of enumerating rooted quotient maps on orbifolds. In addition, we solve ...

Added: December 9, 2019

A.V.Omelchenko, Bogdanov A., Meshkov V. et al., Journal of Knot Theory and Its Ramifications 2012 Vol. 21 No. 7 P. 1–17

The paper addresses the enumeration problem for k-tangles. We introduce the notion of a cascade diagram of a k-tangle projection and suggest an effective enumeration algorithm for projections based on the cascade representation. Tangle projections and alternating tangles with up to 12 crossings are tabulated. We also provide pictures of alternating k-tangles with at most ...

Added: August 30, 2018

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183–189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

Borzykh D., ЛЕНАНД, 2021

Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...

Added: February 20, 2021

В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72–80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017

Красноярск: ИВМ СО РАН, 2013

Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...

Added: November 18, 2013

Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1–16915-18

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...

Added: November 16, 2020

Grines V., Gurevich E., Pochinka O., Russian Mathematical Surveys 2017 Vol. 71 No. 6 P. 1146–1148

In the paper a Palis problem on finding sufficient conditions on embedding of Morse-Smale diffeomorphisms in topological flow is discussed. ...

Added: May 17, 2017