Analytic combinatorics of chord and hyperchord diagrams with k crossings

Pilaud, Vincent; Rué, Juanjo

doi:10.1016/j.aam.2014.04.001

Cited by 7 publications

(6 citation statements)

References 28 publications

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“…The chord diagram 29 , 30 representation is natural in this case because according to the analysis above the fermionic loops yield zero contribution. In order to further facilitate the interpretation of the graphs in frequency space we use color coding for the coefficients entering the GF arguments.…”

Section: Methodsmentioning

confidence: 99%

Padé resummation of many-body perturbation theories

Pavlyukh

2017

Sci Rep

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In a typical scenario the diagrammatic many-body perturbation theory generates asymptotic series. Despite non-convergence, the asymptotic expansions are useful when truncated to a finite number of terms. This is the reason for the popularity of leading-order methods such as the GW approximation in condensed matter, molecular and atomic physics. Appropriate truncation order required for the accurate description of strongly correlated materials is, however, not known a priori. Here an efficient method based on the Padé approximation is introduced for the regularization of perturbative series allowing to perform higher-order self-consistent calculations and to make quantitative predictions on the convergence of many-body perturbation theories. The theory is extended towards excited states where the Wick theorem is not directly applicable. Focusing on the plasmon-assisted photoemission from graphene, we treat diagrammatically electrons coupled to the excited state plasmons and predict new spectral features that can be observed in the time-resolved measurements.

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Section: Methodsmentioning

confidence: 99%

Padé resummation of many-body perturbation theories

Pavlyukh

2017

Sci Rep

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“…Touchard [9] and Riordan [6] enumerated configurations by the total number of crossings, and the limiting Normal distribution was obtained by Flajolet and Noy [2]. More recently Pilaud and Rué [5] have extended the study of crossings in several directions. Kreweras and Poupard [4] enumerated configurations by the number of so-called short pairs, where adjacent vertices Figure 1: The 6 configurations for the case n = 2.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A critical quartet for queuing couples

Young¹

2020

Preprint

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We enumerate arrangements of n couples, i.e. pairs of people, placed in a singlefile queue, and consider four statistics from the vantage point of a distinguished given couple. In how many arrangements are exactly p of the n−1 other couples i) interlaced with the given couple, ii) contained within them, iii) containing the given couple, and iv) lying outside the given couple? We provide generating functions which enumerate these arrangements and obtain the associated continuous asymptotic distributions in the n → ∞ limit. The asymptotic distributions corresponding to cases i), iii), and iv) evince critical phenomena around the value p c = (n − 1)/2, such that the probability that 1) the couple is interlaced with more than half of the other couples, and 2) the couple is contained by more than half of the other couples, are both zero in the strict n → ∞ limit. We further show that the cumulative probability that less than half of the other couples lie outside the given couple is π/4 in the limit, and that the associated distribution is uniform for p < p c .

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“…[11,50]) on crossings and nestings in chord diagrams to 1-terminal diagrams. For example, Pilaud and Rué [50] used analytic combinatorial methods to derive an asymptotic estimate for the number of diagrams with n chords and m crossings; the above theorem can be used to extend this asymptotic estimate to 1-terminal diagrams with n chords and m crossings. The above result uses the fact that connectivity is equivalent to 1-terminality for nonnesting diagrams.…”

Section: Relationship With Other Double Factorial Objectsmentioning

confidence: 99%

“…Since then there has been much focus on the enumeration of various subclasses of diagrams and their statistics (e.g. [51,56,57,45,21,50,15]). One of the most prominent and natural types of diagrams studied are connected diagrams, which are those diagrams C for which there is no proper interval of [2n] that is the ground set of a subdiagram of C. Connectivity can also be defined via the intersection graph G(C) of a diagram C, the directed graph on the chords of C formed by adding edge (c, c ) if c crosses c on the right; C is connected if and only if its intersection graph is weakly connected.…”

Section: Introductionmentioning

confidence: 99%

The combinatorics of a tree-like functional equation for connected chord diagrams

Nabergall

2021

Preprint

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We build on recent work of Yeats, Courtiel, and others involving connected chord diagrams. We first derive from a Hopf-algebraic foundation a class of tree-like functional equations and prove that they are solved by weighted generating functions of two different subsets of weighted connected chord diagrams: arbitrary diagrams and diagrams forbidding so-called top cycle subdiagrams. These equations generalize the classic specification for increasing ordered trees and their solution uses a novel decomposition, simplifying and generalizing previous results. The resulting tree perspective on chord diagrams leads to new enumerative insights through the study of novel diagram classes. We present a recursive bijection between connected top-cycle-free diagrams with n chords and triangulations of a disk with n + 1 vertices, thereby counting the former. This connects to combinatorial maps, Catalan intervals, and uniquely sorted permutations, leading to new conjectured bijective relationships between diagram classes defined by forbidding graphical subdiagrams and imposing connectedness properties and a variety of other rich combinatorial objects. We conclude by exhibiting and studying a direct bijection between diagrams of size n with a single terminal chord and diagrams of size n − 1.

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Analytic combinatorics of chord and hyperchord diagrams with k crossings

Cited by 7 publications

References 28 publications

Padé resummation of many-body perturbation theories

Padé resummation of many-body perturbation theories

A critical quartet for queuing couples

The combinatorics of a tree-like functional equation for connected chord diagrams

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