scholarly journals The combinatorial derivation

2013 ◽  
Vol 14 (2) ◽  
Author(s):  
Igor V. Protasov
Keyword(s):  
Combinatorial Derivation

scholarly journals A Combinatorial Derivation of the PASEP Stationary State

10.37236/1134 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Richard Brak ◽  
Sylvie Corteel ◽  
John Essam ◽  
Robert Parviainen ◽  
Andrew Rechnitzer
Keyword(s):  
Markov Chain ◽  
Stationary State ◽  
Stationary Distribution ◽  
Exclusion Process ◽  
Direct Proof ◽  
Asymmetric Exclusion Process ◽  
Original Process ◽  
Combinatorial Derivation ◽  
Path Models ◽  
Asymmetric Exclusion

We give a combinatorial derivation and interpretation of the weights associated with the stationary distribution of the partially asymmetric exclusion process. We define a set of weight equations, which the stationary distribution satisfies. These allow us to find explicit expressions for the stationary distribution and normalisation using both recurrences and path models. To show that the stationary distribution satisfies the weight equations, we construct a Markov chain on a larger set of generalised configurations. A bijection on this new set of configurations allows us to find the stationary distribution of the new chain. We then show that a subset of the generalised configurations is equivalent to the original process and that the stationary distribution on this subset is simply related to that of the original chain. We also provide a direct proof of the validity of the weight equations.

scholarly journals Constellations and multicontinued fractions: application to Eulerian triangulations

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Marie Albenque ◽  
Jérémie Bouttier
Keyword(s):  
Generating Function ◽  
Nous Obtenons ◽  
Hankel Determinants ◽  
Combinatorial Derivation ◽  
International Audience ◽  
Comme Application

International audience We consider the problem of enumerating planar constellations with two points at a prescribed distance. Our approach relies on a combinatorial correspondence between this family of constellations and the simpler family of rooted constellations, which we may formulate algebraically in terms of multicontinued fractions and generalized Hankel determinants. As an application, we provide a combinatorial derivation of the generating function of Eulerian triangulations with two points at a prescribed distance. Nous considérons le problème du comptage des constellations planaires à deux points marqués à distance donnée. Notre approche repose sur une correspondance combinatoire entre cette famille de constellations et celle, plus simple, des constellations enracinées. La correspondance peut être reformulée algébriquement en termes de fractions multicontinues et de déterminants de Hankel généralisés. Comme application, nous obtenons par une preuve combinatoire la série génératrice des triangulations eulériennes à deux points marqués à distance donnée.

The combinatorial derivation and its inverse mapping

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Igor Protasov
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Inverse Mapping ◽  
Combinatorial Derivation ◽  
Limit Points

AbstractLet G be a group and P G be the Boolean algebra of all subsets of G. A mapping Δ: P G → P G defined by Δ(A) = {g ∈ G: gA ∩ A is infinite} is called the combinatorial derivation. The mapping Δ can be considered as an analogue of the topological derivation d: P X→ P X, A ↦ A d, where X is a topological space and A d is the set of all limit points of A. We study the behaviour of subsets of G under action of Δ and its inverse mapping ∇. For example, we show that if G is infinite and I is an ideal in P G such that Δ(A) ∈ I and ∇(A) ⊆ I for each A ∈ I then I = P G.

scholarly journals Around $P$-small subsets of groups

2014 ◽  
Vol 6 (2) ◽  
pp. 337-341 ◽  
Author(s):  
I.V. Protasov ◽  
K.D. Protasova
Keyword(s):  
Pairwise Disjoint ◽  
Inverse Mapping ◽  
Combinatorial Derivation ◽  
Injective Sequence

A subset $X$ of a group $G$ is called  $P$-small (almost $P$-small) if there exists an injective sequence $(g_{n})_{n\in\omega}$ in $G$ such that the subsets $(g_{n}X)_{n\in\omega}$  are pairwise disjoint ($g_{n}X\cap g_{m}X$ is finite for all distinct $n,m$), and weakly $P$-small if, for every $n\in\omega$, there exist $g_{0}, \ldots ,g_{n}\in G$ such that the subsets $g_{0} X, ..., g_{n} X$ are pairwise disjoint. We generalize these notions and say that $X$ is near $P$-small if, for every $n\in\omega$, there exist $g_{0}, \ldots ,g_{n}\in G$ such that $g_{i}X\cap g_{j}X$ is finite for all distinct $i,j \in\{0,\ldots, n\}$. We study the relationships between near $P$-small subsets and known types of subsets of a group, and the behavior of near $P$-small subsets under the action of  the combinatorial derivation and its inverse mapping.

scholarly journals Truncations of Random Unitary Matrices and Young Tableaux

10.37236/939 ◽  
2006 ◽  
Vol 14 (1) ◽  
Author(s):  
J. Novak
Keyword(s):  
Unitary Group ◽  
Lattice Gauge Theory ◽  
Main Tool ◽  
Unitary Matrix ◽  
Young Tableaux ◽  
Unitary Matrices ◽  
Average Value ◽  
Combinatorial Derivation ◽  
Single Entry ◽  
Random Unitary Matrices

Let $U$ be a matrix chosen randomly, with respect to Haar measure, from the unitary group $U(d).$ For any $k \leq d,$ and any $k \times k$ submatrix $U_k$ of $U,$ we express the average value of $|{\rm Tr}(U_k)|^{2n}$ as a sum over partitions of $n$ with at most $k$ rows whose terms count certain standard and semistandard Young tableaux. We combine our formula with a variant of the Colour-Flavour Transformation of lattice gauge theory to give a combinatorial expansion of an interesting family of unitary matrix integrals. In addition, we give a simple combinatorial derivation of the moments of a single entry of a random unitary matrix, and hence deduce that the rescaled entries converge in moments to standard complex Gaussians. Our main tool is the Weingarten function for the unitary group.

scholarly journals The shifted plactic monoid (extended abstract)

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Luis Serrano
Keyword(s):  
Schur Functions ◽  
Plactic Monoid ◽  
Combinatorial Derivation ◽  
International Audience ◽  
Noncommutative Schur Functions

International audience We introduce a shifted analog of the plactic monoid of Lascoux and Schützenberger, the \emphshifted plactic monoid. It can be defined in two different ways: via the \emphshifted Knuth relations, or using Haiman's mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P-function; a shifted counterpart of the Lascoux-Schützenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.

scholarly journals A Combinatorial Derivation with Schröder Paths of a Determinant Representation of Laurent Biorthogonal Polynomials

10.37236/800 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Shuhei Kamioka
Keyword(s):  
Recurrence Equation ◽  
Combinatorial Proof ◽  
Toeplitz Determinants ◽  
Determinant Representation ◽  
Combinatorial Derivation ◽  
Schröder Paths ◽  
Biorthogonal Polynomials ◽  
Weighted Plane

A combinatorial proof in terms of Schröder paths and other weighted plane paths is given for a determinant representation of Laurent biorthogonal polynomials (LBPs) and that of coefficients of their three-term recurrence equation. In this process, it is clarified that Toeplitz determinants of the moments of LBPs and their minors can be evaluated by enumerating certain kinds of configurations of Schröder paths in a plane.

A NEW COMBINATORIAL STUDY OF THE ROGERS–FINE IDENTITY AND A RELATED PARTIAL THETA SERIES

2009 ◽  
Vol 05 (07) ◽  
pp. 1311-1320 ◽  
Author(s):  
KRISHNASWAMI ALLADI
Keyword(s):  
Theta Series ◽  
Combinatorial Proof ◽  
Combinatorial Derivation

We provide a transparent combinatorial derivation of a variant of the Rogers–Fine identity and a new combinatorial proof of a related partial theta series.

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