How to Draw a Correlation Function

We discuss connection between the XX0 Heisenberg spin chain and some aspects of enumerative combinatorics. The representation of the Bethe wave functions via the Schur functions allows to apply the theory of symmetric functions to the calculation of the correlation functions. We provide a combinatorial derivation of the dynamical auto-correlation functions and visualise them in terms of nests of self-avoiding lattice paths. Asymptotics of the auto-correlation functions are obtained in the double scaling limit provided that the evolution parameter is large.

Entropy Multiparticle Correlation Expansion for a Crystal

As first shown by H. S. Green in 1952, the entropy of a classical fluid of identical particles can be written as a sum of many-particle contributions, each of them being a distinctive functional of all spatial distribution functions up to a given order. By revisiting the combinatorial derivation of the entropy formula, we argue that a similar correlation expansion holds for the entropy of a crystalline system. We discuss how one- and two-body entropies scale with the size of the crystal, and provide fresh numerical data to check the expectation, grounded in theoretical arguments, that both entropies are extensive quantities.

Around $P$-small subsets of groups

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.

The combinatorial derivation and its inverse mapping

Let 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.

Constellations and multicontinued fractions: application to Eulerian triangulations

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.

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

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.

The shifted plactic monoid (extended abstract)

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.

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

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 Combinatorial Derivation of the PASEP Stationary State

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.