Entanglement in quantum graph states is intrinsically linked to rank-width, a graph complexity measure introduced by Oum and Seymour. In this work, we enable the preparation of maximally entangled deterministic graph states in constant depth by developing a general method to derive lower bounds on the rank-width of regular graphs from their edge expansion. By bridging edge-isoperimetric inequalities with the strong chromatic index and Jelínek's approach for lower bounding cut-rank, we systematically establish lower bounds for the rank-width of Cartesian products, including hypercubes, Hamming graphs, and grids. Extending this framework via Boolean function analysis, using a generalization of the Kahn-Kalai-Linial's Theorem, we strengthen the bounds for all Cartesian products by a non-trivial logarithmic factor. These methods result in the discovery of deterministic families of graphs on n vertices with a provably maximum rank-width Θ(n). Our results fill the previous gap in the literature for deterministic graph families of rank-width greater than Θ(√(n)).

In quantum computing, graph states play a crucial role in quantum error correction and measurement-based quantum computing. Preparing these states efficiently on hardware with constrained connectivity is a fundamental challenge. In this work, we establish a universal framework for graph state preparation using only Controlled-Z (CZ) gates along the edges of a given hardware connectivity graph and local complementation operations. We prove that any graph state can be prepared using only these operations, providing a constructive transpilation method that transforms the input circuit into an equivalent one without increasing the number of entangling gates. Additionally, as our approach preserves entangling count and depth of the input circuit, we show that this framework also allows for optimal graph state preparation.

We provide an explicit description of the recurrent configurations of the sandpile model on a family of graphs G_μ ,ν , which we call clique-independent graphs, indexed by two compositions μ and ν . Moreover, we define a delay statistic on these configurations, and we show that, together with the usual level statistic, it can be used to provide a new combinatorial interpretation of the celebrated shuffle theorem of Carlsson and Mellit. More precisely, we will see how to interpret the polynomials ⟨∇ e_n, e_μ h_ν⟩ in terms of these configurations.

This paper studies sorted recurrent configurations of the Abelian sandpile model on the complete split graph. We introduce two natural toppling processes, CTI and ITC toppling, on the recurrent configurations and use these to define two toppling delay statistics, wtopple_CTI and wtopple_ITC. These new toppling delay statistics are time-weighted sums for the number of vertices that topple during each iteration of the toppling processes. We then introduce the bivariate q,t-CTI and q,t-ITC polynomials that are the generating functions of the bistatistics (level,wtopple_ITC) and (level,wtopple_CTI), where level is the well-established sandpile level statistic. We prove the bistatistic (level,wtopple_ITC) maps to a bistatistic (area,bounce) on Schröder paths that was introduced by Egge, Haglund, Killpatrick and Kremer (2003). This establishes equality of the q,t-ITC polynomial and the q,t-Schröder polynomial of those same authors. This connection allows us to relate the q,t-ITC polynomial to the theory of symmetric functions and also establishes symmetry of the q,t-ITC polynomials. We conjecture equality of the q,t-CTI and q,t-ITC polynomials. We also present and prove a characterization of sorted recurrent configurations as a new class of polyominoes that we call sawtooth polyominoes. The CTI and ITC toppling processes on sorted recurrent configurations are proven to correspond to bounce paths within the polyominoes. The main difference between the two bounce paths is the initial direction in which they travel. In addition to this, and building on the results of Aval, D'Adderio, Dukes, and Le Borgne (2016), we present a cycle lemma for a slight extension of stable configurations that allows for an enumeration of sorted recurrent configurations within the framework of the sandpile model.

In [Dugan-Glennon-Gunnells-Steingrimsson-2019], the authors introduce tiered trees to define combinatorial objects counting absolutely indecomposable representations of certain quivers, and torus orbits on certain homogeneous varieties. In this paper, we use Theta operators, introduced in [D'Adderio-Iraci-VandenWyngaerd-Theta-2021], to give a symmetric function formula that enumerates these trees. We then formulate a general conjecture that extends this result, a special case of which might give some insight about how to formulate a unified Delta conjecture [Haglund-Remmel-Wilson-2018].

The one-dimensional three-state cyclic cellular automaton is a simple spatial model with three states in a cyclic "rock-paper-scissors" preypredator relationship. Starting from a random configuration, similar states gather in increasingly large clusters; asymptotically, any finite region is filled with a uniform state that is, after some time, driven out by its predator, each state taking its turn in dominating the region (heteroclinic cycles). We consider the situation where each site in the initial configuration is chosen independently at random with a different probability for each state. We prove that the asymptotic probability that a state dominates a finite region corresponds to the initial probability of its prey. The proof methods are based on discrete probability tools, mainly particle systems and random walks.

De combien de facons peuvent etre distribuees les 32 cartes d’un jeu de belote ? De combien de facons pouvons-nous obtenir 13 en sommant les resultats de 3 des ? De combien de facons peut etre melange un paquet de n cartes ? L’ambition de la combinatoire enumerative est de compter le nombre (fini) de combinaisons dans ce type de situation. La motivation pour envisager tous les cas possibles n’est pas toujours ludique. De combien de facons peut s’executer ce programme ? Combien de temps prend en moyenne cet algorithme de tri ? La probabilite d’un evenement se definit souvent comme le ratio entre le nombre de cas ou il se produit et le nombre de cas possibles. Compter necessite aussi souvent d’avoir trouve une structure sur toutes les combinaisons possibles. C’est souvent un premier pas vers la recherche (efficace) d’une combinaison optimale pour le critere que nous nous donnerons. Cette structure et ce comptage permettent aussi parfois de generer aleatoirement une, ou quelques, combinaison(s) typique(s) et ainsi de calibrer certaines choses. Un exemple de calibrage : observer une centaine d’individus pris au hasard dans une population humaine incite a fixer la hauteur des portes a environ 2,05 m, pour peu qu’on evite les rassemblements de basketteurs professionnels ou bien de gymnastes olympiques. En physique statistique, les combinaisons apparaissent frequemment comme les etats accessibles d’un systeme compose de nombreux elements, par exemple le positionnement des particules d’un gaz. Ce nombre d’etats est souvent lie a la notion d’entropie. Compter est donc souvent pertinent dans l’analyse d’un probleme mais peut etre extremement difficile. Si difficile par exemple que ce type de probleme a ete envisage en 2010 par les chercheurs Scott Aaronson et Alex Arkhipov pour mettre en evidence la superiorite de l’eventuel ordinateur quantique sur l’ordinateur classique. Ce texte propose une promenade dans des exemples d’enumerations combinatoires. Ces exemples illustrent en quoi cette discipline realise parfois un equilibre entre un modele le plus pertinent possible et les possibilites d’y faire une analyse par un calcul faisable voire efficace. En effet, un modele est une description approximative d’un objet d’etude, qui est a la fois fidele a cet objet par certains aspects et mathematiquement confortable pour l’analyse, le calcul. Dans cette communaute, un resultat est donc souvent un modele combinatoire, en general discret et non continu, permettant un calcul combinatoire, parfois qualifie de « beau » lorsque nous laissons s’epancher un exces de sentimentalisme.

For any finite graph, the Tutte polynomial is the generating function of spanning trees counted by their numbers of active external, respectively internal, edges. We consider two restrictions of this definition, either summing over a subset of spanning trees or counting only the activities in a subset of edges. Adding to the (infinite) square lattice one projective vertex in a (rational) direction ~θ, we define the restricted Tutte polynomial T~θ,W×H(q, t) summing over some periodic spanning forests of period W × H and considering only activities on edges of the fundamental domain. Those polynomials are symmetric in q and t by self-duality of square lattice. Our main result is a family of bijections indexed by a finite number of ~θ proving that (T~θ,W×H(q, 1))~θ does not depend on ~θ. Auto-duality preserving the number of trees per period and their common slope, we obtain refinements (T~θ,W×H(w, z; q, t))~θ still symmetric in q and t.

For the sandpile model on the usual two dimensional grid, we propose a weaker version of Dhar criterion to define recurrent configurations among stable biperiodic configurations. We check this new criterion via an algorithm which auto-stabilises to a canonical ultimately periodic behaviour independent of details in its not fully specified initialisation. This leads to ultimately periodic edge/vertex traversals similar to those of Cori-Le Borgne [2] in the case of finite graphs and then to a bijection with some cycle-rooted forests on the torus describing the period. A determinantal formula [5] counts all those forests and the refinement with some monodromy parameters allows to identify in some coefficients the number of recurrent configurations. The Abelian sandpile model was introduced by physicists Bak, Tang and Wiesenfeld in [1] as a model of self-organized criticality. Given a simple, undirected graph ( V ∪ { s } , E ) where we distinguish s as the sink of the graph, we consider configurations in this model which are an assignments η : V (cid:55)→ Z of some grains of sand on each vertex. We say that η is stable at x ∈ V if η ( x ) < deg( x ), and η is stable if it is stable at all x ∈ V . If η is unstable at x , then x is allowed to topple which means that the vertex x sends one grain along each incident edge. This toppling is said legal . A toppling is forced when it is not necessarily legal. Grains arriving at the sink are lost. Given a configuration η , we define a stabilization as a sequence of allowed topplings until a stable configuration is reached. The result of all stabilizations is unique due to commutations of topplings of unstable vertices and is noted stab ( η ).

We introduce two operators on stable configurations of the sandpile model that provide an algorithmic bijection between recurrent and parking configurations. This bijection preserves their equivalence classes with respect to the sandpile group. The study of these operators in the special case of the complete bipartite graph Km,n naturally leads to a generalization of the well-known Cyclic Lemma of Dvoretsky and Motzkin, via pairs of periodic bi-infinite paths in the plane having slightly different slopes. We achieve our results by interpreting the action of these operators as an action on a point in the grid Z2 which is pointed to by one of these pairs of paths. Our Cyclic Lemma allows us to enumerate several classes of polyominoes, and therefore builds on the work of Irving and Rattan (2009), Chapman et al. (2009), and Bonin et al. (2003).

We present an algorithm to compute the rank of a configuration of the sandpile model for the complete bipartite graph K_{m,n} of complexity O(m+n). Furthermore, we provide a formula for the generating function of parking sorted configurations on complete bipartite graphs K_{m,n} according to rank, degree, and the sizes m and n. The results in the present paper are similar to those found by Robert Cori and the second named author for the complete graph K_{n+1}, and they rely on the analysis of certain operators on the stable sorted configurations of K_{m,n} developed in a previous work by the authors together with Jean-Christophe Aval and Mark Dukes.

This chapter looks at some aspects of the Tutte polynomial that result from considering different methods of topplings when applying the burning test to recurrent configurations. It presents three different toppling schedules and the corresponding deterministic algorithms for the burning test. The first toppling schedule is a naive choice, which illustrates simply how to construct a corresponding spanning tree, but this correspondence is not necessarily bijective. The second toppling schedule is attributed to Dhar and Majumdar and is called the parallel update burning test. The third toppling schedule is attributed to Cori and Le Borgne and relates the sandpile model to an evaluation of the classical Tutte polynomial: up to a constant, the distribution of the number of nonsink grains on recurrent configurations is also the distribution of the Tutte external activity on spanning trees.

The paper by M. Baker and S. Norine in 2007 introduced a new parameter on configurations of graphs and gave a new result in the theory of graphs which has an algebraic geometry flavor. This result was called Riemann-Roch formula for graphs since it defines a combinatorial version of divisors and their ranks in terms of configurations on graphs. The so called chip firing game on graphs and the sandpile model in physics play a central role in this theory. In this paper we present an algorithm for the determination of the rank of configurations for the complete graph $K_n$. This algorithm has linear arithmetic complexity. The analysis of number of iterations in a less optimized version of this algorithm leads to an apparently new parameter which we call the prerank. This parameter and the parameter dinv provide an alternative description to some well known $q,t$-Catalan numbers. Restricted to a natural subset of configurations, the two natural statistics degree and rank lead to a distribution which is described by a generating function which, up to a change of variables and a rescaling, is a symmetric fraction involving two copies of Carlitz $q$-analogue of the Catalan numbers. In annex, we give an alternative presentation of the theorem of Baker and Norine in purely combinatorial terms.

We study the statistics $\mathsf{area}$, $\mathsf{bounce}$ and $\mathsf{dinv}$ associated to polyominoes in a rectangular box $m$ times $n$. We show that the bi-statistics ($\mathsf{area}$,$\mathsf{bounce}$) and ($\mathsf{area}$,$\mathsf{dinv}$) give rise to the same $q,t-$analogue of Narayana numbers, which was introduced by two of these authors in a recent paper. We prove the main conjectures of that same work, i.e. the symmetries in $q$ and $t$, and in $m$ and $n$ of these polynomials, by providing a symmetric functions interpretation which relates them to the famous diagonal harmonics.

We consider the parameter rank introduced for graph configurations by M. Baker and S. Norine. We focus on complete graphs and obtain an efficient algorithm to determine the rank for these graphs. The analysis of this algorithm leads to the definition of a parameter on Dyck words, which we call prerank. We prove that the distribution of area and prerank on Dyck words of given length $2n$ leads to a polynomial with variables $q,t$ which is symmetric in these variables. This polynomial is different from the $q,t-$Catalan polynomial studied by A. Garsia, J. Haglund and M. Haiman.

We classify recurrent configurations of the sandpile model on the complete bipartite graph K_{m,n} in which one designated vertex is a sink. We present a bijection from these recurrent configurations to decorated parallelogram polyominoes whose bounding box is a m*n rectangle. Several special types of recurrent configurations and their properties via this bijection are examined. For example, recurrent configurations whose sum of heights is minimal are shown to correspond to polyominoes of least area. Two other classes of recurrent configurations are shown to be related to bicomposition matrices, a matrix analogue of set partitions, and (2+2)-free partially ordered sets. A canonical toppling process for recurrent configurations gives rise to a path within the associated parallelogram polyominoes. This path bounces off the external edges of the polyomino, and is reminiscent of Haglund's well-known bounce statistic for Dyck paths. We define a collection of polynomials that we call q,t-Narayana polynomials, defined to be the generating function of the bistatistic (area,parabounce) on the set of parallelogram polyominoes, akin to the (area,hagbounce) bistatistic defined on Dyck paths in Haglund (2003). In doing so, we have extended a bistatistic of Egge, Haglund, Kremer and Killpatrick (2003) to the set of parallelogram polyominoes. This is one answer to their question concerning extensions to other combinatorial objects. We conjecture the q,t-Narayana polynomials to be symmetric and prove this conjecture for numerous special cases. We also show a relationship between Haglund's (area,hagbounce) statistic on Dyck paths, and our bistatistic (area,parabounce) on a sub-collection of those parallelogram polyominoes living in a (n+1)*n rectangle.

The paper by M. Baker and S. Norine in 2007 introduced a new parameter on configurations of graphs and gave a new result in the theory of graphs which has an algebraic geometry flavour. This result was called Riemann-Roch formula for graphs since it defines a combinatorial version of divisors and their ranks in terms of configuration on graphs. The so called chip firing game on graphs and the sandpile model in physics play a central role in this theory. In this paper we give a presentation of the theorem of Baker and Norine in purely combinatorial terms, which is more accessible and shorter than the original one. An algorithm for the determination of the rank of configurations is also given for the complete graph $K_n$. This algorithm has linear arithmetic complexity. The analysis of number of iterations in a less optimized version of this algorithm leads to an apparently new parameter which we call the prerank. This parameter and the classical area parameter provide an alternative description to some well known $q,t$-Catalan numbers. Restricted to a natural subset of configurations, the two natural statistics degree and rank in Riemann-Roch formula lead to a distribution which is described by a generating function which, up to a change of variables, is a symmetric fraction involving two copies of Carlitz q-analogue of the Catalan numbers.

We give a polyomino characterisation of recurrent configurations of the sandpile model on the complete bipartite graph $K_{m,n}$ in which one designated vertex is the sink. We present a bijection from these recurrent configurations to decorated parallelogram polyominoes whose bounding box is a $m×n$ rectangle. Other combinatorial structures appear in special cases of this correspondence: for example bicomposition matrices (a matrix analogue of set partitions), and (2+2)-free posets. A canonical toppling process for recurrent configurations gives rise to a path within the associated parallelogram polyominoes. We define a collection of polynomials that we call $q,t$-Narayana polynomials, the generating functions of the bistatistic $(\mathsf{area ,parabounce} )$ on the set of parallelogram polyominoes, akin to Haglund's $(\mathsf{area ,hagbounce} )$ bistatistic on Dyck paths. In doing so, we have extended a bistatistic of Egge et al. to the set of parallelogram polyominoes. This is one answer to their question concerning extensions to other combinatorial objects. We conjecture the $q,t$-Narayana polynomials to be symmetric and discuss the proofs for numerous special cases. We also show a relationship between the $q,t$-Catalan polynomials and our bistatistic $(\mathsf{area ,parabounce}) $on a subset of parallelogram polyominoes. Pour le modèle du tas de sable sur un graphe $K_m,n$ biparti complet, on donne une description des configurations rècurrentes à l'aide d'une bijection avec des polyominos parallèlogrammes dècorès de rectangle englobant $m×n$. D'autres classes combinatoires apparaissent comme des cas particuliers de cette construction: par exemple les matrices de bicomposition et les ordres partiels évitant le motif (2+2). Un processus d'éboulement canonique des configurations récurrentes se traduit par un chemin bondissant dans le polyomino parallèlogramme associè. Nous définissons une famille de polynômes, baptisée de $q,t$-Narayana, à travers la distribution d'une paire de statistique $(\mathsf{aire, poidscheminbondissant})$ sur les polyominos parallélogrammes similaire à celle de Haglund définissant les polynômes de $q,t$-Catalan sur les chemins de Dyck. Ainsi nous étendons une paire de statistique de Egge et d'autres à l'ensemble des polynominos parallélogrammes. Cela répond à l'une de leur question sur des généralistations à d'autres objets combinatoires. Nous conjecturons que les polynômes de $q,t$-Narayana sont symétriques et discutons des preuves de plusieurs cas particuliers. Nous montrons ègalement une relation avec les polynômes de $q,t$-Catalan en restreignant notre paire de statistique à un sous-ensemble des polyominos parallélogrammes.

We prove a conjecture of Desrosiers, Lapointe and Mathieu giving a closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of admissible tableaux of the non-symmetric Jack polynomials. In the final step of the proof appears an identity on weighted sums of partitions that we demonstrate using the methods of Gessel-Viennot.

Using computer enumerations and a rational approximant method of series analysis, we conjecture an expression for the first perimeter moment of directed animals on the square lattice which are confined in a strip of a given width with open boundary conditions. When the width tends to infinity, the conjecture leads to an algebraic series for the first perimeter moment of directed animals on one-half of the square lattice, similar to Conway's earlier conjecture for the whole square lattice.