The notion of a word-representable graph has been studied in a series of papers in the literature. A graph G = (V, E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if xy is an edge in E. If V = {1, . . . , n}, this is equivalent to saying that G is word-representable if for all x, y ∈ {1, . . . , n}, xy ∈ E if and only if the subword w {x,y} of w consisting of all occurrences of x or y in w has no consecutive occurrence of the pattern 11.In this paper, we introduce the study of u-representable graphs for any word u ∈ {1, 2} * . A graph G is u-representable if and only if there is a labeled version of G, G = ({1, . . . , n}, E), and a word w ∈ {1, . . . , n} * such that for all x, y ∈ {1, . . . , n}, xy ∈ E if and only if w {x,y} has no consecutive occurrence of the pattern u. Thus, word-representable graphs are just 11-representable graphs. We show that for any k ≥ 3, every finite graph G is 1 k -representable. This contrasts with the fact that not all graphs are 11-representable graphs.The main focus of the paper is the study of 12-representable graphs. In particular, we classify the 12-representable trees. We show that any 12-representable graph is a comparability graph and the class of 12-representable graphs include the classes of cointerval graphs and permutation graphs. We also state a number of facts on 12-representation of induced subgraphs of a grid graph.

In this paper, we introduce a new method for computing generating functions with respect to the number of descents and left-to-right minima over the set of permutations which have no consecutive occurrence of τ where τ starts with 1. In particular, we study the generating function n≥0

Abstract. In this paper, we study the distribution of the number occurrences of the simplest frame pattern, called the µ pattern, in n-cycles. Given an n-cycle C, we say that a pair i, j matches the µ pattern if i < j and as we traverse around C in a clockwise direction starting at i and ending at j, we never encounter a k with i < k < j. We say that i, j is a nontrivial µ-match if i + 1 < j. We say that an n-cycle C is incontractible if there is no i such that i + 1 immediately follows i in C. We show that number of incontractible n-cycles in the symmetric group Sn is Dn−1 where Dn is the number of derangements in Sn. We show that number of n-cycles in Sn with exactly k µ-matches can be expressed as a linear combination of binomial coefficients of the form n−1 i where i ≤ 2k + 1. We also show that the generating function N T In,µ(q) of q raised to the number of nontrivial µ-matches in C over all incontractible n-cycles in Sn is a new qanalogue of Dn−1 which is different from the q-analogues of the derangement numbers that have been studied by Garsia and Remmel and by Wachs. We will show that there is a rather surprising connection between the charge statistic on permutations due to Lascoux and Schüzenberger and our polynomials in that the coefficient of the smallest power of q in N T I 2k+1,µ (q) is the number of permutations in S 2k+1 whose charge path is a Dyck path. Finally, we show that N T In,µ(q)| q ( n−12 )−k and N Tn,µ(q)| q ( n−12 )−k are the number of partitions of k for sufficiently large n.

International audience In this paper, we develop a new method to compute generating functions of the form $NM_τ (t,x,y) = \sum\limits_{n ≥0} {\frac{t^n} {n!}}∑_{σ ∈\mathcal{lNM_{n}(τ )}} x^{LRMin(σ)} y^{1+des(σ )}$ where $τ$ is a permutation that starts with $1, \mathcal{NM_n}(τ )$ is the set of permutations in the symmetric group $S_n$ with no $τ$ -matches, and for any permutation $σ ∈S_n$, $LRMin(σ )$ is the number of left-to-right minima of $σ$ and $des(σ )$ is the number of descents of $σ$ . Our method does not compute $NM_τ (t,x,y)$ directly, but assumes that $NM_τ (t,x,y) = \frac{1}{/ (U_τ (t,y))^x}$ where $U_τ (t,y) = \sum_{n ≥0} U_τ ,n(y) \frac{t^n}{ n!}$ so that $U_τ (t,y) = \frac{1}{ NM_τ (t,1,y)}$. We then use the so-called homomorphism method and the combinatorial interpretation of $NM_τ (t,1,y)$ to develop recursions for the coefficient of $U_τ (t,y)$. Dans cet article, nous développons une nouvelle méthode pour calculer les fonctions génératrices de la forme $NM_τ (t,x,y) = \sum\limits_{n ≥0} {\frac{t^n} {n!}}∑_{σ ∈\mathcal{lNM_{n}(τ )}} x^{LRMin(σ)} y^{1+des(σ )}$ où τ est une permutation, $\mathcal{NM_n}(τ )$ est l'ensemble des permutations dans le groupe symétrique $S_n$ sans $τ$-matches, et pour toute permutation $σ ∈S_n$, $LRMin(σ )$ est le nombre de minima de gauche à droite de $σ$ et $des(σ )$ est le nombre de descentes de $σ$ . Notre méthode ne calcule pas $NM_τ (t,x,y)$ directement, mais suppose que $NM_τ (t,x,y) = \frac{1}{/ (U_τ (t,y))^x}$ où $U_τ (t,y) = \sum_{n ≥0} U_τ ,n(y) \frac{t^n}{ n!}$ de sorte que $U_τ (t,y) = \frac{1}{ NM_τ (t,1,y)}$. Nous utilisons ensuite la méthode dite "de l'homomorphisme'' et l'interprétation combinatoire de $NM_τ (t,1,y)$ pour développer des récursions sur le coefficient de $U_τ (t,y)$.

Abstract. The Schur functions in superspace s Λ ands Λ are the limits q = t = 0 and q = t = ∞ respectively of the Macdonald polynomials in superspace. We prove Pieri rules for the bases s Λ ands Λ (which happen to be essentially dual). As a consequence, we derive the basic properties of these bases such as dualities, monomial expansions, and tableaux generating functions.