We introduce square diagrams that represent numerical semigroups and we obtain an injection from the set of numerical semigroups into the set of Dyck paths.

the first ($pi_1$) and the second $(pi_2$) multiplicative zagreb indices of a connected graph $g$, with vertex set $v(g)$ and edge set $e(g)$, are defined as $pi_1(g) = prod_{u in v(g)} {d_u}^2$ and $pi_2(g) = prod_{uv in e(g)} {d_u}d_{v}$, respectively, where ${d_u}$ denotes the degree of the vertex $u$. in this paper we present a simple approach to order these indices for connected graphs on ...

The Khovanov–Lauda–Rouquier (KLR) algebra arose out of attempts to categorify quantum groups. Kleshchev and Ram proved a result reducing the representation theory of these algebras of finite type to the study of irreducible cuspidal representations. In type A, these cuspidal representations are included in the class of homogeneous representations, which are related to fully commutative elements...

This thesis concerns the enumeration of pattern-avoiding permutations with respect to certain statistics. Our first result is that the joint distribution of the pair of statistics ‘number of fixed points’ and ‘number of excedances’ is the same in 321-avoiding as in 132-avoiding permutations. This generalizes a recent result of Robertson, Saracino and Zeilberger, for which we also give another, ...

P. Erdiis [4] communicated the conjecture that this is the only situation in which (1) holds for all edges e in G. Conjecture. If (1) holds for every edge e of a graph G, then G contains no triangle. A counterexample is given in this note. Unfortunately the origin of the conjecture is not known to us. The symbol (p, q} denotes the regular tessellation of a simply connected surface into p-gons, ...

This paper is devoted to characterize permutations with forbidden patterns by using canonical reduced decompositions, which leads to bijections between Dyck paths and Sn(321) and Sn(231), respectively. We also discuss permutations in Sn avoiding two patterns, one of length 3 and the other of length k. These permutations produce a kind of discrete continuity between the Motzkin and the Catalan n...

Our starting point is one of the main results of [BBFP], which we are going to recall in the next lines. Denote by Dn, NC(n) and Sn(312) the sets of Dyck paths of length 2n, noncrossing partitions of [1, n] and 312-avoiding permutations of [1, n], respectively, where [1, n] is the set of positive integers less than or equal to n. For our purposes, the following notations will be particularly us...