###### Colourability and word-representability of near-triangulations

## Abstract

A graph *G* = (*V;E*) is word-representable if there is a word *w* over the alphabet *V* such that *x* and *y* alternate in *w* if and only if the edge (*x; y*) is in *G*. It is known [6] that all 3-colourable graphs are word-representable, while among those with a higher chromatic number some are word-representable while others are not.

There has been some recent research on the word-representability of polyomino triangulations. Akrobotu et al. [1] showed that a triangulation of a convex polyomino is word-representable if and only if it is 3-colourable; and Glen and Kitaev [5] extended this result to the case of a rectangular polyomino triangulation when a single domino tile is allowed.

It was shown in [4] that a near-triangulation is 3-colourable if and only if it is internally even. This paper provides a much shorter and more elegant proof of this fact, and also shows that near-triangulations are in fact a generalization of the polyomino triangulations studied in [1] and [5], and so we generalize the results of these two papers, and solve all open problems stated in [5].

###### The Deletion-Insertion model applied to the genome rearrangement problem

## Abstract

Permutations are frequently used in solving the genome rearrangement problem, whose goal is finding the shortest sequence of mutations transforming one genome into another. We introduce the Deletion-Insertion model (DI) to model small-scale mutations in species with linear chromosomes, such as humans. Applying one restriction to this model, we obtain the transposition model for genome rearrangement, which was shown to be NP-hard in [4]. We use combinatorial reasoning and permutation statistics to develop a polynomial-time algorithm to approximate the minimum number of transpositions required in the transposition model and to analyze the sharpness of several bounds on transpositions between genomes.

###### Enumeration of permutations avoiding a triple of 4-letter patterns is almost all done

## Abstract

This paper basically completes a project to enumerate permutations avoiding a triple *T* of 4-letter patterns, in the sense of classical pattern avoidance, for every *T*. There are 317 symmetry classes of such triples *T* and previous papers have enumerated avoiders for all but 14 of them. One of these 14 is conjectured not to have an algebraic generating function. Here, we find the generating function for each of the remaining 13, and it is algebraic in each case.

######
Generalized Jacobsthal numbers and restricted *k*-ary words

## Abstract

We consider a generalization of the problem of counting ternary words of a given length which was recently investigated by Koshy and Grimaldi [10]. In particular, we use finite automata and ordinary generating functions in deriving a *k*-ary generalization. This approach allows us to obtain a general setting in which to study this problem over a *k*-ary language. The corresponding class of *n*-letter *k*-ary words is seen to be equinumerous with the closed walks of length *n* − 1 on the complete graph for *k* vertices as well as a restricted subset of colored square-and-domino tilings of the same length. A further polynomial extension of the *k*-ary case is introduced and its basic properties deduced. As a consequence, one obtains some apparently new binomial-type identities via a combinatorial argument.

###### On the exhaustive generation of generalized ballot sequences in lexicographic and Gray code order

## Abstract

A generalized (resp. *p*-ary) ballot sequence is a sequence over the set of non-negative integers (resp. integers less than *p*) where in any of its prefixes each positive integer *i* occurs at most as often as any integer less than *i*. We show that the Reected Gray Code order induces a cyclic 3-adjacent Gray code on both, the set of fixed length generalized ballot sequences and *p*-ary ballot sequences when *p* is even, that is, ordered list where consecutive sequences (regarding the list cyclically) differ in at most 3 adjacent positions. Non-trivial efficient generating algorithms for these ballot sequences, in lexicographic order and for the obtained Gray codes, are also presented.

###### On the largest part size and its multiplicity of a random integer partition

## Abstract

Let λ be a partition of the positive integer *n* chosen uniformly at random among all such partitions. Let *L _{n}* =

*L*

*(λ) and*

_{n}*M*

*=*

_{n}*M*

*(λ) be the largest part size and its multiplicity, respectively. For large*

_{n}*n*, we focus on a comparison between the partition statistics

*L*

*and*

_{n}*L*

_{n}*M*

*. In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of*

_{n}*L*

_{n}*M*

*–*

_{n}*L*

*grows as fast as*

_{n}###### The Logical Sustainability Theory for pension systems: the discrete-time model in a stochastic framework under variable mortality

## Abstract

The aim of this work is to provide the logical sustainability model for defined contribution pension systems (see [1], [2]) in the discrete framework under stochastic financial rate of the pension system fund and stochastic productivity of the active participants. In addition, the model is developed in the assumption of variable mortality tables.

Under these assumptions, the evolution equations of the fundamental state variables, the pension liability and the fund, are provided. In this very general discrete framework, the necessary and sufficient condition of the pension system sustainability, and all the other basic results of the logical sustainability theory, are proved.

In addition, in this work new results on the efficiency of the rule for the stabilization over time of the level of the unfunded pension liability with respect to wages, level that is defined as β indicator, are also proved.

###### Eigenvalues asymptotics for Stark operators

## Abstract

We give the eigenvalues asymptotics for the Stark operator of the form −Δ+*F x*, *F >* 0 on L^{2}([0, *d*]). This is given in the case when *F* is small enough or sufficiently large. We impose various boundary conditions. The proof is based on the asymptotics of the specialized Airy functions.

######
Approach of *q*-Derivative Operators to Terminating *q*-Series Formulae

## Abstract

The *q*-derivative operator approach is illustrated by reviewing several typical summation formulae of terminating basic hypergeometric series.

###### Geometry of Mus-Sasaki metric

## Abstract

In this paper, we introduce the Mus-Sasaki metric on the tangent bundle *T M* as a new natural metric non-rigid on *T M*. First we investigate the geometry of the Mus-Sasakian metrics and we characterize the sectional curvature and the scalar curvature.