<abstract xmlns="http://www.w3.org/1999/xhtml"> Since the last two decades huge systems (such as giant graphs, big data structures, . . . ) have played a central role in computer science, and with the technology improvements, those large objects are now massively used in practice. In order to handle them we need to analyse some typical properties of models of large objects. One way to study typical behaviours consists in generating random objects to get some experimental results on their properties. A new technique has been introduced ten years ago: the Boltzmann sampling. It has been presented by Duchon et al, and is based on automatic interpretation in terms of samplers of the specification of the combinatorial objects under study.One of the core problem in Boltzmann sampling lies in the distribution of the object sizes, and the choice of some parameters in order to get the more appropriate size distribution. From this choice depends the efficiency of the sampling. Moreover some additional ideas allows to improve the efficiency, one of them is based on some anticipated rejections, the other one on the combinatorial differentiation of the specification. Anticipated rejection consists during the recursive building of a random object to kill the process as soon as we are sure to exceed the maximum target size, rather than waiting until the natural end of the process. In the original paper, while both approaches have been presented, and used on the same kind of structures, the methods are not compared. We propose in this paper a detailed comparison of both approaches, in order to understand precisely which method is the more efficient. </abstract>

Laboratoire d’Informatique de Paris-Nord CNRS UMR 7030 Université Paris-Nord 93430 Villetaneuse, France

Sorbonne Universités UPCM Univ. Paris 06, CNRS, LIP6 UMR 7606 75005 Paris, France

Pointed versus singular Boltzmann samplers: a comparative analysis

<abstract xmlns="http://www.w3.org/1999/xhtml"> We establish an alternating sum identity for three classes of singleton-free set partitions wherein the number of elements minus the number of blocks is fixed: (i) permutations, that is, partitions into cycles, (ii) unrestricted partitions, and (iii) contents-ordered partitions. Both algebraic and combinatorial proofs are given, the latter making use of a sign-changing involution in each ease. As a consequence, combinatorial proofs are found of specific cases of recent identities of Gould et al. involving both kinds of Stirling numbers. </abstract>

Department of Statistics University of Wisconsin Madison, WI 53706, USA

Department of Mathematics University of Haifa 3498838 Haifa, Israel

Department of Mathematics University of Tennessee Knoxville, TN 37996 USA

Some identities for derangement and Ward number sequences and related bijections

<abstract xmlns="http://www.w3.org/1999/xhtml"> The aim is to correct part of the Remark 3 of my paper “On regular, intra-regular ordered semigroups” in Pure Math. Appl. (PU.M.A.) 4, no. 4 (1993), 447-461. On this occasion, some further results and the similarity between the po-semigroups and the le-semigroups are discussed. </abstract>

Remark on quasi-ideals of ordered semigroups

<abstract xmlns="http://www.w3.org/1999/xhtml"> Two well known results on semigroups, one of them clue to Calais, the other to Lajos, are generalized in case of hypersemigroups in the way indicated in the present paper. We prove that a nonempty subset B of a regular hypersemigroup H is a bi-ideal of H if and only if it is represented in the form B = A * C where A is a right ideal and C a left ideal of H. We also show that an hyper semigroup H is regular if and only if the right and the left ideals of H are idempotent, and for every right ideal A and every left ideal B of H, the product A * B is a quasi-ideal of H. </abstract>

On hypersemigroups

<abstract xmlns="http://www.w3.org/1999/xhtml"> A permutation of length n may be represented, equivalently, by a sequence a1a2 • • • an satisfying 0 &lt; ai &lt; i for all z, which is called an inversion sequence. In analogy to the usual case for permutations, the pattern avoidance question is addressed for inversion sequences. In particular, explicit formulas and/or generating functions are derived which count the inversion sequences of a given length that avoid a single pattern of length three. Among the sequences encountered are the Fibonacci numbers, the Schröder numbers, and entry A200753 in OEIS. We make use of both algebraic and combinatorial methods to establish our results. An explicit Injection is given between two of the avoidance classes, and in three cases, the kernel method is used to solve a functional equation satisfied by the generating function enumerating the class in question. </abstract>

Department of Mathematics University of Haifa 31905 Haifa, Israel

Pattern avoidance in inversion sequences

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