Abstract. AMotzkin path of length n is a lattice path from (0, 0) to (n, 0) in the plane integer lattice Z × Z consisting of horizontal-steps (1, 0), up-steps (1, 1), and down-steps (1,−1), which never passes below the x-axis. A u-segment (resp. h-segment ) of a Motzkin path is a maximum sequence of consecutive up-steps (resp. horizontal-steps). The present paper studies two kinds of statistics...

We introduce two possible generalizations of the classical Blissard problem and we show how to solve them by using the second order and multi-dimensional Bell polynomials, whose most important properties are recalled. © 2010 Elsevier Ltd. All rights reserved.

In this paper, we study the relationship between the number of n-vertex graphs in a hereditary class X , also known as the speed of the class X , and boundedness of the clique-width in this class. We show that if the speed of X is faster than n!cn for any c, then the clique-width of graphs in X is unbounded, while if the speed does not exceed the Bell number Bn, then the clique-width is bounded...

We introduce a family of polynomials that generalizes the Bell polynomials, in connection with the combinatorics of the central moments of the Poisson distribution. We show that these polynomials are dual of the Charlier polynomials by the Stirling transform, and we study the resulting combinatorial identities for the number of partitions of a set into subsets of size at least 2.

We establish an extension of Spivey’s Bell number formula and its associated Bell polynomial extension by using Hsu-Shiue’s generalized Stirling numbers. By means of the extension of Spivey’s Bell number formula we also extend Gould-Quaintance’s new Bell number formulas.

The labeled factorizations of a positive integer n are obtained as a completion of the set of ordered factorizations of n. This follows a new technique for generating ordered factorizations found by extending a method for unordered factorizations that relies on partitioning the multiset of prime factors of n. Our results include explicit enumeration formulas and some combinatorial identities. I...

Abstract The n th r -extended Lah–Bell number is defined as the of ways a set with $n+r$ n + r elements can be partitioned into ordered blocks such that distinguished have to in distinct blocks. aim this paper introduce incomplete polynomials and complete respectively multiva...