Volume 9, Issue 4

Functions counting the number of subsets of {1, 2, . . . , n } having particular properties are defined by Nathanson. Here, generalizations in two directions are given.

In this paper we study some combinatorial properties of the antichains of a garland , or double fence (see Figure 1 for an example). Specifically, we encode the order ideals and the antichains in terms of words of a regular language, we obtain several enumerative properties (such as generating series, recurrences and explicit formulae), we consider some statistics leading to Riordan matrices, we study the relations between the lattice of ideals and the semilattice of antichains, and finally we give a combinatorial interpretation of the antichains as lattice paths with no peaks and no valleys.

We show that 1, 2 and 3 are the only Fibonacci numbers whose Euler functions are also Fibonacci numbers.

Qualified residue difference sets of power n are known to exist for n = 2, 4, 6, as do similar sets that include the zero element, while both classes of set are known to be nonexistent for n = 8 and n = 10. Both classes of set are proved nonexistent for n = 12.

Graph pebbling is a game played on a connected graph G . A player purchases pebbles at a dollar a piece and hands them to an adversary who distributes them among the vertices of G (called a configuration) and chooses a target vertex r . The player may make a pebbling move by taking two pebbles off of one vertex and moving one of them to a neighboring vertex. The player wins the game if he can move k pebbles to r . The value of the game ( G, k ), called the k -pebbling number of G and denoted π k ( G ), is the minimum cost to the player to guarantee a win. That is, it is the smallest positive integer m of pebbles so that, from every configuration of size m , one can move k pebbles to any target. In this paper, we use the block structure of graphs to investigate pebbling numbers, and we present the exact pebbling number of the graphs whose blocks are complete. We also provide an upper bound for the k -pebbling number of diameter-two graphs, which can be the basis for further investigation into the pebbling numbers of graphs with blocks that have diameter at most two.

It is shown that the conjugacy classes of integral matrices with a given irreducible characteristic polynomial is in bijection with the class group of a corresponding order in an algebraic number field.

In 2005, Sloane and Sellers defined a function b ( n ) which denotes the number of non-squashing partitions of n into distinct parts. In their 2005 paper, Sloane and Sellers also proved various congruence properties modulo 2 satisfied by b ( n ). In this note, we extend their results by proving two infinite families of congruence properties modulo 4 for b ( n ). In particular, we prove that for all k ≥ 3 and all n ≥ 0, b (2 2 k +1 n + 2 2 k –2 ) ≡ 0 (mod 4) and b (2 2 k +1 n + 3 · 2 2 k –2 + 1) ≡ 0 (mod 4).

Let p be a prime, n be an integer, k | p n –1, and γ ( k, p n ) be the minimal value of s such that every number in 𝔽 p n is a sum of s k th powers. A known upper bound is improved to γ ( k, p n ) ≪ nk 1/ n and generalizations of Heilbronn's conjectures are proven for an arbitrary finite field.

In this paper, we show that the only triple of positive integers a < b < c such that ab + 1, ac + 1 and bc + 1 are all members of the Lucas sequence ( L n ) n ≥0 is ( a , b , c ) = (1, 2, 3).