The Link between Scrambling Numbers and Derangements

The group equation abcdef = dabecf can be reduced to the equation xcde = dxec. In general, we are interested in how many variables are needed to represent group equations in which the right side is a permutation of the variables on the left side. Scrambling numbers capture this information about a permutation. In this paper we present several facts about scrambling numbers, and expose a striking relationship between permutations that cannot be reduced and derangements

Invariants on primary abelian groups and a problem of Nunke

If $G$ is an arbitrary abelian $p$-group, an invariant $K_G$ is defined which measures how closely $G$ resembles a direct sum of cyclic groups. This invariant consists of a class of finite sets of regular cardinals, and is inductively constructed using filtrations of various subgroups of $G$; $K_G$ can also be considered to be a measure of the presence of non-zero elements of infinite height in $G$. This construction is particularly useful when the group has final rank less than the smallest weakly Mahlo cardinal; and in this case, $G$ is a direct sum of cyclics iff $K_G$ is empty. These deliberations are then used to place several of the most significant results relating to direct sums of cyclics into a significantly broader context. For example, $G$ is shown to be almost a direct sum of cyclics iff every set in $K_G$ has at least two elements. Finally, $K_G$ is used to give a more complete and concrete answer to a classical problem of Nunke, which asks when the torsion product of two abelian $p$-groups is a direct sum of cyclics

Journal of Integer Sequences, Vol. 10 (2007), Article 07.7.6

In this paper, we define a class of semiorders (or unit interval orders) that arose in the context of polyhedral combinatorics. In the first section of the paper, we will present a pure counting argument equating the number of these interesting (connected and irredundant) semiorders on n + 1 elements with the nth Riordan number. In the second section, we will make explicit the relationship between the interesting semiorders and a special class of Motzkin paths, namely, those Motzkin paths without horizontal steps of height 0, which are known to be counted by the Riordan numbers. 1 Counting Interesting Semiorders We begin with some basic definitions. Definition 1. A partially ordered set (X, ≺) is a semiorder if it satisfies the following two properties for any a,b,c,d ∈ X. • If a ≺ b and c ≺ d, a ≺ d or c ≺ b. • If a ≺ b ≺ c, then d ≺ c or a ≺ d. Semiorders are also known as unit interval orders in the literature. This name come

THE REPRESENTATION POLYHEDRON OF A SEMIORDER

Let a finite semiorder, or unit interval order, be given. All its numerical representations (when suitably defined) form a convex polyhedron. We show that the facets of the representation polyhedron correspond to the noses and hollows of the semiorder. Our main result is to prove that the coordinates of the vertices and the components of the extreme rays of the polyhedron are all integral multiples of a common value. The result follows from the total dual integrality of the system defining the representation polyhedron. Total dual integrality is in turn derived from a particular property of the oriented cycles in the directed graph of noses and hollows of a strictly upper diagonal step tableau. Our approach delivers also a new proof for the existence of the minimal representation, a concept originally discovered by Pirlot (1990). Finding combinatorial interpretations of the vertices and extreme rays of the representation polyhedron is left for future work

The Polyhedron of all Representations of a

In many practical situations, indifference is intransitive. This led Luce (1956) to base a preference model on the following principle: an alternative is judged better than another one only if the utility value of the first alternative is significantly higher than the value of the second alternative. Here, ‘significantly higher ’ means higher than the value augmented by some constant threshold. The resulting relations are called “semiorders ” by Luce. Their axiomatic description is established by Scott and Suppes (1958)—see below for details. Given a semiorder P, we form the collection of all numerical representations of P. This collection R happens to be a convex set, although not a closed or open one. It is naturally turned into (and approximated by) a polyhedral set Rε consisting of all ε-representations. We first explain the facets of Rε: in general, they bijectively correspond to the “noses ” and “hollows ” of the semiorder P. Noses and hollows were introduced by Pirlot (1991) as a tool for proving the existence of the “minimal ε-representation ” of P. They were further investigated by Doignon and Falmagne (1997). Next, we impose that the ε-representations of the semiorder P are nonnegative, and denote with R + ε the collection of such representations. Understanding the vertices and extreme rays of R + ε seems to be a more difficult problem. The minimal ε-representation of P is always a vertex of R + ε, and sometime the only one. We provide examples with other vertices, even with vertices involving another threshold than the one in the minimal ε-representation. We then offer some partial results on the vertices and extreme rays of R + ε