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Completely dissociative groupoids

Milton Braitt, David Hobby, Donald Silberger (2012)

Mathematica Bohemica

In a groupoid, consider arbitrarily parenthesized expressions on the k variables x 0 , x 1 , ⋯ x k - 1 where each x i appears once and all variables appear in order of their indices. We call these expressions k -ary formal products, and denote the set containing all of them by F σ ( k ) . If u , v ∈ F σ ( k ) are distinct, the statement that u and v are equal for all values of x 0 , x 1 , ⋯ x k - 1 is a generalized associative law. Among other results, we show that many small groupoids are completely dissociative, meaning that no generalized associative law holds...

ECO species.

Ferrari, Luca, Leroux, Pierre (2009)

Séminaire Lotharingien de Combinatoire [electronic only]

General numeration I. Gauged schemes.

D. W. Dubois (1982)

Revista Matemática Hispanoamericana

The paper deals with special partitions of whole numbers in the following form: given a sequence of pairs {[Gi;Di]} of positive integers in which the Gi form a strictly increasing sequence, sums of the form ∑niGi, with 0 ≤ ni ≤ Di, are considered. The correspondence[nk ... n0] → ∑i≤k niGidefines then a mapping α from a set M of numerals, called Neugebauer symbols, satisfying 0 ≤ ni ≤ Di, into the set W of all non-negative integers. In M, initial zeros are supressed and M is ordered in the usual...

Graphs S ( n , k ) and a variant of the Tower of Hanoi problem

Sandi Klavžar, Uroš Milutinović (1997)

Czechoslovak Mathematical Journal

For any n ≥ 1 and any k ≥ 1 , a graph S ( n , k ) is introduced. Vertices of S ( n , k ) are n -tuples over { 1 , 2 , ... , k } and two n -tuples are adjacent if they are in a certain relation. These graphs are graphs of a particular variant of the Tower of Hanoi problem. Namely, the graphs S ( n , 3 ) are isomorphic to the graphs of the Tower of Hanoi problem. It is proved that there are at most two shortest paths between any two vertices of S ( n , k ) . Together with a formula for the distance, this result is used to compute the distance between two vertices in...

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