Combinatorial proof of a curious -binomial coefficient identity.
We study the problem of consistent and homogeneous colourings for increasing families of dyadic intervals. We determine when this problem can be solved and when it cannot.
In a groupoid, consider arbitrarily parenthesized expressions on the variables where each appears once and all variables appear in order of their indices. We call these expressions -ary formal products, and denote the set containing all of them by . If are distinct, the statement that and are equal for all values of is a generalized associative law. Among other results, we show that many small groupoids are completely dissociative, meaning that no generalized associative law holds...