Concave Compositions

Abstract

Concave compositions are compositions (i.e. ordered partitions) of a number in which the parts decrease up to the middle summand(s) and increase thereafter. Perhaps the most surprising result is for even length, concave compositions where the generating function turns out to be the quotient of two instances of the pentagonal number theorem with variations of sign. The false theta function discoveries also lead to new facts about concatenatable, spiral, self-avoiding walks.