Determine Unknown Counts of Arrow-Type Semigroupoids

Determine the number of isomorphism classes of arrow-type semigroupoids—equivalently, transitively closed directed graphs without parallel edges—for each (number of arrows, number of objects) pair corresponding to the grey cells in Table 1 that are marked as currently unknown. Concretely, for every indicated pair (n, m), compute the count of distinct arrow-type semigroupoids with n arrows and m objects up to isomorphism.

Background

Arrow-type semigroupoids encode the composability structure of arrow-types (domain–codomain pairs) of a semigroupoid. They can be viewed as transitively closed directed graphs without parallel edges, and their enumeration provides blueprints of possible type-connectivity patterns. The paper presents a table summarizing known counts of such structures parameterized by the number of arrows and objects.

The authors indicate that several entries in this table remain undetermined and are shown in grey, reflecting ongoing computation. Computing these counts requires handling isomorphism classes of directed graphs (graph canonization), which is computationally challenging. The open task is to complete the enumeration by determining the unknown values for the specified parameter pairs.