U.L.B. c.p. 216,
Bd du triomphe,
1050 Bruxelles,

Counting structures of preference

A set $A$ of $n$ alternatives has $n!$ distinct linear orderings but only $1$ type of linear ordering. This illustrates two possible viewpoints for counting specific relations on $A$: either we see $A$ as a labelled set, or we consider relations up to isomorphism. We first briefly review some known results and open questions about counting specific preference structures on $A$, such as weak orders, semiorders, or interval orders. Then we present new findings about biorders (also called Guttman scales or Ferrers relations) from a set $X$ of $m$ elements to a set $Y$ of $n$ elements. Counting biorders up to isomorphism is easily done. For labelled sets $X$ and $Y$, a recurrence formula as well as two explicit formulas involving Stirling numbers of the second kind are obtained for the number of biorders from $X$ to $Y$. These findings are taken from joint work with Julie Christophe and Samuel Fiorini (a paper is accepted in the Journal of Integer Sequences).