Absorbing States of Binary Trust Gossip Are Counted by Plane Partitions

We study an opinion dynamics model in which $n$ agents hold directed trust or distrust opinions about one another, represented as a matrix $M\in \{0,1{\}}^{n\times n}$ in which 1 represents trust and 0 represents distrust. A gossip event $(a,z,y)$ causes agent $z$ to adopt agent $a$’s opinion of $y$, provided that $z$ trusts $a$. We characterize the absorbing states of this process, i.e. the states in which no further opinion change can take place: we find that they are the states in which agents are partitioned into isolated factions, each faction containing a subset of core members who share mutual trust, while the remaining peripheral members trust all core members but receive no trust in return. This structure establishes a bijection between absorbing states on $[n]$ and pairs consisting of a set partition $\pi$ of $[n]$ together with a choice of non-empty subset of each faction of $\pi$. The number of such absorbing states is therefore given by OEIS A143405, with exponential generating function $\exp (\exp (x)\cdot (\exp (x)-1))$. In addition, up to isomorphism, the count equals the number of plane partitions of $n$, given by OEIS A000219, recovering MacMahon’s classical product formula ${\prod }_{k\ge 1}1\mathrm{/}(1-x^k{)}^k$. Exhaustive computation for $n\le 7$ confirms both counts.