The gold partition conjecture and the Lexicographic sum of posets
We prove that if a finite poset $Q$ satisfies the Gold Partition Conjecture, and $P$ is a finite poset, then for any $i\in P$ the lexicographic sum $P\circ_iQ$ of $P$ with $Q$ on the point $i$, satisfies the Gold Partition Conjecture. Let $L(P)$ be the set of linearizations of $P$, $e(P)=\#L(P)$, $ \mathbb{P}(x<y)=\frac{\#\{f\in L(P)|x<_f y\}|}{e(P)}$, and $δ(P)=\max_{x,y\in P}\min\{\mathbb{P}(x<y),$ $\mathbb{P}(y<x)\}$. We describe the behavior of those numbers when $P$ is a lexicographic sum. Namely, $\prod_{i=1}^n e(Q_i)\,$ divides $\,e(P( Q_1,\dots,Q_{|P|}))$, and $δ(P(Q_1,\dots,Q_n))\geq \max_{1\leq i\leq n}\{ δ(Q_i)\}$.
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