Tags: #definition

differential poset

A differential poset is a ranked poset with unique minimum element $\hat{0}$ satisfying:

1. the number of ways to get from $a$ to $b$ via going up and then down is equal to the number of ways to get from $a$ to $b$ with down and then up.
2. if there are $n$ items below some $a$ in your poset, then there are $n+1$ items above it in the Hasse diagram

Equivalently, there exist up $U$ and down $D$ operators satisfying

acting on the vector space $V$ consisting of basis elements $\{v_x\mid x\in P\}$ defined via

• $U:v_a\mapsto \sum _{b⋗a}{v}_b$
• $D:v_a\mapsto \sum _{b⋖a}{v}_b$

Equivalently, for all $x,y\in P$ of the same rank,

• if $x\ne y$, the number of elements covered by both $x$ and $y$ are equal to the number of elements covering both $x$ and $y$
• if $x=y$, then the number of elements covering $x$ is one more than the number of elements covered by $x+1$

Properties

• The square over all $x\in P$ of rank $n$ of the number of saturated chains from 0 to $x$ is equal to $n!$
  • $\sum _{r(x)=n}(\mathrm{\#}\text{chains }\hat{0}\to x{)}^2=n!$
  • proof is the same as proof 2 of sum over squares of number of SYTs is n!

Examples

• Young's lattice
• Fibonacci's lattice