MIT Combinatorics Seminar

Posets with the Same Number of Order Ideals of Each Cardinality: A Problem from Stanley's Enumerative Combinatorics

Jonathan David Farley(Harvard University)

Wednesday March 2, 2005,   4:15 pm    Room 2-338


In Richard P. Stanley's 1986 text, {\sl Enumerative Combinatorics\/}, the following problem is posed: Fix a natural number $k$. Consider the posets $P$ of cardinality $n$ such that, for $0<i<n$, $P$ has exactly $k$ order ideals (down-sets) of cardinality $i$. Let $f_k(n)$ be the number of such posets. What is the generating function $\sum f_k(n) x^n$?

We solve this problem. (Joint work with Ryan Klippenstine.)