The ACO Seminar (2018–2019)

Let $S(n,k)$ be a Stirling number of the second kind, the number of ways to partition an $n$-element set into $k$ subsets. Similarly $s(n,k)$ is a Stirling number of the first kind and $L(n,k)$ is a Lah number. They count the number of ways to partition an $n$-element set into $k$ cycles and $k$ total orders, respectively. Let $S$ be the matrix $[S(n,k)]$ with $S(n,k)$ in the $n$th row and $k$th column of $S$. Similarly $s = [s(n,k)]$ and $L = [L(n,k)]$. The inverses of $S$, $s$, and $L$ are classically known: $S^{-1} = [(-1)^{n-k} s(n,k)]$, $s^{-1} = [(-1)^{n-k} S(n,k)]$ and $L^{-1} = [(-1)^{n-k} L(n,k)]$.

Let $R$ be a subset of the positive integers. What about $S(R,n,k)$, $s(R,n,k)$, and $L(R,n,k)$, the numbers that count only the respective structures that have all part sizes in $R$? Just like $S$, $s$, and $L$, do the inverses of $S(R) = [S(R,n,k)]$, $s(r) = [s(R,n,k)]$ and $L(R) = [L(R,n,k)]$ also have inverses whose entries up to sign are the counts of certain combinatorial structures? The answer is yes when $R$ contains $1$ (as it must for the inverses to exist) and the other endpoints of the intervals comprising $R$ are even.

This is joint work with John Engbers of Marquette University and David Galvin of the University of Notre Dame.

Before the talk, at 3:10pm, there will be tea and cookies in Wean 6220.