Monday, January 23, 2017 at 3:00 PM in SAS 4201

Clifford Smyth, UNC Greensboro

Restricted Stirling and Lah numbers and their inverses

Let $s(n,k)$, $S(n,k)$, and $L(n,k)$ be the Stirling numbers of the first kind, Stirling numbers of the second kind, and the Lah numbers, respectively. It is well known that the inverse of the infinite lower triangular matrix $[s(n,k)]_{n,k geq 1}$ is $[(-1)^{n-k} S(n,k)]_{n,k geq 1}]$, the inverse of $[S(n,k)]_{n,k}$ is $[(-1)^{n-k} s(n,k)]_{n,k}$, and the inverse of $[L(n,k)]_{n,k}$ is $[(-1)^{n-k} L(n,k)]_{n,k}$.

More generally, we consider restricted versions of these numbers, $s(n,k,R)$, $S(n,k,R)$, and $L(n,k,R)$, for arbitrary subsets $R$ of natural numbers. These are defined to be the number of ways of partitioning a set of size $n$ into $k$ non-empty cycles (for Stirling numbers of the first kind), subsets (for Stirling numbers of the second kind) or ordered lists (for Lah numbers) with the size of each cycle, subset, or list in $R$. (To recover $S(n,k)$, $s(n,k)$, and $L(n,k)$, take $R$ to be the set of natural numbers.)

If $R$ contains $1$ then the matrices $[S(n,k,R)]_{n,k}$, $[s(n,k,R)]_{n,k}$ and $[L(n,k,R)]_{n,k}$ are all invertible and their inverses have integer entries. We obtain combinatorial formulas for the entries of these inverse matrices, expressing

each such entry as the difference between the sizes of two explicitly defined sets of forests. (Note, that some of these entries must indeed be negative.)

For $S(n,k,R)$ and $L(n,k,R)$ and for certain $R$ we can do better: each $(n,k)$ entry of the inverse will have a predictable sign, $sigma(n,k)$, and its magnitude will be the size of a single explicitly defined set of forests. Among these special $R$s are those which contain $1$ and $2$ and which have the property that for all odd $n$ in $R$ with $n geq 3$ we have $n-1$ and $n+1$ in $R$. (For these $R$, the sign of the $(n,k)$ entry is $(-1)^{n-k}$, as when $R$ is the set of the natural numbers.)

Our proofs depend in part on two combinatorial interpretations of the coefficients of the compositional inverse of a power series. This is joint work with David Galvin of the University of Notre Dame and John Engbers of Marquette University.

Monday, January 30, 2017 at 3:00 PM in SAS 4201

Seth Baldwin, UNC Chapel Hill

TBA

Monday, February 6, 2017 at 3:00 PM in SAS 4201

Uladzimir Shtukar, NC Central University

Canonical bases, subalgebras, reductive pairs of Lie algebras, and possible applications

Subalgebras of Lie algebra of Lorentz group will be discussed as the basic examples at the beginning of the report. The corresponding analysis is performed by canonical bases for subspaces of a vector space. All canonical bases for 5-dimensional and 4-dimensional subspaces of a 6-dimensional vector space are found, and they are utilized to find the corresponding subalgebras of 6-dimensional Lie algebra of Lorentz group.

The examples will show that each canonical basis is associated with matrix in reduced row echelon form. This relation allows us to generalize previous results, and canonical bases for (n-1)-dimensional subspaces and for (n-2)-dimensional subspaces of n-dimensional vector space will be found. Meanwhile all reduced row echelon forms of matrices and matrices will be classified also.

Prospective applications for canonical bases will be described as the final part of the report.

Monday, February 20, 2017 at 3:00 PM in SAS 4201

Jamie Pommersheim, Reed College

TBA

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Seminar Organizers: Bojko Bakalov and Ricky Liu