Congruences for the Fishburn Numbers

When/Where:

February 25, 2014, at 1:55pm, in LIT 368

Abstract:

This talk will present joint work with James Sellers. The Fishburn numbers, \(\xi(n)\), have many interpretations and combinatorial applications. For example, \(\xi(n)\) equals the number of upper triangular matrices with nonnegative integer entries and without zero rows or columns such that the sum of all entries equals \(n\). For example, when \(n = 3\) we have the following matrices:

$$\begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{pmatrix},
\begin{pmatrix}
2 & 0\\
0 & 1
\end{pmatrix},
\begin{pmatrix}
1 & 0\\
0 & 2
\end{pmatrix},
\begin{pmatrix}
1 & 1\\
0 & 1
\end{pmatrix},
\text{ and }
\begin{pmatrix}
3
\end{pmatrix}.
$$
Thus \(\xi(3) = 5\).

In this talk I shall describe many of the interpretations of \(\xi(n)\) and will prove an infinite family of congruences for \(\xi(n)\).  In particular, \(5 \mid \xi(5n+3)\).