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\vskip 1cm{\LARGE\bf {The Half-Perimeter Generating Function of\\
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Gated and Wicketed Ferrers Diagrams}
}
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\large
Arvind Ayyer\\
Department of Physics\\
Rutgers --- The State University \\
136 Frelinghuysen Rd\\
Piscataway, NJ 08854 \\
USA \\
\href{mailto:ayyer@physics.rutgers.edu}{\tt ayyer@physics.rutgers.edu}
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\begin{abstract}
We show that the half-perimeter generating functions for the number of
wicketed and gated Ferrers diagrams is algebraic. Furthermore, the
generating function of the wicketed Ferrers diagrams is closely related
to the generating function of the Catalan numbers. The methodology of
the experimentation, as well as the proof, is the umbral transfer matrix
method.
\end{abstract}
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\section{Introduction}
Motivated by the recent interest in the enumeration of staircase
polygons with a single staircase puncture (see \seqnum{A057410} in
\cite{sloane}) and conjectures of a holonomic solution \cite{gutt,gutt2}, we
investigate the simpler problem of Ferrers diagrams with Ferrers
punctures of different kinds---wickets and gates. Since Ferrers
diagrams form a subset of staircase polygons, we simple-mindedly expect
a holonomic solution here too. Fortunately for us, this na\"{\i}ve
expectation turns out to be true and the generating function in both
cases is not only holonomic, but also algebraic and moreover, the
degree of the algebraic equation satisfied by the generating function
is two!
The umbral transfer matrix method \cite{zeil} is a technique to
calculate terms in the series expansion of generating functions. For a
given combinatorial building, the umbral operator is essentially the
architectural plan for the structure. At any stage of the construction,
it tells the builder exactly how to proceed from there on. The power of
the method is twofold: it helps in generating terms in the sequence and
secondly, once an ansatz is in place, it is very easy to prove or
disprove the ansatz. It is in that spirit that the proofs here must be
read, namely as simple exercises in algebra. The techniques here are
more important than the proofs.
We also emphasize the experimental nature of the paper. The proofs here
are completely computerizable. For this problem, the solution is simple
enough for everything to be done by hand. For more complex problems of
this type, however, pen-and-paper calculations would be far too long
and error-prone to be efficient. This method can be generalized to more
complex problems such as the conjecture of punctured staircase polygons
\cite{gutt,gutt2}.
The plan of the paper is as follows: We first introduce the method by
applying it to the simple case of standard Ferrers diagrams. We then go
on to apply it to the case of interest deriving the various umbral
operators that arise and proving the main theorems. Finally, we offer
hope for a bijective proof of the main theorem.
\section{Standard Ferrers Diagrams}
\begin{defn}
A \emph{Ferrers diagram} is a collection of $n$ rows of blocks, the $i$th row of which contains $m_i$ blocks, the first row at the bottom and the last row on top. All the rows are left aligned and such that if $1 \leq i