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%The Goulden-Jackson Method for Infinitely Many Mistakes
%a Plain TeX file by Doron Zeilberger (10 pages)
 
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\bf
\centerline
{
THE UMBRAL TRANSFER-MATRIX METHOD.
}
\centerline
{V. The Goulden-Jackson Cluster Method for Infinitely Many Mistakes}
\rm
\bigskip
\centerline{ {\it 
Doron ZEILBERGER
}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019, USA.
%\break
{\eighttt zeilberg@math.rutgers.edu} , 
%\hfill \break
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/.}
First version: May 21, 2001 (written while the author was still at
Temple University). This version: Feb. 13, 2002. 
Accompanied by the Maple packages {\tt UGJ},
{\tt SymUGJ} and {\tt SiPerUGJ},
available from Zeilberger's homepage, or from
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/utm.html.}
Supported in part by the NSF.
} 
}

{\bf Abstract:} This is the fifth, and last, installment of the
saga on the Umbral Transfer-Matrix
method, based on Gian-Carlo Rota's seminal notion of the umbra.
Here we extend the powerful Goulden-Jackson Cluster method,
that enables one to compute generating functions enumerating
words that avoid, as factors, any given {\it finite} set of
``mistakes'', to the case where there are infinitely many
mistakes. This infinite set of mistakes should be a union
of finitely many
finite-parameter families,  given symbolically in frequency
notation. We illustrate the method by introducing a new
`toy model' for self-avoiding walks, that is much more interesting
and complex than finite-memory approximations, yet much simpler
than the (probably intractable) ``real thing".

{\bf Required Reading}

The reader is expected to be familiar with [Z1],
and the classical Goulden-Jackson Cluster Method ([GJ1],[GJ2]), lucidly
described in [NZ].

{\bf How Likely is a Monkey to Type YHVH?}

It is a sin to utter the name of the Lord in vain, or even
to have it written down, or printed, except in a holy text.
Ignoring spaces, it is also bad luck to have the letters
Yod, Heh, Vav, Heh consecutively. The classical Goulden-Jackson
Cluster Method answers the following question. What is the probability
that a random word will not contain, {\it consecutively},
the word YHVH?. Or for that matter, any other given word, or
{\it finite} set of words, for example 
the set of words $\{YHVH, DORON, DAVID, IAN\}$.

It is immediate that the enumerating generating function is
always a rational function, since the `language' of `clean words' is regular,
and can be modeled by a finite automate. This also implies an algorithm
for computing the generating function that, however, is impractical.
The amazing Goulden-Jackson method is an {\it efficient} method
for computing  such generating functions.

But a naughty monkey may type YYYHHHHVVVVHHHH, which does not contain
YHVH, so by the {\it letter} of the law it should not be punished,
but by the {\it spirit} of the law it is committing an even graver
sin. So what we really want to avoid is any factor of the
form $Y^{a+1}H^{b+1}V^{c+1}H^{d+1}$, for any $a,b,c,d$
{\it non-negative integers}. Of course, now we have {\it infinitely}
many `mistakes' to avoid, and the classical Goulden-Jackson method
will take infinite time and infinite space.

$Y^{a+1}H^{b+1}V^{c+1}H^{d+1}$ is an example of 
what we will call {\it symbolic word}.

{\bf Symbolic Words}

{\bf Definition}:
Given a {\it finite} alphabet $W$, and a {\it finite} set of symbols
$\{a_1,a_2, \dots, a_s\}$, that represent generic non-negative integers,
an (s-parameter) symbolic word is a creature of the form
$$
w_1^{L_1} w_2^{L_2} \dots w_r^{L_r} \quad,
$$
where for all $i$, $1 \leq i \leq r$, $w_i \in W$, and each
$L_i$ is an affine linear combination, with {\it non-negative coefficients}
of the $a_i$. In other words, each $L_i$ can be written as
$$
L_i=c_0+\sum_{j=1}^{s} c_j a_j \quad,
$$
where the $c_j$ are {\it specific} non-negative integers,
and $c_0>0$.

{\bf An Example of A Symbolic Word:} If the alphabet is
$\{1,2,3\}$ and the set of generic non-negative integers
is $\{a,b,c\}$, then $1^{a+1}2^{a+b+1}3^{b+c+1}1^{c+1}$ is
an example of a symbolic word. Its embodiment is the
(triply) {\it infinite} set of mistakes
$$
\{1^{a+1}2^{a+b+1}3^{b+c+1}1^{c+1} \vert a,b,c \geq 0\}
$$
i.e. $\{1231,112231,11122231, \dots ,122331, \dots \}$.

A plain word is also symbolic, since it can
be viewed as a $0$-parameter symbolic word.
For example the word $3112$ is written $3^11^22^1$.

{\bf The GOAL}

Given a finite alphabet and a finite set of symbolic words, 
to be called {\it mistakes}, in
the letters of the alphabet, compute the generating function
$$
f(t):=\sum_{n=0} b_n t^n \quad,
$$
where $b_n$ is the number of $n$-letter words in the alphabet
that do not contain, as {\it factors}, any instance of any
of these (symbolic) mistakes.

What do we mean by compute? Since $f(t)$ is not going to be a rational
function, there is no {\it a priori} reason to expect that $f(t)$ can be
represented {\it explicitly}, whatever that means. 

So we have to be content to find some kind of {\it equation}:
algebraic, differential, functional, integral, or what have you
(or even of mixed type), that would be satisfied by $f(t)$, and that
would enable computing the first $n$ terms of $f(t)$ in time
(and space) polynomial in $n$. I confess that I can't do it,
and I am almost sure that it can't be done.

Failing this, something almost as good can be achieved. 
I will show that there exist
finitely many {\it catalytic variables} (corresponding to
the set of generic integers), $x_1, x_2, \dots, x_s$, and a finite set of
{\it generating functions} (corresponding to the symbolic mistakes)
$F_1, \dots, F_r$ that depend on $t$ and $x_1, \dots, x_s$,
such that {\it collectively} 
$$
F_1(t;x_1, \dots, x_s) \quad , \quad F_2(t;x_1, \dots, x_s) \quad , 
\quad \dots \quad ,  \quad
F_r(t;x_1, \dots, x_s)
$$
satisfy a {\it system} of (inhomogeneous) {\it linear-functional equations},
and such that $f(t)$ can be written as
$$
f(t)={{1} \over {1-dt- \sum_{i}^{r} F_i(t;1, \dots, 1)}} \quad,
$$
where $d$ is the number of letters in the alphabet.
Note that the system can be used to iteratively find higher and
higher order terms (in $t$) of the $F_i$'s, and hence, after plugging
in all the $x_i$'s to be $1$, will yield successive terms of $f(t)$.

{\bf An Important Assumption}

No symbolic mistake can ever have an instance that is a factor
of an instance of (the same or another) symbolic mistake.

For example the input 
$Y^{a+1}H^{b+1}V^{c+1}H^{d+1}$ is not legitimate, since, for example,
$YHHVVH$ is a factor of $YYYHHVVHH$. But we do not lose
any generality by insisting on this.
All we have to do is consider
the minimal symbolic word $Y^{1}H^{b+1}V^{c+1}H^{1}$, that has
the desired property, and of course the original problem is
equivalent to the problem of enumerating words that avoid
$Y^{1}H^{b+1}V^{c+1}H^{1}$ as factors, and the
modified problem is even simpler, since it
only involves a 2-parameter symbolic mistake.

{\bf The Clusters}

The Classical Goulden-Jackson method goes verbatim to the present case,
except that now the entries of the clusters are 
{\it symbolic} mistakes, and each cluster really represents
infinitely many possible instances.

A cluster of $m$ mistakes can be represented as a sequence
$$
M_1 \, , \, i_1 \, , \, M_2 \, , \, i_2,\, \dots,\, 
M_{m-1} \, , \, i_{m-1} \, , \, M_m \quad
$$
where $M_1, M_2, \dots, M_m$ are symbolic mistakes, and
$i_1, i_2, \dots, i_{m-1}$ are integers such that
$i_k \leq length(M_k)$ (for $k=1, \dots, m-1$), that
indicate at what part of $M_k$ starts the overlap with
the next mistake $M_{k+1}$.

For example, if 
$$
M_1=1^{a+1}2^{a+b+1}3^{a+b+1}1^{b+1} \quad ,\quad
M_2=3^{a'+1}1^{a'+b'+1}2^{a'+b'+1}3^{b'+1} \quad ,\quad
M_3=2^{a''+1}3^{a''+b''+1}1^{b''+1} \quad ,
$$
the following is a cluster:
$$
(M_1, 3, M_2,3,M_3) \quad .
$$
It is being assumed that the overlapping sections coincide.
It may well happen that a symbolic cluster can't be realized,
i.e. its set of instances is empty.

There is still some choice involved in instantiating a symbolic cluster.
It is the decision where exactly does the first (symbolic) letter
of $M_{k+1}$ start under the $i_k$-th letter of $M_k$?
That distance is also symbolic, let's denote it by $c_k$.
Another decision to make is: how much does the symbolic letter of
$M_{k+1}$ that lies right under the last letter of $M_k$ stick out?
Let's call this distance $d_k$.

For a symbolic cluster to be realizable, 
the respective lengths (`frequencies') of overlapping letters
that lie on top of each other must match, getting a system of
linear diophantine equations. If they are all satisfied, then
the whole cluster has a well-defined symbolic length:
$length(cluster)$. If the last mistake in the cluster
is the $s$-parameter mistake $M_m$, that
has parameters $a_1,a_2, \dots, a_s$, say, then 

{\bf Definition}:
$$
Weight(M_1,i_1,M_2,i_2, \dots, i_{m-1}, M_m):=-t^{length(cluster)}
x_1^{a_1} x_2^{a_2} \dots x_s^{a_s} \quad.
$$

{\bf The Umbral Evolution}

To each symbolic mistake associate a  vertex. It is clear that
each symbolic cluster is a walk in that graph, with the
obvious adjacency pattern, where there is an edge between
mistake $M_i$ and mistake $M_j$ for each of the possible
interfaces, i.e. for each of the times a proper suffix of
$M_i$ coincides with a proper prefix of $M_j$, and
their set of instances is non-empty.

Suppose that we have an already constructed cluster 
$$
M_1 \,,\, i_1 \, ,\, M_2 \, , \, i_2 \, ,\dots, \, M_{m-1}
\, , \, i_{m-1} \, , \, M_m \quad .
$$
In how many ways can one add another mistake $M_{m+1}$?
Of course, all that matters is $M_{m}$, so things are
Markovian.

Let's try to understand the edges and the Rota operators.
Let $M_m=A$ and $M_{m+1}=B$. How can $B$ follow $A$?
Suppose $A$ is 
$$
A=A_1^{L_1} A_2^{L_2} \dots A_r^{L_r} \quad ,
$$
where $L_i$ are affine linear combinations of a set
of discrete variables $\{a_1, \dots, a_R\}$
and
$$
B=B_1^{N_1} B_2^{N_2} \dots B_s^{N_s} \quad,
$$
where $N_i$ are affine linear combinations of a set
of discrete variables $\{b_1, \dots, b_S\}$,
and the letters of both $A$ and $B$ belong to the alphabet $W$,
i.e. $A_i \in W, i=1, \dots, r$ and
$B_i \in W, i=1, \dots, s$.

Let's call the {\it underlying word} of a symbolic
word $SymWord:=A_1^{L_1} A_2^{L_2} \dots A_k^{L_k}$ the plain word
$A_1 A_2 \dots A_{k-1} A_k$, and denote this by $U(SymWord)$.
{\bf Question}: What are the potential edges between $A$ and $B$? 
{\bf Answer:} 
Whenever a proper suffix of $U(A)$ coincides with a proper prefix of
$U(B)$. In other words, whenever
there is a $u$ such that
$$
A_{r-u+1}=B_1 \quad , \quad A_{r-u+2}=B_2 
\quad , \quad \dots \quad , \quad A_r=B_u \quad  .
$$
In addition, the following system of {\it compatibility conditions},
in the variables 
$$
\{a_1, \dots, a_R\} \cup 
\{b_1, \dots, b_S\} \cup 
\{c,d\}
$$
must be satisfied:
$$
L_{r-u+1}=c+N_1 \quad , \quad L_{r-u+2}=N_2, \quad \dots \quad , \quad
L_{r-u+i}=N_i \quad , 
$$
$$
\dots,  \quad L_{r-1}=N_{u-1}
\quad , \quad L_r+d=N_u \quad.
\eqno(Ian)
$$
Here $c$ denotes the `vacuum' before $B_1^{N_1}$ that lies
under $A_{r-u+1}^{L_{r-u+1}}$, and
$d$ denotes the `vacuum' after $A_r^{L_r}$ that lies
above $B_u^{N_u}$.

The set of equations $(Ian)$ needs to be modified when
$u=r$, when $u=1$, and when $s=1$. When $u=r$, the $c$ in the
first equation of $(Ian)$ needs to be replaced by $c+1$, since
when $c=0$ we won't have a proper cluster. 
When $s=1$, i.e. $U(B)$ consists of just one letter, then
the $d$ has to be replaced by $d+1$.
Finally, when
$u=1$, $(Ian)$ only contains one equation: $L_{r}+d=c+N_1$.

If $(Ian)$ has solutions with all the
unknowns $a_i,b_i,c,d$ {\it non-negative} integers,
then there is an edge between $A$ and $B$ corresponding to
this particular interface. If the set of solutions of
$(Ian)$ is empty, then there is no edge.

For any two symbolic mistakes $A$ and $B$, there may be
zero, or several, edges, each corresponding to
the scenario where a proper suffix of $U(A)$ coincides with
a proper prefix of $U(B)$ and the corresponding set of
Linear Diophantine Equations, $(Ian)$, is satisfied.

But how do we determine whether the system $(Ian)$ is solvable?
More generally, how do we solve it? Luckily, there are
algorithms for doing this. First, there was the theoretical
seminal work of Richard Stanley (see [S], section 4.6), that had
an algorithm implicit in it. 
However, for our purposes, it is most convenient to adapt
the beautiful work of 
George Andrews, Peter Paule and Axel Riese,
in resurrecting, extending,
and implementing MacMahon Partition Analysis[APR]. 
In particular,
the Mathematica package {\tt OMEGA}, written by
Mathematica Wizard Axel Riese, and available
at the Risc-Linz website, is exactly what we need.

Alas, I have two problems with {\tt OMEGA}. The first one 
is {\it linguistic}. I am a Maple person, who does not,
and {\it will not} use Mathematica. The other problem is 
{\it cultural}, perhaps even
``{\it theological}''. 
Andrews, Paule, and Riese justify the algorithm using the dubious
notion of {\it analytic convergence},
and their power series represent `analytic' functions
(whatever that means). 

I ``remedy'' both defects in [Z6], that is accompanied by a Maple
package {\tt LinDi} that is a Maple analog of Riese's {\tt OMEGA}.
The parts of {\tt LinDi} that were needed for the
present project were transported to the Maple packages
{\tt UGJ, SymUGJ, SiPerUGJ}.

Let's
introduce `continuous' variables $x_1, \dots, x_R$ and $y_1, \dots, y_S$
corresponding to the discrete variables $a_1, \dots, a_R$,
$b_1, \dots, b_S$ respectively, plus $z$ and $w$ corresponding to
the `slack' variables $c$ and $d$.
Consider the formal power series
$$
F(x_1, \dots, x_R; y_1, \dots, y_S; z,w):=
\sum x_1^{a_1}x_2^{a_2} \dots x_R^{a_R} \cdot
y_1^{b_1}y_2^{b_2} \dots y_S^{b_S} \cdot
z^c w^d \quad,
\eqno(Richard)
$$
where the sum ranges over the set of solutions of $(Ian)$ with all
the $a_i$, $b_j$, $c$ and $d$ being
{\it non-negative} integers.

By the celebrated theorem of Stanley ([S], Theorem 4.6.11), $F$ must
be a {\it rational function} of its arguments. 
Furthermore the numerator has positive coefficients, and
the denominator is a product of terms of the form
$(1- monomial)$.
Using {\tt OMEGA}, or my Maple analog, {\tt LinDi}, one
can compute this rational function $F$ explicitly.

Finally we are ready for the Umbral Evolution. The Pre-Umbra is
$$
x_1^{a_1} \dots x_R^{a_R} \rightarrow
ConstantTerm_{x_1, \dots, x_R} 
{ {F(x_1, \dots, x_R; y_1, \dots, y_S; z,w)}
\over
{x_1^{a_1}x_2^{a_2} \dots x_R^{a_R}}
} \quad .
$$
The right side is an expression in the remaining variables
$y_1, \dots, y_s$ and $z$ and $w$. Once we find it, we plug
in $z=1$ and $w=1$.

However, this operation only takes care of the catalytic variables.
The most important variable, $t$, carrying the length of
the cluster, has not been mentioned yet. 
First, we must have a minus sign in front, since the sign of a cluster
is $(-1)$ to the power of the number of mistakes participating
in it, and adding the mistake $B$ changes the sign.
Next we have to multiply by $t$ raised to the power of the
`newly acquired length', which, in the above notation is
$d+N_{u+1}+N_{u+2}+\dots+N_s$. So what we {\it really} need to compute
is the generating function
$$
G(x_1, \dots, x_R; y_1, \dots, y_S; z,w):=
\sum x_1^{a_1}x_2^{a_2} \dots x_R^{a_R} \cdot
y_1^{b_1}x_2^{b_2} \dots y_S^{b_S} \cdot
z^c w^d t^{d+N_{u+1}+N_{u+2}+\dots+N_s}
\quad,
\eqno(George)
$$
where the summation set in $(George)$ is the same as in $(Richard)$.

It is easy to see (and compute) $G$ in terms of $F$.
Indeed if the coefficient of a variable $b_i$ in
the affine linear combination $d+N_{u+1}+N_{u+2}+\dots+N_s$ is,
say, $e_i$, and its free term is $e_0$, then, obviously
$$
G(x_1, \dots, x_R; y_1, \dots, y_S; z,w;t):=
-t^{e_0}F(x_1, \dots, x_r; t^{e_1}y_1, \dots, t^{e_S} y_S; z,wt)
\quad .
$$
The full Pre-Umbra is
$$
x_1^{a_1} \dots x_r^{a_R} \rightarrow
ConstantTerm_{x_1, \dots, x_R} 
{ {G(x_1, \dots, x_r; y_1, \dots, y_s; 1,1;t)}
\over
{x_1^{a_1}x_2^{a_2} \dots x_r^{a_R}}
} \quad .
\eqno(GianCarlo)
$$

This is unlike the situations in [Z2],[Z3], and [Z4], where our Maple
packages
found explicit expressions for pre-umbras, that were then converted
to Umbral Schemes with 
procedures transported from the general package {\tt ROTA}.

However, it turns out that the Pre-Umbra given in the form
$$
x_1^{a_1} \dots x_r^{a_r} \rightarrow
ConstantTerm 
{{H(x_1, \dots, x_r; y_1, \dots, y_s)} \over
{x_1^{a_1} \dots x_r^{a_r}}} \quad,
$$
can also be converted, often, into what we called Rota Operators
in [Z1]. And the central tool of {\tt OMEGA} and {\tt LinDi},
viz. {\it partial fraction decomposition}, comes to the rescue once
again.

Let's illustrate this with a simple example in one variable.
Consider the pre-umbra
$$
x^n \rightarrow Coeff_{x^n} {{a} \over {1-Ax}} \quad,
$$
yielding the pre-umbra $x^n \rightarrow aA^n$,
which gives the umbra $f(x) \rightarrow af(A)$.

Since the denominators of the rational functions of interest are
products of terms of the form $(1-monomial)$,
we can write:
$$
R(x)={{P(x)} \over 
{ (1-A_1x) (1-A_2 x ) \cdots (1-A_mx)}} \quad,
$$
where $x$ is the current variable,
with the degree of $P(x)$, in $x$, being less than $m$.
Performing a partial-fraction decomposition splits $R(x)$ into
a sum of terms of the form $a/(1-Ax)$, which was considered above.

In the generic case, things should work out, even when we have
to constant-termize with respect to several variables. But once in
a while, we might get stuck, in which case there is no Umbral
Scheme. It is possible to also treat the non-generic case,
but since in all the natural examples that I tried things
worked out, I did not bother.

{\bf A Very Simple Example}

Let's count the number of words in the alphabet $\{1,2,3\}$,
that avoid any factor of the form
$A:=12^{a+1}3^{a+1}1$, i.e. you can't have $\{1231,122331,12223331, \dots \}$
as factors. Even in this simple case we already have
to avoid infinitely many mistakes.

The Clusters must be of the form
$(A,4,A,4,A,4, \dots, A)$. Let's call the
cluster generating function $F(x;t)$, i.e.
the weight-enumerator of all clusters with the weight
$(-1)^m t^{length} x^a$, where the last $A$ was
$12^{a+1}3^{a+1}1$.
The set of clusters consisting of
one $A$ has weight-enumerator 
$-t^4/(1-t^2 x)$, while the rest has weight-enumerator 
$-t^3F(1;t)/(1-t^2x)$.
It follows that  $F(x;t)$ satisfies the functional equation:
$$
F(x;t)=
{{-t^4} \over {1-t^2x}}-
{{t^3} F(1;t)\over {1-t^2x}} \quad.
$$
The desired generating function is $1/(1-3t-F(1;t))$.
In this trivial case, we can actually {\it solve} the
Umbral Scheme. Plugging in $x=1$ yields
$$
F(1;t)={{-t^4} \over {1-t^2}}-{{t^3 F(1;t)} \over {1-t^2}} \quad,
$$
and hence $F(1;t)=-t^4/(1-t^2+t^3)$, and the generating function
is $f(t)=1/(1-3t-F(1;t))=(1-t^2+t^3)/(1-3t-t^2+4t^3-2t^4)$.

{\bf The Maple Package UGJ}

This is implemented in the Maple package {\tt UGJ}.
In order to use it, download it from my website, keeping its name
{\tt UGJ}. Now get into Maple, by typing
{\tt maple} (or the appropriate command on your system), followed by 
{\tt ENTER}. Once in Maple,
assuming that you are in the same directory as {\tt UGJ}, type
{\tt read UGJ:}, and follow the instructions given there.
In particular, to see the list of the main procedures, type
{\tt ezra();}, and for help with any given procedure,
type {\tt ezra(ProcedureName);}. For example, try
{\tt ezra(UmSc);}.

Given a list of symbolic words
ListSW, {\tt UmSc(ListSW,x,t)} 
constructs an Umbral Scheme for the Cluster
generating function for enumerating words that avoid
instances of the words of {\tt ListSW} as factors.
For example, to get automatically the trivial example
above, type:

{\tt UmSc( [ [[[1,1],[2,a+1],[3,a+1],[1,1]],[a]]],x,t);}
 
Another main procedure is {\tt UGJseries}.
{\tt UGJseries(ListSW,NumLetters,L)}
finds the first {\tt L} terms of the series expansion
enumerating words that avoid any instances  of the mistakes of ListSW,
where {\tt NumLetters} is the number of letters.
For example, type:

{\tt UGJseries( [ [[[1,1],[2,a+1],[3,a+1],[1,1]],[a]]],3,10);}

In order to check the program empirically, I wrote the
procedure {\tt CheckU}. {\tt CheckU} uses the
classical (finite) Goulden-Jackson method to check the
validity of {\tt UGJseries}, and hence of {\tt UmSc}, since
the former depends on the latter. 
The syntax of {\tt CheckU} is:
{\tt CheckU(ListSW,Alphabet,L);}, where {\tt Alphabet} is the
alphabet. For example type:

{\tt CheckU( [ [[[1,1],[2,a+1],[3,a+1],[1,1]],[a]]],$\{1,2,3\}$,10);}.

You should get the same output.


{\bf The Symmetric Case}

If the set of symbolic mistakes is symmetric with respect
to the action of the symmetric group, then it is much more
efficient to only consider each representative. This
is implemented in the analogous Maple package {\tt SymUGJ}.
However, we had to slightly modify the notion of Umbral
Scheme, to that of {\it Weighted Umbral Scheme}, to take care
of the multiplicities resulting from each equivalence class.
Hence the main procedure of {\tt SymUGJ} is {\tt WtUmSc}.
Everything else is analogous.

{\bf Symmetry Under the Action of the Group of Signed Permutations}

If the alphabet is $\{-1,1,-2,2, \dots, -n,n \}$, then a stronger
symmetry is possible. Namely, the action of the group of
signed permutations. If our set of symbolic mistakes is
invariant under the action of this group, we can have even a
bigger saving in time and memory. In this case, this means
that if, say, $2^1 1^{a+1} 2^1$ is a symbolic mistake, then so 
are $2^1 (-1)^{a+1} 2^1$, $(-2)^1 1^{a+1} (-2)^1$,
$(-2)^1 (-1)^{a+1} (-2)^1$, and the four other symbolic mistakes
obtained by transposing $1$ and $2$. If we use {\tt UGJ}, then
we need an Umbral Matrix of size $8$. If we use {\tt SymUGJ}, we
still need the Umbral Matrix to be of size $4$. But if we
exploit the full symmetry, then we would only need an Umbral
Matrix of size $1$.

This is implemented in the package {\tt SiPerUGJ}, that is
completely analogous.

{\bf Fancy Toy Models for Self-Avoiding Walks}

John Noonan [N] used the finite Goulden-Jackson method,
as extended in [NZ], to derive new upper bounds for
the so-called connective constants for self-avoiding walks.
He did this by considering the finite-memory approximation,
that considers random walks on the lattice that do not
visit any site visited in the previous $m$ steps. In other words,
the walker only remembers what she did in the previous $m$ steps,
and avoids the sites visited then, but she may return to sites
visited more than $m$ steps ago.

It is easy to see that a finite ($m$) memory walk 
on the $d$-dimensional lattice can be modeled in terms of
a Goulden-Jackson type problem of enumerating the
words in the alphabet $\{1,-1,2,-2, \dots, d, -d \}$ that
avoid all the walks one obtains by starting anywhere on
one of the (finitely many) self-avoiding polygons of length
$\leq m$, and traversing it in either direction.
This set of `mistakes' is obviously invariant under the
group of signed permutations, and John Noonan took advantage of
this.

Even though finite-memory approximation give relatively good
upper bounds, they are not theoretically interesting, since the
generating functions are always rational, giving
a boring asymptotics of the form $C \mu^n$ rather than
the more realistic $C \mu^n n^\theta$ ($\theta \neq 0$).

A natural toy problem is to consider walks that avoid any
rectangular subwalks (in addition to retracing), or, 
those that avoid retracing, rectangular, 
and hexagonal subwalks, etc.. Now we have infinitely many
mistakes, and {\tt SiPerUGJ} is ideally suited for the task.
Procedures {\tt SawUm4} and {\tt SawUm6} compute the
Umbral Schemes for these problems, respectively, and
{\tt Tony4} and {\tt Tony6} find the `series expansions'.
Unfortunately, even though the generating functions are
(probably) not rational, the asymptotics seems to
still be $C\mu^n$ (i.e. $\theta=0$), 
like in the Markovian, finite-memory case.
So even though we excluded infinitely many kinds of bad events, 
apparently this was not enough. Also the empirically derived
connective constants do not compete with the ones found by
Noonan.

By combining rectangles and hexagons with larger,
specific (not symbolic) mistakes, it should be possible to
get improved upper bounds for connective constants of self-avoiding walks.

The webpage for this article:
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/mamarim/mamarimhtml/umbV.html} contains 
sample input and output files for the three Maple packages
{\tt UGJ, SymUGJ} and {\tt SiPerUGJ}, with Umbral Schemes and
series expansions.

{\bf Acknowledgement:}  Many thanks are due to the referee for
his careful reading and many useful remarks and suggestions.

{\bf REFERENCES}

[APR] George E. Andrews, Peter Paule, and Axel Riese,
{\it MacMahon's partition analysis III: The Omega package},
Europ. J. Comb. {\bf 22} (2001), 887-904.
The accompanying Mathematica package is available at 
{\eighttt
http://www.risc.uni-linz.ac.ac/research/combinat/risc/software/Omega/}.


[GJ1] Ian Goulden  and David M. Jackson,
{\it An inversion theorem for cluster decompositions of
sequences with distinguished subsequences},
J. London Math. Soc.(2){\bf 20} (1979), 567-576.

[GJ2] Ian Goulden  and David M. Jackson,
{\it "Combinatorial Enumeration"}, John  Wiley, 1983, New York.

[N] John Noonan, {\it New upper bounds for the connective constants
of self-avoiding walks}, J. Stat. Physics {\bf 91} (1998), 
871-888.

[NZ] John Noonan and Doron Zeilberger,
{\it The Goulden-Jackson Cluster Method: Extensions, Applications, and
Implementations}, J. Difference Eq. Appl. {\bf 5}, 355-377, (1999).

[S] Richard Stanley, {\it ``Enumerative Combinatorics'', vol. 1},
Wadsworth, Monterey, 1986. Reprinted by Cambridge University Press.

[Z1] Doron Zeilberger,
{\it The Umbral Transfer-Matrix Method. I. Foundations},
J. Comb. Theory Ser. A {\bf 91} (2000), 451-463.
[Rota memorial issue].

[Z2] Doron Zeilberger,
{\it The Umbral Transfer-Matrix Method. II. Counting Plane 
Partitions}, Personal Journal of Ekhad and Zeilberger,
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/pj.html   .}

[Z3] Doron Zeilberger,
{\it The Umbral Transfer-Matrix Method. III. Counting Animals},
submitted. Available from
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/papers1.html   .}

[Z4] Doron Zeilberger,
{\it The Umbral Transfer-Matrix Method. IV. Counting Self-Avoiding
Polygons and Walks}, submitted. Available from
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/papers1.html   .}

[Z5] Doron Zeilberger,
{\it The Umbral Transfer-Matrix Method. V. The Goulden-Jackson
Cluster Method for Infinitely Many Mistakes}, this article.

[Z6] Doron Zeilberger, {\it George Andrews's Resurrection of
MacMahon's Partition Analysis Done Right: Let's Be fORmAL, not ANALytic},
in preparation.
\bye
 

