Catwalks Sandsteps Pascal Pyramids

Richard K Guy

For Frank Harary

Bill Sands where the problem is stated without the result noticed that the number

of dierent walks of n steps b etween lattice p oints each in a direction N S E or W starting

from the origin and remaining in the upp er halfplane is

n

w

n

n

and asked for a neat pro of What is wanted is a simple choice argument any oers This

is sequence in my rst attempt at any sort of pro of was by induction from the

formula

w w c

n n1 n

since a walk of length n is one more step in one of the four directions N S E or W than a

walk of length n except that a southerly step is not allowed if the walk of length n

terminated on the xaxis and its well known that the number of such walks is the nth

Catalan number

But its not well known It do esnt o ccur among the manifestations listed by Kuchinski

nor can we immediately see any simple corresp ondence b etween the walks and any of the

manifestations However rst lets assume that its true and that holds with n in

place of n Then

n n

w c

n1 n

n n n

n n

nn nn

n

fnn ng

nn

nn n

nn n

What is well known is that the number of walks of n steps each N or E from to

n n which dont cross the diagonal y x or the number of walks of n steps from

to n n which stay strictly ab ove the diagonal is c the nth Catalan number

n

This is clearly the same as the number of walks of n steps on the p ositive xaxis starting

and nishing at the origin

Let us lo ok at this onedimensional analog of the Sands problem We can exhibit the

numbers of walks w n x of n unit steps starting at and ending at x x in a

Pascal semitriangle Fig

Columns x and x contain the Catalan numbers as already earnested column

x also o ccurs in connexion with partitioning a p olygon Columns x

n k Total

Figure Numbers of walks w n x on the p ositive xaxis

are sequences in Sloane they are Laplace transform

co ecients more precisely w n k is denoted in by C which is dened by

k

n

X

2n

C sink cos sin

k

k =0

Presumably there is an analogous formula for w n k compare equation b elow

The rst table in Cayleys pap er is for the number of partitions of an r gon into k parts

by nonintersecting diagonals His column k is our main diagonal and his column k

is our third diagonal starting at n x More generally his column k is our diagonal

starting at k except that his entries contain an extra factor x k x k

a generalized Catalan number in fact for x k it is c Cayley attributes his results

k 2

to and the latter pap er gives some history mentioning Terquem Lame Ro drigues

Binet Catalan

(4)

A near miss for column x is Stirling numbers of the second kind S but the entries

n

not shown here for rows are decient by resp ectively

We omit zero values of w n x from our table its fairly obvious that w n x if n

and x are of opp osite parity or if x n Its not to o dicult to nd formulas for the rst

few diagonals

w n n w n n n w n n nn w n n nn n

In fact there is a comparatively simple formula for all the entries in Figure

n n

w n x

r r

1

where r n x Indeed the formula also works in the ap o cryphal cases mentioned

2

n

ab ove if we take the reasonable interpretation that if r or if n r or if r is not

r

an integer We shall do this note that the usual formulas such as

n n n

r r r

and still hold in these cases Formula is easily proved by induction since

w n x w n x w n x

n n n n

r r r r

n n

r r

The well known result that we mentioned is the sp ecial case

n n n

w n c

n

n n n n

The total number w n of walks of length n is

w n n w n n w n n

n n n n n n n

k k k

where n k or according as n is even or o dd ie

k k

or

k k

Here it is clear that the number of walks of even length is just twice the number of walks

of o dd length one less

k k k k

k k k k k

Is there a simple choice argument for walks of o dd length If you know the Catalan

number result then we can use a device similar to formula

w k w k c

k

k k

k k k

k k k

k k k

but this has an air of circularity ab out it or at b est may b e using a sledgehammer to crack

a nut

Return to the original problem we can solve it if we go into more detail than most p eople

would deem desirable The numbers w x y of walks of n steps from to x y which

n

remain in the halfplane y may b e exhibited in a Pascal semipyramid whose layers

are shown in Fig

If we sum the rows in the layers of Fig we obtain the numbers w y of walks of n

n

steps which start at and end at distance y from the xaxis These are shown in Fig

We shall see that a sp ecial case is as we have already earnested

w c

n n+1

n n n n n n

n n

Figure Layers of a Pascal semipyramid values of w x y

n

w w w w w w w w w w w w n

n n n n n n n n n n n n

Figure Sums of rows of Fig values of w y

n

In turn the row sums of Fig are the total numbers w of Sandstype walks of length n

n

They are listed in column two of Fig and we will conrm another of our earlier statements

n

w

n

n

At risk of losing some interesting heuristics we again leap to the conclusion

n n n n

w x y

n

r s r s

1 1

n x y s n x y where r

2 2

The obvious symmetry w x y w x y is reected in formula since changing

n n

the sign of x is equivalent to interchanging r and s It is also clear that

a if n x y is o dd then r s are not integers and

b if jxj y n then at least one of r s is negative

so that in either of these cases

w x y

n

We can prove inductively from the recursion which states that the last step was

either N S E or W

w x y w x y w x y w x y w x y

n n1 n1 n1 n1

Notice that the sums of the three arguments in the ve terms are all of the same parity

If this is o dd then all the terms are zero But if r s are integers then the corresp onding

values for the four terms on the right of are

r s r s r s r s

and if we assume that formula holds with n in place of n then yields

n n n n n n n n

w x y

n

r s r s r s r s

n n n n n n n n

r s r s r s r s

which b ecomes formula after some more or less tedious manipulation dep ending on ones

ingenuity or symbol manipulator

To nd w y sum over x

n

2(ny )

x=ny

X X

w y w x y w x y

n n n

x=n+y

r =0

n n n n n n

n y n y n y

n n n n n n

n y n y n y

ny ny 2 n

The two brackets are the co ecients of t and of t in the expansion of t

n

t so that

n n

w y

n

n y n y

which may b e rewritten as

n n

w y

n

n y n y

On comparing this with we see that

w y w n y

n

the number of o dd length onedimensional walks which nish of course at an o dd distance

from the origin

In particular is the same as the number of walks from to n n which b egin

with a northward step and do not cross the line joining start to nish w n

n n n

w n n

n

n n nn

n n n

c

n+1

n n n n

0

Figure Layers of a Pascal quarterpyramid values of w x y

n

the n th Catalan number

Finally summing from y to y n gives

We could ask similar questions concerning walks which do not stray outside the p osi

tive quadrant The numbers of such walks now form a Pascal quarterpyramid which is

exhibited in Fig

The entries in Fig are given again without motivation by

n n n n

0

w x y

n

r s r s

1 1

n x y s n x y as b efore Notice that interchange of x and y keeps where r

2 2

s xed and replaces r by n r So the symmetry

0 0

w y x w x y

n n

follows from the symmetries

n n n n

and

r n r r n r

0

x y also satises the relation We may prove as we proved since w

n

A remarkable coincidence is that

0

w c c

k k +1

2k 1

is the number of inequivalent Hamiltonian ro oted maps on k vertices sequence in

although Tutte do esnt give the formula in terms of Catalan numbers Is there yet

another opp ortunity for a pure combinatorial pro of

0

Figure is obtained by summing the rows of Fig and we may nd w y the number

n

of walks in the p ositive quadrant which nish at distance y from the xaxis by summing

0 0 0 0 0 0 0 0 0 0

n w w w w w w w w w w

n n n n n n n n n n

0

Figure Sums of rows of Fig values of w y

n

from x to x n y

n n n n

0

y w

n

n y n y

n n n n

n y n y

n n n n

1 1 1 1

n y n y n y n y

2 2 2 2

if n y is even but with the last term replaced by

n n n n

1 1 1 1

n y n y n y n y

2 2 2 2

if n y is o dd

Put n y k or k and

n+1 n n+1 n

if n y k

0

k k k k 1

w y

n+1 n n+1 n

n

if n y k

k k +1 k +1 k 1

In particular if y

k k k k

0

w or

n

k k k k k k

k k

ie c or c

k k +1

k k

according as n k or n k where c is the k th Catalan number

k

For walks in the p ositive quadrant its more natural and symmetrical to ask for the

numbers of walks which terminate at various distances from the origin using the Manhattan

metric x y n s Figure shows the sums of the diagonals of Fig

The entries in Fig are

X

0 0 00 0

w x y w n s w x y

n n n

x+y =n2s

n n n n

s s s n s

n n n n

s s s n s

0

w n

n

0 0

Figure Sums of diagonals of Fig values of w x y

n

Except for small values of s the truncated binomial expansions do not seem to have a

simple closed form

0 0 n

w n

n

00 n

w n n

n

00 2 n 2

w n n n n n

n

2 n 3 2 00

nn n nn n n w n

n

00 2 n 4 3 2

w n nn n n nn n n n n

n

0 0

An amusing curiosity is that w n is twice the genus of the n dimensional cub e

n

or see Theorem in

0

The total number of walks w the left hand column in Fig has on the other hand

n

the comparatively simple formula

n n

0

w

n

bn c bn c

which again seems to b eg for a simple pro of

If there is no restriction on the twodimensional walks ie if they may wander on either

side of the x and y axes then it is fairly easy to see that their number of length n from

to x y is

n n

r s

where r and s are as b efore but calculated using the absolute values of x and y

n

Of course the total number of walks of length n is

Although we certainly havent found the most aesthetic pro ofs the comparative simplicity

of the nal results tempts us to ask what happ ens in three dimensions Let W x y z b e

n

the number of walks of n steps each in a direction N S E W up or down from

to x y z which never go b elow the x y plane We will not attempt to depict the four

dimensional Pascal semipyramid but the sums of its layers now give W z the number

n

of walks terminating at height z ab ove the x y plane and this satises the recurrence

W z W z W z W z

n n1 n1 n1

which may b e used to pro duce the array of Fig

Each entry in Fig is the sum of four times the entry immediately ab ove it and the two

neighbors of that entry eg

W

5

n W

n

Figure Walks in three dimensions values of W z

n

We again suppress the details of discovery of the general formula and of its inductive

pro of these details seem to b e more complicated than b efore and we found no obvious

manifestation of the Catalan numbers The simplest expression for W z that we have so

n

far found is not in closed form

n n n

W n v a a a

n n0 n1 nt

v v v t

where t bv c and the co ecients a are of shap e

nu

v u v u

2 v 2u

a

nu

u u

although there are of course closed form expressions for small values of v

W n

n

W n n

n

W n n n

n

nn n W n

n

2

W n nn n n

n

2

W n nn n n n

n

3 2

nn n n n n W n

n

3 2

W n nn n n n n n

n

We have not found a closed expression for W the total number of walks of n steps which

n

do not go b elow the x y plane nor have we had an opp ortunity to examine the pap er

which may contain such an expression and may overlap the present pap er in other ways The

n

total number of nstep walks in d dimensions without restriction is of course d

REFERENCES

Arthur Cayley On the partitions of a p olygon Proc London Math Soc

Coll Math Papers

C Domb On the theory of co op erative phenomena in crystals Advances in Physics

Frank Harary Topological concepts in graph theory in Harary Beineke A Seminar

on Graph Theory Holt Reinhart Winston New York London pp

TP Kirkman On the k partitions of the r gon and r ace Phil Trans

Christian Krattenthaler Counting lattice paths with a linear b oundary I Osterreich

Akad Wiss MathNatur Kl Sitzungsber II

Mike Kuchinski Catalan Structures and Corresp ondences MSc thesis West Virginia

University Morgantown WV

Athanasios Papoulis A new metho d of inversion of the Laplace transform Q App

Math

Gerhard Ringel Farbungsprobleme auf Flachen und Graphen VEB Deutscher Verlag

der Wissenschaften Berlin

Bill Sands Problem Crux Mathematicorum Feb

Neil JA Sloane A Handb o ok of Integer Sequences Academic Press New York

London

HM Taylor RC Rowe Note on a geometrical theorem Proc London Math Soc

WT Tutte A census of Hamiltonian p olygons Canad J Math

ADDENDUM

As a result of corresp ondence with Christian Krattenthaler and Bruce Sagan and receipt

of a referees rep ort this pap er will not seek formal publication but will revert to its original

intent to form the basis of a solution to the problem of Bill Sands

Christian Krattenthaler writes that the classes of lattice paths you have considered

do not seem to have b een treated b efore On the other hand b oth he and the referee p oint

out that formula app ears in Theorem in D The referee also notes that formula

0

is in Fe and that formulas and that for w y follow easily from and the

n

reexion principle However the exciting news is that Krattenthaler supplies combinatorial

pro ofs of almost all of the formulas of the kind app ealed for and that Bruce Sagan also gives

very similar pro ofs The main item that is missing is a combinatorial pro of of the formulas

0

for w y

n

I paraphrase the pro ofs b elow and leave their originators to publish them as they think t

I will ask Bill Sands to list Krattenthaler and Sagan among the solvers of Crux Problem

unless they write to one of us requesting that their names do not app ear The list already

contains the names of Harvey Abb ott and George Szekeres incidentally but they didnt give

pure combinatorial pro ofs

Pro of of Set up a corresp ondence b etween NSEW paths p and pairs p

1

p of NE paths if the mth step of p is N S E W then the mth step of p is resp ectively

2 1

N E E N and that of p is N E N E Then the NSEW paths of n steps from to

2

x y are in corresp ondence with pairs p p of NE paths where p runs from

1 2 1

1 1

n x y and s n x y as to r n r and p from to s n s where r

2

2 2

b efore

Algebraic detail If the numbers of N S E W steps are resp ectively a b c d then

n a b c d x c d y a b r b c n r a d s b d n s a c and

r s are as stated

k +l

But the number of NE paths from to k l is so the number of NSEW

k

paths from to x y is

n n

r s

ie formula

Pro of of To count NSEW paths of n steps from to x y which dont go

b elow the xaxis use the reexion principle We must subtract o the number of paths which

cross the xaxis Each of these has a rst p oint say P for which y Relect the initial

p ortion OP in the line y giving a corresp ondence b etween paths which cross the

xaxis and paths from to x y Their number is the same as the total number of

paths already counted except that y is replaced by y ie r and s are each decreased by

This gives formula

n n n n

w x y

n

r s r s

Pro of of Reexion may also b e used to count the number of NSEW paths

which stay in the p ositive quadrant A second relexion in x together with the inclusion

exclusion principle shows that this number is

fpaths from to x y g fpaths from to x y g fpaths from

to x y g fpaths from to x y g

n n n n n n n n

r s r s r s r s

which easily manipulates into formula

Pro of of To count the number of Sandstype paths p which nish at height

y use another corresp ondence with NE paths p of twice the length If the mth step of p

is N S E W then the m th and mth steps of p are resp ectively NNEENEEN

This sets up a bijection b etween NSEW paths with n steps from to height y which do

not cross the xaxis and NE paths from to n y n y which do not cross the line

y x To enumerate these use the reexion principle again A path which crosses the

line y x has a rst p oint Q for which y x Reect the p ortion OQ of such a path

in the line y x This gives a corresp ondence with NE paths of n steps from

2n

to n y n y whose number is This conrms the formula displayed just b efore

ny 2

formula To see itself adjoin a single Nstep to the b eginning of path p and again

apply the reexion principle

Note that if we also adjoin a nal Estep to the path p we see that the number of Sands

type paths with n steps from which nish on the xaxis is the same as the number

of NE walks from to n n which do not cross the line y x and this is

wellknown to b e c

n+1

Pro of of First use the corresp ondence of to map p Sandstype n steps from

=

p

NE path n steps from to to height y not crossing the xaxis onto

=

p

with y x n y n y not crossing y x Consider the last meeting p oint of

The next step is an Nstep which we change to an Estep obtaining a new path from

to n y n y Rep eat this pro cedure this time considering the last meeting p oint

with the line y x Next consider the line y x c After y changes we arrive

at a path from to n n Since this sets up a bijection b etween NE paths of n

steps starting from and not crossing the line y x and NE paths from

to n n last meeting p oints b ecome rst crossing p oints we obtain the total number of

Sandstype paths of n steps

n

n

AUGMENTED AND ANNOTATED BIBLIOGRAPHY

Items referred to by pure numbers p ertain to the original article When a nal selection

of appropriate references is made they are easily renumbered by query replace starting at

the end eg replace bf Ze by bf say

A D Andre Solution directe du probleme resolu par M Bertrand CR Acad Sci

Paris Jbuch

cf B and Ze

A George E Andrews Catalan numbers q Catalan numbers and hypergeometric series

J Combin Theory Ser A MR f

P

n1

The Catalan numbers may b e dened as solutions to C C C C n

1 n

k nk

k =1

n

The author introduces a new q analog of the Cats via C q q q C q

1 n

P

n

k 2n 1 2 2 2

C q C q q and the more general numbers C a q q aq q q q

n n n

k nk

k =1

P

n

1 j

C a q C a q aq n He also establishes the and gives two pro ofs of

j nj

j =0

2n1

explicit formula C q C q He notes that his q Cats are the only known

n n

q analog of the Cats which have b oth a simple representation as a nite pro duct and satisfy

1

an exact q analog of He also interprets C a q and C a q as generating functions of

n n

numbers of certain partitions Mourad EH Ismail

B T Bertrand Solution dun probleme CR Acad Sci Paris

cf A and Ze

B MTL Bizley Derivation of a new formula for the number of minimal lattice paths

from to k m k n having just t contacts with the line my nx and having no p oints

ab ove this line and a pro of of Grossmans formula for the number of paths which may touch

but do not rise ab ove this line J Inst Actuar MR d

Write k t for the lattice paths with t contacts as describ ed in the title and following

G write

j m j n

1

F j m n

j

j m

then the authors new formula may b e stated as

X

k 2 t

k tx expF x F x

2

P

The corresp onding formula for k t namely

k

X

k 2

k x expF x F x

1 2

is equivalent to Grossmans formula J Riordan presumably k and the summa

k

tion is over t

B David Blackwell JL Ho dges Elementary path counts Amer Math Monthly

Zbl c

Consider a sequence x x with entries x and let s x x b e the

0 n i i 0 i

partial sums The authors furnish an elementary combinatorial pro of of two known theorems

concerning the enumeration of such sequences relative to two parameters the sum s and

n

the lead or number of indices i for which b oth s and s are nonnegative HH Crapo

i1 i

Ca L Carlitz J Riordan Two element lattice p ermutation numbers and their q

generalization Duke Math J MR

A twoelement lattice p ermutation can b e describ ed in a twodimensional lattice as a path

leading from to a p oint m n where m n with the conditions that the path has

minimum length viz m n and that it do es not contain p oints a b with a b It can

also b e describ ed as an election for two candidates A B with nal vote n m which is such

that none of the partial results gives a ma jority for B in the pap er it is less accurately said

that all partial results correctly predict the winner The number a of such twoelement

nm

lattice p ermutations was determined in by J Bertrand B see Ma as

n m

1

a n mn

nm

m

P

n

m

a x and show that this nth degree p olynomial is The authors consider a x

nm n

m=0

n+1

characterized by the prop erty that x a x has the form xP x x where P

n n n

P

n

m

a u A number of relations is again an nth degree p olynomial in fact P u

mm n

m=0

P

1

n

x a y are derived for the generating function ax y

n

n=0

The authors consider these formulas as the sp ecial case q of a q generalization They

dene the nth degree p olynomial a x q by the condition that there exist c c such

n 0 n

that

n

X

m

x a x q x c q x x

n+1 n m m

m=0

k 1

where x denotes x q x q x The co ecients a q of this p olynomial

nm

k

are studied in several ways

fThe reviewer remarks that a q has the following combinatorial interpretation It is

nm

k ++k

m

1

the sum of all q where k k are integers sub ject to the conditions k

1 m 1

k n k k mg NG de Bruijn

m 1 m

Arthur Cayley On the partitions of a p olygon Proc London Math Soc

Coll Math Papers

Ci J Cigler Some remarks on Catalan families European J Combin

MR a

r n

The author rst gives a simple pro of that the r Catalan number r n is

n

the number of ways of parenthesizing an r ary pro duct of r n factors or equivalently

the number of r ary trees with n p oints Next he shows using a variant of the Dvoretzky

Motzkin cycle lemma that the number of r ary trees with n p oints and t edges from a

i

P

n n n

p oint to its ith subtree where t n is n From this formula he

i

t t t

r

1 2

deduces that the number of nonnegative paths from to r n with k p eaks using the

(r 1)n

n

steps and r is the generalized Runyon number n Finally he

n1k

k

discusses the connexion b etween these results noncrossing partitions and Sp erners theorem

Ira Gessel

C E Csaki On the number of intersections in the onedimensional random walk

Magyar Tud Akad Mat Kutato Int Kozl MR

Consider a discrete onedimensional random walk in which with the usual notation

(k )

1

p p and let b e the number of passages through the p oint k in a walk

ii+1 ii1

j

2

(0)

of j steps Write The following results are proved i P l P

j 2n1 2n

j

2n1 2n

2n+2 2n+1

l for l and is if l ii For xed p ositive even k

n+l n

(k ) (k ) (k )

2n

2n+1

l P l l P iii For xed p ositive o dd k P

2n1 2n 2n

n+l +k 2

(k )

2n+1

2n

The pro ofs are entirely combinatorial J Gil lis P l

1

2n+1

(k +1) n+l +

2

C Endre Csaki Sri Gopal Mohanty Some joint distributions for conditional random

walks Canad J Statist MR k

Joint distributions of maxima minima and their indices are determined for certain con

ditional random walks called Bernoulli excursion and Bernoulli meander The distribution

of the lo cal time of these pro cesses is treated by a generating function technique Limiting

distributions are also given providing some partial results for Brownian excursion and mean

der For instance the authors conjecture the joint limit distributions of the lo cal time and the

maximum for these two pro cesses Similar investigations are carried out for the unconditional

random walk and for the Bernoulli bridge G Louchard Brussels

C Endre Csaki Sri Gopal Mohanty Jagdish Saran On random walks in a plane Ars

Combin Not yet seen

C E Csaki I Vincze On some problems connected with the Galton test Magyar

Tud Akad Mat Kutato Int Kozl MR

The authors give the probability of the number of waves relative to the horizontal line

in the sequence of the sum S of the rst i members of a random sequence of n s and n

i

s where i n with limiting distribution in n and the joint probability

distribution for the number of waves mentioned ab ove and the Galton statistic in the sequence

with the limiting distribution Then they give the joint probability distribution for the number

of waves relative to the height k from the horizontal line and the length of time sp ent

ab ove this height expressed by the number of p ositive members in the welldened sequence

with the limiting distribution They suggest the statistical tests based on these theorems in

a twosample problem C Hayashi Tokyo

C E Csaki I Vincze On some distributions connected with the arcsine law Russian

summary Magyar Tud Akad Mat Kutato Int Kozl MR

Several results extending those of the reviewer and Feller and generalized by ES Ander

sen are given The combinatorial formulae are obtained by onetoone corresp ondence from

which asymptotic ones are computed For instance let s i b e the successive

i

(2k )

sums in the cointossing game with stakes s and let denote the number of

0

2n

indices i i n for which either s k or s k but s k where k is a

i i i1

nonnegative integer then

g n g

(2k )

g n k P g

2n

2n

g n g k

k

X

n

(2k )

P

2n

2n

n j

j =k

KL Chung

De Nachum Dershowitz Schmuel Zaks The cycle lemma and some applications

European J Combin not yet seen

D Duane W DeTemple Jack M Rob ertson Equally likely xed length paths in

graphs Ars Combin MR h

give equation in the original pap er The authors investigate the unique sto chastic

pro cess whose realizations are the set of paths of given length joining two given vertices of a

given graph and which has the prop erty that all such paths are equally likely to o ccur There

is an application to the design of exp eriments GR Grimmett Bristol

D Duane W DeTemple CH Jones Jack M Rob ertson A correction for a lattice

path counting formula Ars Combin MR i

gives equation in the original pap er In D DeT R gave the formula

a k b d d

dab da+b

d

2 2

for the number of lattice paths in the plane length d with unit steps in the p ositive and

negative co ordinate directions starting at the origin and ending at a b which touch the line

x k y only at the initial p oint In the present pap er the authors note that this formula is

correct if d a b a k b or k in other cases it is an upp er b ound

The authors use the metho d of cyclic p ermutation or p enetrating analysis due to Dv

A metho d which yields an exact but complicated formula for this kind of problem was

describ ed in G Ira Gessel

C Domb On the theory of co op erative phenomena in crystals Advances in Physics

didnt nd this in MR or in Zbl I think I must have got it from Sloane and I can

no longer nd a reference to it in the text so lets forget it

Dv A Dvoretzky Th Motzkin A problem of arrangements Duke Math J

MR e

In an election candidates P and Q receive p and q votes resp ectively required the

probability that the ratio of the ballots for P to those for Q will throughout the counting

b e larger than larger than or equal to

E O Engleb erg On some problems concerning a restricted random walk J Appl Prob

ability MR

The restricted random walk in question is on a line vertical for convenience with unit

steps up and down and prescrib ed numbers of each In the terminology of Feller Fe such a

walk is a p olygonal path from to n m n m it is also in onetoone corresp ondence

with a twocandidate election return with nal vote n m In election return terms the

authors main results are as follows If cx n m is the enumerator of election returns with

nal vote n m n m by number of changes of lead if tx n m is the corresp onding

P

m

k

enumerator by number of ties then cx n m a x n m cx n n

n+k mk

k =0

P

m

k

cx n n tx n m a x where

n1mk

k =0

n m n m n m n m

a

nm

n m m m

The numbers a are the oldest ballot numbers For n m the enumerators in the two

nm

cases by symmetry are those ab ove with n and m interchanged a result not noticed by the

author These results are used with obvious summations to verify the results of Feller Fd

for the enumeration of unrestricted random walks by number of axis crossings and by number

of zeros of their p olygonal paths Also the limiting distribution functions of tie returns

n m are determined Finally the application of the results to the comparison of two

empirical distributions is sketched

fIn equation there are two typographical slips whose corrections will probably b e

evident to most readersg J Riordan

Fd W Feller The number of zeros and of changes of sign in a symmetric random walk

Enseignement Math MR

Let S X X X where the X are indep endent and assume the values

n 1 2 n j

1

with probability The author derives for this symmetric random walk explicit formulas for

2

the probability distribution of the number of returns to the origin the number of changes of

sign and other related quantities The derivations are of a very elementary nature and the

pap er is selfcontained A more exhaustive treatment app ears in Chapter III of the nd ed

of Fe JL Snel l

Fe W Feller An Introduction to Probability Theory and its Applications Vol Wiley

p

2k 2n2k

The reference gives equation in the original pap er On p it is noted that

k nk

is the number of paths of length n with exactly k steps lying ab ove y x On p is

given a bijection b etween paths from to k k and paths of length k which do not pass

b elow y x

Fl Philipp e Fla jolet Combinatorial asp ects of continued fractions Discrete Math

Ann Discrete Math MR fab

referred to in review of G

Fu J F urlinger J Hofbauer q Catalan numbers J Combin Theory Ser A

MR e

P

1

2n

1

n

C z The Catalan numbers C are dened by C and satisfy z

n n n

n=1

n+1 n

P

1

2n n n

z and z C z z This pap er surveys several q analogs of the Catalan

n

n=1

numbers The authors rst consider the q Catalan numbers of Ca These satisfy

1

n

X

n1

z

2

z q C q

n

2n1

z q z q z

n=1

and

1

X

n n1

z C q z z q z q z

n1

n=1

There is no simple explicit formula for C q but there is a combinatorial interpretation in

n

terms of inversions of certain sequences

Next the authors consider the q Catalan numbers given by

n

X

2

n n

k +k

q c q

n

k k

n

k =0

n

n

is in which the terms are q Runyon numbers Here one denes n q q and

k

the q binomial co ecient For these reduce to c q C These q Catalan numbers

n n

satisfy

1

n

X

c q z

n

z

n

n n

2

q q z q z

n

n=1

n1

where a a q a q a and they have a combinatorial interpretation in

n

terms of ma jor indices and descents of sequences

A generalization which includes b oth types of q Catalan numbers is considered next These

satisfy

n

1

n

X

2

a C x a bz

n

z

1 n n

a z a z xbz xb z

n=1

and are also given a combinatorial interpretation They also include as a sp ecial case the

q Catalan numbers studied in Po Ira Gessel

Fv Harry Furstenberg Algebraic functions over nite elds J Algebra

MR

referred to in review of G

G Ira M Gessel A factorization for formal Laurent series and lattice path enumeration

J Combin Theory Ser A MR j

A p owerful and striking factorization for certain formal Laurent series is proved namely

that the series is a pro duct of a constant a series in only negative p owers and a series in only

p ositive p owers Lagranges formula for series reversion is treated as an application Other

applicationa are to the problems of enumerating restricted lattice paths a novel interpretation

of Laurent series in combinatorial theory and to H Furstenbergs theorem Fv that the

diagonal of a rational p ower series in two variables is algebraic giving a new formal metho d

of showing that certain series are algebraic DG Rogers

G Ira M Gessel A probabilistic metho d for lattice path enumeration J Statist Plann

Inference MR h

Some lattice path counting problems may b e converted into problems of deriving distri

butions on random walks which give rise to functional equations Solutions of these equations

provide a probabilistic approach to the lattice path enumeration problems The approach is

illustrated by a few examples Sri Gopal Mohanty

G Ira M Gessel A combinatorial pro of of the multivariable Lagrange inversion formula

J Combin Theory Ser A MR h

Using the exp onential generating function the author gives a combinatorial pro of of one

form of the multivariable Lagrange inversion formula MLIF An outline of the pro of is

interpret the dening functional relations as generating functions for colored trees

interpret the desired co ecient as the generating function for functions from a set to a larger

set decomp ose the functional digraph from into two types of connected comp onents

whose generating functions give the MLIF Lab elle had given such a pro of in one variable L

The author also gives a useful survey of forms of the MLIF given by Jacobi Stieltjes

Go o d Joni and Abhyankar He shows that Jacobis form implies Go o ds form and gives a

simple form generalizing that of Stieltjes Joni and Abhyankar

The pap er includes some historical information on the Jacobi formula for matrices det

exp A exp trace A Dennis Stanton

Go Henry W Gould Final analysis of Vandermondes convolution Amer Math

Monthly MR c

referred to in review of Ra

GJ IP Goulden DM Jackson Path generating functions and continued fractions J

Combin Theory Ser A MR i

The authors consider paths along the nonnegative integers in which each step consists

of an increase of altitude of a rise a level or a fall The paths are weighted to

record the number of rises and levels at each altitude The main result of the pap er answers

the following question What is the sum of the weights of all paths with given initial and

terminal altitudes and with given b ounds on maximum and minimum altitudes The answer

is expressed in terms of continued fractions and extends P Fla jolets combinatorial theory

of continued fractions Fl Some classical identities for continued fractions are obtained as

corollaries Ira Gessel

G Dominique GouyouBeauchamps G Viennot Equivalence of the twodimensional

directed animal problem to a onedimensional path problem Adv in Appl Math

MR c

A set P of lattice p oints in the plane is called a directed animal if there is a subset of P

whose elements are called ro ot p oints such that the ro ot p oints lie on a line p erp endicular to

the line y x and every p oint of P can b e reached from a ro ot p oint by a path in P using

only north and east steps This pap er is concerned with the enumeration of compactro oted

animals which are animals in which the ro ot p oints are consecutive Animals which dier

only by translation are considered to b e equivalent

The main result is a bijection b etween compactro oted animals of size n and paths

of length n on the integers with steps and This bijection implies that there are

n

compactro oted animals of size n Moreover the bijection allows the compactro oted

animals to b e counted according to the number of ro ot p oints Further consequences are that

p

1

2

t t t the generating function for directed animals with one ro ot p oint is

2

and that the number of directed animals of size n ro oted at the origin and contained in the

rst o ctant x y is the Motzkin number m

n1

The pap er also contains many references to work by physicists on problems of counting

animals which arise in studying thermo dynamic mo dels for critical phenomena and phase

transitions Ira Gessel

G Howard D Grossman Fun with lattice p oints Scripta Math MR

d

Supp ose an election results in k m votes for A and k n for B In how many orders may

votes b e cast so that As vote is always at least mn times B s The author gives without

pro of the formula

X

k k

1

1 2

p F F k k

1 2

k

1 2

j m+j n

1 1

with k k k F j m n and the sum over all partitions of k He

1 2 j

j n

also gives a short introduction to enumerations in threedimensional and derives a solution

to a corresp onding election problem noting its agreement with MacMahons J Riordan

H BR Handa Sri Gopal Mohanty Enumeration of higherdimensional paths under

restrictions Discrete Math MR b

The authors consider the problem of counting lattice paths in k dimensional space under

restrictions They obtain k dimensional analogs of the familiar results in two dimensions

DP Roselle

H BR Handa Sri Gopal Mohanty On a prop erty of lattice paths J Statist Plann

Inference MR i

The authors give an algebraic discussion of the implications on lattice paths of the fol

lowing fact increasing sequences of integers such that the j th is at least bj greater than

the j st and is at most aj are in onetoone corresp ondence to increasing sequences

such that the j th is at most aj minus the sum of the rst j bk s DJ Kleitman

Frank Harary Topological concepts in graph theory in Harary Beineke A Seminar

on Graph Theory Holt Reinhart Winston New York London pp

a noncereference on p of the original pap er

Hi Terrell L Hill Steadystate kinetics of a linear array of interlocking reactions Statis

tical Mechanics Statistical Metho ds in Theory and Applications Ro chester NY Plenum

New York pp MR

referred to in review of Sh the MR reference gives no further information

J Andre Joyal Une theorie combinatoire des series formelles Adv in Math

MR d

We present a combinatorial theory of formal p ower series The combinatorial interpre

tation of formal p ower series is based on the concept of sp ecies of structures A categorical

approach is used to fomulate it A new pro of of Cayleys formula for the number of lab elled

trees is given as well as a new combinatorial pro of due to G Lab elle of Lagranges inversion

formula Polyas enumeration theory of isomorphism classes of structures is entirely renewed

Recursive metho ds for computing cycle index p olynomials are describ ed A combinatorial

version of the implicit function theorem is stated and proved The pap er ends with general

considerations on the use of coalgebras in combinatorics

K S Karlin G McGregor Coincidence probabilities Pacic J Math

MR

among Gessels references but may b e marginal cf immediately preceding pap er and

review

TP Kirkman On the k partitions of the r gon and r ace Phil Trans

v p l of original pap er

Kl Daniel Kleitman A note on some subset identities Studies in Appl Math

MR

gives combin pro of of

n

X

k n k

n

k n k

k =0

of the ErdosPosa problems see Mi

Christian Krattenthaler Counting lattice paths with a linear b oundary I Osterreich

Akad Wiss MathNatur Kl Sitzungsber II

K Christian Krattenthaler Enumeration of lattice paths and generating functions for

skew plane partitions Manuscripta Math

K G Kreweras H Niederhausen Solution of an enumerative problem connected with

lattice paths European J Combin MR d

The authors consider lattice paths of p horizontal and q vertical steps from q to p

Let W C denote the number of paths b elow C in the dominance partial ordering Theauthors

prove

X

p q p q

2

W C

p p q q

SG Wil liamson

Mike Kuchinski Catalan Structures and Corresp ondences MSc thesis West Virginia

University Morgantown WV

L Gilb ert Lab elle Une nouvelle demonstration combinatoire des formules dinversion de

Lagrange Adv in Math MR e

The purp ose of this pap er is to examine connexions b etween two classical results the

Lagrange inversion formula for p ower series and Cayleys formula for the number of lab elled

trees on n vertices in the light of the combinatorial theory of formal series recently presented

by J and founded on the notion of sp ecies of structures Some combinatorial op erations are

dened over sp ecies and corresp ond to analytic op erations over their generating functions

hence prop erties of the op erations over sp ecies yield identities for formal series The authors

introduce two canonical constructions which asso ciate a new sp ecies arb orescence R

enrichie and endofonction R enrichie resp ectively to any given sp ecies R and proves

some deep isomorphism results These yield as simple corollaries some generalized versions

of Cayleys formula and the Lagrange inversion formula Andrea Brini

L Jacques Lab elle YeongNan Yeh Dyck paths of knight moves Discrete Appl Math

MR g

This pap er enumerates lattice paths from the origin to a p oint along the xaxis which do

not go b elow the xaxis where the allowable moves are knight moves from left to right The

resulting generating function satises a fourthdegree p olynomial equation This compares

with socalled Dyck paths where the allowable moves are onestep diagonal moves to the

right In this classical case the resulting generating function satises a quadratic p olynomial

equation whose solution yields the Catalan number generating function

The authors apply their metho ds to paths with r s knight moves and obtain p olynomial

equations of higher degree Dennis White

L Jacques Lab elle YeongNan Yeh Generalized Dyck paths Discrete Math

Not yet seen

L Jack Levine Note on the number of pairs of nonintersecting routes Scripta Math

Zbl a

Soit p ermutations X x x x et Y y y y de p elements a et q elements b

n 1 2 n n 1 2 n

p q n Si le nombre des a est dierent du nombre des b dans chaque paire de sequences

partielles corresp ondantes x x x et y y y p our k n les p ermuta

1 2 1 2

k k

tions sont dites une paire de p ermutations non intersectantes p our une raison qui provient

dune interpretation graphique Les paires X Y et Y X sont a considerer comme

n n n n

equivalentes LA obtient le nombre suivant des paires distinctes de telles p ermutations de

n2 n2

n1

p elements a et q elements b p q n N p q p n q n

n

pq p1 q 1

S Bays The Strens Collection has a run of Scripta Math if youd like me to send a copy of

the pap er

Li N Linial A new derivation of the counting formula for Young tableaux J Combin

Theory Ser A MR m

The wellknown ho oklength formula for the number of standard Young tableaux of a

given shap e can b e written in determinantal form Frobenius see eg DE Knuth The Art

of Computer Programming Vol pp AddisonWesley A short pro of of this

result is given by observing that the expansion of the determinant yields an alternating sum

of mutinomial co ecients which obviously satises the dierence equation for the numbers in

question together with the initial conditions Volker Strehl

Ly RC Lyness Al Cap one and the death ray Math Gaz

neither this nor Pe app ears to have made it into MR or Zbl I enclose photo copies of

each of these as they predate many other pap ers on the sub ject I dont have immediate

access to Math Gaz or to Chess Amateur mentioned by Pe but they may

also b e of some historical interest

Ma Ma jor PA MacMahon Combinatory Analysis Vol I Section I I I Chapter V

Cambridge Univ Press Cambridge

referred to in review of Ze and p erhaps G

Mi EC Milner Louis Posa a mathematical pro digy Nabla Bul l Malayan Math

Soc Solutions to the ErdosPosa problems I I I ibid

Problem was to prove the identity mentioned under Kl

M Sri Gopal Mohanty Lattice Path Counting and Applications Probability and

Mathematical Statistics Academic Press xi pp

p gives the reexion principle whereby equations and the formula at

the fo ot of p of the original pap er all follow from equation I havent had access to

Mohantys b o ok there may b e several other relevant references therein

M Sri Gopal Mohanty On some generalization of a restricted random walk Studia Sci

Math Hungar MR

The author considers the paths of a restricted random walk starting from the origin

which at each step moves either one unit to the right or p ositive integer units to the left

and reaches the p oint m n in m n steps Random walks are considered schematically

by representing each movement of the particle to the right or to the left by a horizontal or

a vertical unit so that the restricted random walk corresp onds to the minimal lattice paths

the particle describ es from the origin to m n Expressions are obtained for total numbers

of distinct paths under certain further conditions as follows For a given path say C of such

a random walk the total number of paths is found which after each step do not lie to the

left of the corresp onding p oint of C and which touch C in a prescrib ed way in exactly r of

the last s left steps Expressions are obtained also for the number of paths crossing r times

but not necessarily reaching a p oint for the number of paths reaching r times

and concerning the joint distribution of the numbers of times and steps in the region to the

right of The last is shown to lead to a result connected with a ballot theorem of L Takacs

T The author mentions also related results due to E Csaki C E Csaki I Vincze C

C K Sen Se and O Engleb erg E CJ Rid lerRowe

Athanasios Papoulis A new metho d of inversion of the Laplace transform Q App

Math MR e

nds Legendre co ecients done b efore by Widder Duke Math J and

by Shohat ibid MR

P J Peacock On Al Cap one and the Death Ray Note Math Gaz

see note under Ly

Po G Polya On the number of certain lattice p olygons J Combin Theory

MR

A closed p olygon without double p oints that consists of segments of length one joining

neighboring lattice p oints is called a lattice p olygon Two lattice p olygons are considered as

not dierent if and only if there exists a parallel translation sup erp osing one to the other

The number of dierent lattice p olygons is indicated by dlp A closed plane curve without

double p oints is termed convex with resp ect to the direction d if for the intersection of the

closed domain surrounded by the curve with any straight line of direction d only three cases

are p ossible The intersection is either the empty set or consists of just one p oint or of just

one segment A curve is convex in the usual sense if and only if it is convex with resp ect to

all directions a square is not convex under this strict denition RKG

Let a denote the number of dlp convex with resp ect to the vertical direction with area

m



m b the number of dlp convex with resp ect to the direction with p erimeter n and

n



c the number of dlp convex with resp ect to the direction with area m and p erimeter

mn

P

n evidently c b In this pap er explicit expressions are given for the numbers a

mn n m

m

P

1

m 2 3 4 5 3

a x x x x x x x x b and c For example

m n mn

1

2n

2 3

x x x b n The pro ofs of the results stated will b e presented in

n

n

a subsequent pap er AL Whiteman

Ra George N Raney Functional comp osition patterns and p ower series reversion Trans

Amer Math Soc MR

P

Let a a b e an innite sequence of natural numbers such that a is

1 2 i

nite The author denes the numbers L Ln a a combinatorially and shows that

1 2

P

1

a n

i

i=0

Q

Ln a a

1 2

1

a m

i

i=0

P P

1 1

where m n ia a n i a and L if m n He then derives

i 0 i

i=1 i=1

some identities involving the numbers L and uses them to prove a Lagrange inversion formula

on formal p ower series and a convolution formula given by Go Rimhak Ree

Gerhard Ringel Farbungsprobleme auf Flachen und Graphen VEB Deutscher Verlag

der Wissenschaften Berlin

R Don Rawlings The EulerCatalan identity European J Combin

MR g

This pap er is an attempt to unify the various generalizations of Catalan and Eulerian

numbers by using q theory The author denotes the generating function for p ermutations by

descents ma jor index inversions and patterns by

X

d( ) m( ) i( ) a( ) b( )

An t q p u v t q p u v

2s

Denoting A t An t q p u v the author shows that the recurrence

n

n

X

n

k k k (nk ) k +1

A t A tq t q p u A tA tq

n+1 n

k nk

k

k =1

holds Dening C n t q p An t q p and K n t q p An t q p the author

deduces recurrences for C and K from which two classic q Catalan numbers dened by Ca

follow by taking t q Taking u v in leads to a new recurrence involving

E n t q p An t q p which denes generalized Eulerian numbers A conjecture

involving the q Catalan numbers is p osed at the end of Section RN Kalia

R John Riordan Combinatorial Identities

Bill Sands Problem Crux Mathematicorum Feb

Se Kanwar Sen On some combinatorial relations concerning the symmetric random

walk Magyar Tud Akad Mat Kutato Int Kozl MR

The author considers sequences of n k s and n k s in other words

1 2 2n

the p olygonal paths from to n k through the p oints i s with s

i i 1 2 i

in a rectangular co ordinatesystem where each p ossible array path has the same probability

In connexion with this restricted random walk there are joint distribution laws and joint

limiting distribution laws determined ofr the number of intersections number of changes of

sign of the s and the number of p ositive steps as well as for the number of intersections

i

at the height r the number of changes of sign of s r and the number of steps ab ove

i

the height r Applying the results obtained the author proves some known relations for the

unrestricted case also The pro ofs of the theorems are of a combinatorial character some of

them involving onetoone corresp ondences of paths The pap er has some p oints in common

with E I Vincze

Sh Louis W Shapiro A lattice path lemma and an application in enzyme kinetics J

Statist Plann Inference MR j

Consider the rectangular lattice from to a b Cho ose z integers a a

1 2

a a and z integers b b b b and place stones on the squares

z 1 2 z

a b a b The author rst shows that if the integers are chosen randomly and a

1 1 z z

random path from to a b with unit steps east and north is chosen then the probability

that the path passes b eneath all the stones is z

The author next considers a mo del for enzyme kinetics closely related to that studied

by Hi and by SZ In these mo dels the states are all sequences of length M Each

or represents an enzyme which is either reduced or oxidized The transition rules

essentially allow a subsequence to b ecome Using the result of the rst part of the

pap er the author computes the steadystate distribution under transition probabilities which

are dierent from those used in the earlier pap ers Ira Gessel

SZ Louis W Shapiro Doron Zeilb erger J Math Biol MR

f

The authors investigate a continuous time Markov chain mo delling the diusion of a

ligand across a membrane The states of the chain are strings of s and s of length M

and the transitions are where all the transition rates are

equal The authors give a formula for the steady state probabilities that the r th comp onent

of the string is zero Petr K urka

Neil JA Sloane A Handb o ok of Integer Sequences Academic Press New York

London

Su Rob ert A Sulanke A recurrence restricted by a diagonal condition generalized

Catalan numbers Fibonacci Quart MR c

This pap er contains a pro of via lattice paths of the Lagrange inversion theorem for ordi

nary generating functions It also includes many examples of the CatalanMotzkinSchroder

sequence variety A translation from lattice paths to planar trees is given along with several

planar tree examples

Basically this pap er considers paths where each step is of the form x y x j y

j f g Such a path from to k l is go o d if after leaving all p oints on

the path must lie ab ove the line y x ZZ

The number of go o d paths from to k d sic is shown to b e d k of all

paths b etween the same p oints These paths are then weighted and generating functions are

introduced leading eventually to a combinatorial pro of of the Lagrange inversion theorem

Other combinatorial pro ofs include those of Ra ordinary generating functions L exp o

nential generating functions and G Louis Shapiro

Sv Marta Sved Math Intel ligencer

cf Kl Mi and see Gessel corresp ondence

T La jos Takacs Ballot problems Z Wahrscheinlichkeitstheorie und verw Gebiete

MR

The author proves a discrete variant of Spitzers lemma to the eect that P f

n

j j g n where is the number of p ositive partial sums

1 n n 1 m

m n and the are integervalued with p ermutationsymmetric distribution He uses this

i

to prove that if all p ossible voting sequences are equally likely and the two candidates get



a and b votes with a b and if denotes the number of the a b votecount times

ab



when the ratio of votes is ab then P f j g a b The probability that the

ab

amount by which the rst candidate is ahead stays strictly b etween c d and c where c and

d are p ositive integers c d and c d b a c is shown to b e

1

X

a b a b a b

a k d a c k d a

k

The results extend those of many authors some of the more recent b eing Chung Feller

Ho dges Gnedenko Rvaceva J Kiefer

T La jos Takacs The distribution of ma jority times in a ballot Z Wahrscheinlichkeit

stheorie und verw Gebiete MR

In this sequel to a previous pap er T further probabilities are calculated concerning the

number of times one candidate leads over another during the successive stages of a ballot

The combinatorial pro ofs are based on lemmas which are relevant also to uctuation theory

order statistics and the theory of queues The author also gives a generalization to pro cesses

with indep endent increments FL Spitzer

Ta La jos Takacs Some asymptotic formulas for lattice paths J Statist Plann Inference

MR k

The author proves asymptotic formulas for the area under lattice paths starting at

with unit steps in the p ositive x and y directions Two of the simpler formulas are as follows

Let n b e the area under a random path of length n Then

p

1

2

32

j n

n

8

p

P f n j g lim

32

n!1

n

p

2

1

2 x 2

n c where x e uniformly in j for j b

2

Let a b b e the area under a random path from to a b Then

1

j ab

2

lim P f a b j g

ab

ab

a

b

1

uniformly in j for jj abj where is a p ositive real number and

ab ab

2

p

aba b

Let w n j b e the number of lattice paths from to n n with area j which stay

2n

b elow the line y x for x n Let C b e the Catalan number n and let

n

n

b e the discrete random variable whose probability distribution is given by P f j g

n n

n

w n j C The author gives empirical evidence that suggests

n1

2

Z

x

a

n

u a1

P x e u du

n

n 4

a

0

8C 2

n1

where a Ira Gessel

Im sure there are misprints in MR check against original pap er RKG

HM Taylor RC Rowe Note on a geometrical theorem Proc London Math Soc

WT Tutte A census of Hamiltonian p olygons Canad J Math

MR

number of inequivalent ro oted maps of n vertives is

n+1

n

nn

and number of inequivalent Hamiltonian ro oted maps of n vertices is

nn

2

nn n

W Toshihiro Watanabe Sri Gopal Mohanty On an inclusionexclusion formula based

on the reection principle Discrete Math MR d

A rst used the refexion principle to count paths in the plane with unit steps in the

p ositive horizontal and vertical directions that never touch the line x y Supp ose that

lattice p oints p and q lie on the same side of this line The number of go o d paths from p

to q is the total number of paths from p to q minus the number of bad paths from p to q

those which touch the line By reecting in the line x y the segment of a bad path from

its starting p oint to its rst meeting with this line we nd that the number of bad paths

0 0

from p to q is equal to the total number of paths from p to q where p is the reexion of p

in x y

The authors show here how Andres reexion principle can b e used to solve the multi

dimensional generalization of the ballot problem which is equivalent to counting paths not

touching the hyperplanes x x i j n Here all reexions in the hyperplanes

i j

x x are used and there are n terms in the resulting formula with alternating signs

i j

corresp onding to the n p ermutations of the co ordinates generated by the reexions in the

hyperplanes A similar pro of was given by Ze the authors pro of describ es in more detail the

successive reexions applied to a path Ira Gessel

W Toshihiro Watanabe On a determinant sequence in the lattice path counting J

Math Anal Appl MR g

The number of lattice paths connecting two given lattice p oints and staying b etween upp er

and lower b oundaries can b e expressed by a determinant involving binomial co ecients due

to G Kreweras The recursive nature of this problem leads to a system of dierence equations

and the same type of solution determinants involving p olynomials of binomial type applies

to a much larger class of op erator equations The author makes a new approach by asso ciating

such determinants with random tableaux He obtains determinant sequences which satisfy

a convolution identity similar to sequences of binomial type The theory is then applied to

Hills enzyme mo del repro ducing a result of Shapiro and Zeilb erger Heinrich Niederhausen

cf K Hi SZ

W Toshihiro Watanabe On a generalization of p olynomials in the ballot problem J

Statist Plann Inference MR j

x+m

The ballotp olynomials p x x mx m are Sheer p olynomials for the

m

m

backwards dierence op erator r They are the solutions of the system of op erator equations

rp x p x uniquely determined by the initial conditions p x for all x and

m m1 0

p m for all m Using his earlier results on multivariate umbral calculus the

m

author shows how to solve the ndimensional system P s x s x for all i

i m me

i

n where fP P g is a delta set and e stands for the ith unit vector The

1 n i

general solution for such a system is then sp ecialized to give an ndimensional version of the

ballot p olynomials

A generalized version of the classical ballot problem attributed to Takacs is solved by

the p olynomials

P

r

x n x

k k

k =1

P

r

x n n n

1 r

k k

k =1

The ndimensional analog is obtained from a very general setting leading to a socalled

multinomial basic p olynomial sequence Heinrich Niederhausen

W Toshihiro Watanabe On the Littlewoo dRichardson rule in terms of lattice path

combinatorics Pro c First Japan Conf Graph Theory Appl Discrete Math

MR b

This pap er gives a pro of of the Littlewoo dRichardson rule for multiplying Schur functions

by using the characterization of Schur functions as collections of nonintersecting lattice paths

The pro of is based on Robinsons recomp osition rule for transforming nonlattice paths into

lattice paths Dennis White

Ze Doron Zeilb erger Andres reection pro of generalized to the manycandidate ballot

problem Discrete Math MR g

The ncandidate ballot problem is the problem of counting those lattice walks from the

origin to the p oint m m m m m which never touch any of the

1 n 1 2 n

hyperplanes x x The problem is equivalent to that of counting Young tableaux

i i+1

m +m m +m

1 2 1 2

of a given shap e D Andre A showed that for n the answer is as

m m +1

1 1

m +m

1 2

follows counts all paths from to m m If a path touches x x

1 2 1 2

m

1

reect the initial segment up to the rst such touch in this line to get a path from to

m m This gives a bijection b etween bad paths from to m m and all paths

1 2 1 2

from to m m which proves the formula

1 2

The author generalizes this argument to obtain the determinant formula due to Frobenius

and MacMahon m m det m i j Here a term corresp onding to a

1 n i

p ermutation counts paths from n n to m m The bad paths

1 n

are cancelled in pairs by reexion in the hyperplanes x x A related approach

i i+1

using recurrences instead of reexion was recently given by N Linial Li Ira Gessel