journal of statistical planning

Journal of Statistical Planning and and inference ELSEVIER Inference 54 (1996) 87 100

Symmetric Sheffer sequences and their applications to lattice path counting

H e i n r i c h N i e d e r h a u s e n

Department q/'Mathematics. Florida A thmtic U, iversi(v. Boca Raton. FL 3343l. USA

Received 18 February 1994: revised 10 December 1994

Abstract

A sequence of Sheffer polynomials is symmetric, if the value of the nth degree polynomial at any natural number m agrees with the mth degree polynomial at n. While the above property sounds rather esoteric, symmetric Sheffer sequences frequently describe the elegant results of standard lattice path enumeration. We characterize all symmetric Sheffer sequences, and explain their role from the initial value problem point of view. Applications occur in the enumeration of nonintersecting weighted lattice paths, and the discussion of certain correlated random walks.

AMS classilication: Primary 60J 15; secondary 05A40

Keywords: Lattice path; Correlated random walk; Sheffer sequence; Umbral calculus

1. Introduction

A sequence of ho r i zon ta l -+ and vertical T unit steps will be called a (standard)

lattice path in this paper . Deno te by D(m, n) the number of such pa ths s tar t ing at (0, 0)

and reaching (m,n), perhaps under some constra ints . Obvious ly , D(m,n)-=

D(m - 1,77)+ D ( m , n - 1). W e are main ly interested in cons t ra in ts that a l low us to

view D(m, n) as the so lu t ion of the system of difference equa t ions

VmD(m,n):= D(m,n) - D(m - 1,n) = D(m,n 1)

with some b o u n d a r y condi t ions , reflecting the constra ints . The condi t ion

D(m, 71) - 0 for negat ive n

will a lways be imposed in the sequel. The 'unres t r ic ted ' case can be descr ibed by the

init ial values D ( - 1, n) = 6o.,,; of course, D(m, n) = (" +m). Fig. 1 shows some values of

D(m, n) under two different restr ict ions. The bold numbers are initial values, descr ib-

ing the ( redundant ) vert ical b o u n d a r y m = - 1 in the first case, and the two-piece

0378-3758/96/$15.00 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 3 7 5 8 ( 9 5 ) 0 0 1 5 9 - X

88 l-t Niederhausen /Journal o f Statistical Planning and Inference 54 (1996) 87 100

m : - 1 0 1 2 3 4

n = 4

n = 3

' r~--2

n = O

0 1 5 15 35 70

0 ] 4 10 20 35

0 1 3 6 10 15

0 1 2 3 4 5

1 1 1 1 1 1

"Unres t r i c t ed" case.

m : 1 0 1 2 3 4

n = 7 15 20 - 2 0 0 75 275

n - - 6 - 5 5 0 20 75 200

n 5 1 0 5 20 55 125

n = 4 0 1 5 i5 35 70

n = 3 0 1 4 10 20 35

, r , - 2 0 1 3 6 10 15

n = l 0 1 2 3 4 5 n = 0 1 1 1 1 1 1

B o u n d a r y m, m a x { - 1 , n - 5}.

Fig. 1.

boundary m = m a x { - l ,n - 5} for n = 0, 1 . . . . in the second case. The latter bound- ary prohibits the paths from touching the line x = y - 5 . The numbers in italics are

obtained from D(m, n) by polynomial extension - - they occur in an area of the lattice that is off-limits for the paths and therefore do not represent counts. We call these polynomials d.(m) and say they support the count D(m, n).

The above presentation in terms of supporting polynomials and difference equations de-emphasizes specific combinatorics tools in favor of a more algebraic approach. We reduce the problem to a system of certain operator equations on

polynomial sequences {d.(x)}, say,

Bd.(x) = d._l(x) for all n = 0, 1 . . . . , (1)

where the recurrence B is a degree reducing and translation invariant linear operator like V (and like differentiation). Translation invariant means that B(p(x + a))= (Bp)(x + a) for all polynomials p and constants a. Boundary conditions make the solutions unique. In Fig. 1 they are of the form

d . ( a n + b ) = 6 o . , for a l l n = 0 . . . . ,L,

d . ( c n + d ) = O for a l l n > L ,

where L ~> 0. The (umbral) calculus that handles such systems for translation invari- ant recurrences prints out the solution

L c k + d - a k - b x - c n - d d.(x) k~oL ck +-3Z[~ bk(ck + d - b ) x _ c k _ d b,_k(x ck d), (2)

where {bn(x)} solves the system (1) with initial values b,(0) = 6o., (the basic poly- nomial sequence for B, see Niederhausen (1980, Corollary 2.3)). In the context of path enumeration we substitute V for B, and consequently (n+~-l) for b,(x). In the examples above we used boundaries of the form dn (max{- 1, n - L}) = 60,n, where L > 0. Hence, there are

L ~ I ( 2 k - L n L ( n D(m,n) = d.(m) = k m k + n - k (3)

k=O

H. Niederhausen /Journal q/'Statisticcd Planning and h!l~,rence 54 ¢19961 <~7 lOO , ' ;9

paths that start at the origin and reach the point (re, n), m > t7 - L. without cw:r touching the line x = y - L.

But this is not the way the problem is attacked in Combinatorics. In Combinatorics. paths are reflected (Andre!) and rotated, and after a few elegant arguments the much easier answer

o,,,,,,, (,,;,,,) (::+,:) ,4, is obtained. (The disappointed problem solver may find some consolation in the lack of combinatorial tools for improz, ing on formula (2) for slopes different from 1 and 0.) Proving algebraically that the long form (3) equals the short (4) does not unveil the mystery. Looking at (4) from the boundary value problem point of view, it is obvious that d , , ( x ) _ ( , + x ) _ ( ~ + [ ) satisfies the right recurrence 1 7 d , , ( x t - d , , l(x). But the correct boundary values are only taken on 'by accident':

Foralln>._-L, d"(n - L) = ( n ÷ n - - ( n + n - n L

= ( n + n - - L ) t 7 - - ( n / = 0 .

In other words, the polynomial sequence {w,dx) - (,,+x)~ has the amazing properly

w,,(m) = win(n) for all m, n = O. 1 . . . .

Argument and degree can be interchanged! We call this the s y m m e t r y propert .v .

Solutions to (1) are called Sheffer sequences. As a tool, ShelTer sequences stand between purely combinatorial arguments and generating function methods. Symmet- ric Sheffer sequences are destined to help solving boundary problems of the form d , , ( m a x { - l , n - L } ) = 6 o . , , in the most elegant way. But where are they in the literature'? Or is {t,,+x,~ ,, ,j the only one? We show in this paper (Section 2) that the class of symmetric Sheffer sequences is not large, as expected. They count the number of unrestricted standard lattice paths with weighted left turns. In other words, such lattice paths enumeration is characterized by its use of symmetric Sheffer sequences! In a probabilistic interpretation, the general symmetric Sheffer sequences describe lin Section 3.1) the probability that a certain Markov chain reaches the point (m, nk generalizing the concept of a random walk to correlated random walks. We show in Section 3 how symmetric Sheffer sequences efficiently generate the standard enumer- ation results (cf, Mohanty, 1979) in this area.

For a bivariate application of symmetric Sheffer polynomials, let A(i, j + 1: k + 1, Ii be the number of all pairs of paths that go from (0, 1} to (i,/" + 1), and from (1,01 1o (k + 1, I), respectively. Obviously,

90 I~ Niederhausen / Journal of Statistical Planning and Inference 54 (1996) 87--100

j + l o

1 0 0 ~ 0

T l • ~ o ~ o o

0 0 • --+ 0 ~ 0

0 1 i k + l

Fig. 2. Nonintersecting lattice paths.

Thus, the number of pairs is supported by bivariate Sheffer polynomials, which are again symmetric in the unrestricted case. This symmetry can be exploited if we determine the cardinality R(i, j + 1; k + 1, l) of the subset of nonintersecting pairs of paths (Fig. 2).

If j + 1 ~>l>~O and k + 1 ~>i~>O one can show that R( i , j+ l ; k + l , l ) is sup- ported by a bivariate Sheffer sequence {ri,~(x,y)} satisfying both partial difference equations

Vjr i , l ( j , k) = r i , t ( j , k) -- ri, t( j -- 1, k) = ri_ l , t ( j , k)

and

Vkrl, t( j , k) = r id ( j , k) -- ri, t ( j , k -- 1) = r i , t - l ( j , k). (5)

Using the symmetry of {w,(x)} and the boundary condition

R(k+ 1, j + 1;k + 1, j + 1)=O=rj+nk+l(k , j )

for all nonnegative integers j and k, we get

(6)

for all j + 1 ~> l/> 0 and k + 1 /> i ~> 0 (Narayana's formula, see Narayana (1955) for a special case). The case i = k, j = l has already been solved by MacMahon (1916, Vol. 2, No. 242).

The above derivation of Narayana's formula is sketchy; there is a combinatorial proof in Sulanke (1993). But the result is only a special case of nonintersecting lattice paths with weighted turns. In Section 4 we give a detailed proof for the weighted case, which requires a different approach. In 'Counting pairs of nonintersecting lattice paths with respect to weighted turns' by Krattenthaler and Sulanke (1996) this result was derived using the 'rotation method', and more references are given there. It was their work that renewed my interest in symmetric Sheffer sequences and motivated the

H. Niederhausen/'Journal of Statistical Planning and [nfi, rence 54 (1!)96) 87 100 91

search for their complete characterization. A q-analog of Narayana ' s formula is also

considered in Krat tenthaler and Sulanke (1996). In that case, a turn at (i, j ) gets the

weight q~+J. The method we use in Section 4 can be extended to such weight functions, but there are some addit ional technical difficulties to deal with. We leave this case for

a later paper (Niederhausen, 1994).

2. C l a s s i f i c a t i o n o f s y m m e t r i c Sheffer s e q u e n c e s

A Sheffer sequence for [4 and p is a polynomial sequence f • tan(X);n=o,1, in ~ [x ] , i . c . deg(a,(x)) = n, such that

a,(x)t" = p(t)e ~a('), (7) n>~O

where

• D ( t ) = Do + P l t + P2 t2 + "'" is a power series in t of order 0 (i.e., Po ¢ 0),

• [4(t) = fi~t +/~2t 2 + ... is a power series of order 1 with [41 = t. The recurrence B on Nix ] that maps a, into a, ~ (n = 1,2 . . . . ) is called a delta operator , and has the power series expansion B = [4-1(D), where D is the deriw~tive

(delta) opera tor D : x" ~ nx"- ~ and [4-1 is the composi t ional inverse of [4.

The polynomial sequence {w,(x)} is symmetric iff

w a n ) = w.(m) (~i

for all n and m in No, the set of nonnegat ive integers. The following theorem is trivial but useful.

T h e o r e m 1. I f {w,(x)} is a symmetric Sheffer sequence for [4 and p, then p(t)=:

~/(1 - t ) /o r some constant ~ :/: O.

Proof. Let n = 0. F r o m (8), win(0)= wo(m) follows for all mE No. But wo(m) is

constant , hence all polynomials win(x) have the same constant term ~, say, and therefore

- ~ w.(0 ) t" = p( t ) 1 -- t , ~ o

(see (7)). [ ]

We want to determine the power series oJ(t) in the generating function

9( Z Wn(X)tn-- eX°(t)

.>1o 1 - - t

92 H. Niederhausen /Journal of Statistical Planning and InJerence 54 (1996) 87-100

of s y m m e t r i c Sheffer sequences . W e f ind the fo l l owing func t i ona l e q u a t i o n for ~o:

~ E sn emO(o; '(t))= E Sn ~ Wk(n)fD l ( t )k 1 - c o - l ( t ) l - sd ,>~o 1 - c o - l ( t ) n>lO k>~O

= ~ Z SnWn(k)(D--l(t)k= E (D-k(t) ek°(s) k>~O n>~O k>~O 1 -- S

1

- 1 - s 1 - co-a(t)e°(S)"

R e p l a c i n g s by co- 1 (s) gives

(1 - co-X(s))(l - co-l( t)eS) = (1 - ~o l(t))(1 - co-l(s)et) .

There fore , we can s e p a r a t e the v a r i a b l e s s a n d t i n to fac to rs

co 1(0(1 - e S + co i ( s )&) = co - l ( s ) (1 - e ' + co l ( t )e ' ) .

Hence , e i the r 1 - e t + co- l ( t )e t = 0, i.e., c o - l ( t ) = 1 - e - t , o r c o - l ( t ) is p r o p o r t i o n a l

to 1 - e ~ + ¢o- l ( t )e t for s o m e p r o p o r t i o n a l i t y f ac to r 1/(1 - / ~ ) , say, 0 # i t # 1. So lv ing

for c o - l ( t ) gives in the l a t t e r case

CO_l(t ) __ l l-u(1 - - e ' ) _ 1 - e ' 1 -- ~ - u e t 1 - - / t -- e r"

p = 1 cove r s the first case. W e cal l the de l t a o p e r a t o r O(~'1:= ~o- I(D) the weighted path

recurrence, b e c a u s e it is a s s o c i a t e d wi th Sheffer s equences (see R o t a et al., 1973) t ha t

p l a y a ro le in the e n u m e r a t i o n of l a t t i ce p a t h s wi th w e i g h t e d tu rns (see Sec t ion 3). T h e

w e i g h t e d p a t h r e c u r r e n c e can be e x p r e s s e d in t e r m s of the f ami l i a r f o r w a r d s di f ference

o p e r a t o r A = E 1 -- 1 = e D - 1,

e D - 1 1 A ~(,,) = c o l ( O ) _ - _ y ~ (-#)-"~".

p + e D - - 1 t t 1 + 4 / # ,>~ 1

/ ~ = 1 gives f2 ~ l ) = l - e - ° = V, the b a c k w a r d s d i f ference o p e r a t o r (in this case,

w,(x) = ~( '+x)) . I t is i n t e r e s t i ng to reverse the r e l a t i o n s h i p a n d exp res s A a n d V in

t e r m s of (2 (~,),

A = ~tO(")/(1 -- ~2 (''~)

V = ( p e o + V)f2(~,) = p e - O ~ (f2(~,l)i = p e ° K 2 ( ~ ' ) / ( 1 - g2(~')) . i>~l

(9)

2.1. Explicit symmetr ic Sheffer polynomials

A b o v e o u r o r i g i n a l s y m m e t r i c Sheffer p o l y n o m i a l s {(,+x)} we f o u n d a o n e - p a r a -

m e t e r c lass of Sheffer s equences {w,(x)} wi th g e n e r a t i n g func t ions (~/(1 - t))e x°'(°

H. Niederhausen / Journal of Statistical Planning and ln[i'rence 54 (1996) 87 lO0 93

where ~o(t) is the composi t ional inverse of ~ ) - l ( t ) = (1 - e')/(l - / ~ - et) - Thus. e ~'~t> = 1 + la / ( l - t). We expand for all k ~

tl 1 + 1 _ t ; E t" .~>o l :o + k llz' (10)

and obtain for k = 1 the symmetric Sheffer polynomials

w~,,"°(x) = ~ ~ td. (1 l) / : 0

The symmetry is obvious. If k = 0 we get from (10) the basic sequence {h,,(x)} for the weighted path recurrence O~')

b,,(.\-): ~ ( ' ~ ) C 5 ; ) / ~ ' ~ = ~ for positive integers n. (1121

We expressed in (9) the backwards difference opera tor as a power series in (2 u'>. This shows that any Sheffer sequence {s.(x)} is associated with the weighted path recur- fence iff it satisfies one or both of the expansions

s , , ( x ) - s . ( x - 1 ) = ( / ~ - l)s. l ( .x ' - l ) + s . l(X). (131

n - I

s . ( x ) - s . ( x - 1 ) = t z ~, s i ( x - 11. t141 i 0

In the next section we present a count ing problem that is governed by the abovc recurrence.

3. Lattice paths with weighted left turns

A lattice path makes a left turn if a vertical step follows a horizontal step: --,T A path starting at (0, 1) is uniquely determined by its end point (m, n + 1), say, and its sequence ( ~ , ql) . . . . . (~, rh) of left turning points

1 < ~ ~ ~ < . . . < ~ t <<, t n , l <<, T l l < . . . < ~ h <~ n . 115)

A lattice paths that starts at (0, 1), and reaches the point (m, n + 1) with exactly / left turns gets the weight I{. Let Dp,(m, n + 1; l) denote the sum of tile weights of all such paths. In the unrestricted case,

D . ( m , n + 1):= ~ o D . ( m , n + 1; l) is the number of paths with /z-weiqhted Iq/t turns that start at (0, 1) and reach (m,n + 1). In the unrestricted case, D,(m,n + l) can be represented by the symmetric Shefl'er sequence {w~"~(x)} (see (11)) with ~ - - 1 . We Check the weighted path recurrence relation (13) in Fig. 3.

94 H. Niederhausen /Journal of Statistical Planning and Inference 54 (1996) 87-100

In the backwards difference nota t ion ,

VmDu(m,n + 1) = (/t - 1)Du(m -- 1, n) + Du(m,n ). (17)

It is i m p o r t a n t to realize tha t this recurrence al lows us to find Du(m, n + 1) even if

some b o u n d a r y values are given. F o r example , if the pa ths are res t r ic ted as in Fig. 4,

the number s D,(m, n + 1) can still be recursively ca lcu la ted from the above formula.

U n d e r n e a t h a m o n o t o n e increas ing bounda ry , Du(m, n + 1) is suppo r t ed by a poly-

nomia l dt,")(x) with deg d~,U~(x) = n. The sequence {dt,")(x)} is a Sheffer sequence for the

weighted pa th recurrence f2 l") because of (17) and (13).

3.1. The t~-weighted path as a correlated random walk

We can view a/~-weighted lattice path as a 'Correlated random walk' W (Mohanty, 1979, Ch. 5.2; Renshaw and Henderson, 1981), a Markov chain with states 'horizontal' (= h) and 'vertical' (= v), and transition matrix

J

f r o m \ t o : h pV

h Phlh h (18)

V Phlv Pvlv

For example, a vertical step is taken with probability P<h if the previous step was horizontal (a left turn). In Renshaw and Henderson (1981), Phth = P < [ . = P , and hence

P~qh = Phl~ = 1 -- p =:q.

D,.(ra, n + i) =

n + l o -4 o

' rD

D~,(m - 1,r~ + 1)

r z + l o

T Th 0 ---)" @

#

Tn

+#Du(m - 1, rQ

r z + l o

T 7"/ - - + 0

t

+ D,,(rn, .,)

n~+ I o

t

m

- O , ( m - l,n)

Fig. 3.

6

5

4

3

2

1

Tt~:

0 0 • 0 •

0 1

0 0

0 •

• I

2 3

0 • •

@ • •

• • @

@ ,411 @

456

Fig. 4.

H. Niederhausen/ 'Journal o[Statist ical Planning and In/brence 54 (1996) 87 lO0 ( 5

n + 1

II.

! Phil:

1 c

T Po 0 o

0

P~.~ l" © , •

Pt~lv Pt~l~,

P~ h

[)l~ L

[ P'I'~' 5:

~Lqh

1 . . . . m m + 1

Fig. 5. m = 4, n = 4~ / = 2: Pr(W} = POP{I,,P~,,P~,IhP~!,'

T h e ana lys i s gets eas ie r if we res t r ic t o u r a t t e n t i o n to r a n d o m wa lks W tha t s t a r t

wi th a ver t i ca l s tep ( f rom (0, 0) to (0, 1), wi th p r o b a b i l i t y P0), a n d end wi th a h o r i z o n t a l

s tep ( f rom ( m , n + 1) to (m + 1,n + 1)); see Fig. 5.

N o w the first a n d las t t u rn mus t be r igh t turns , a n d the re fo re we have / + I r ight

t u rns if the p a t h t akes l left turns :

l t+ l . . -1 - m ,, / \ |P, ' ihPhl"/l P r ( W ) = P o P v l h P h l v Phlh P n l , ' l = PoPhl~,PhlhP,' i , , ~ 1 - - " \ P h l h P , : l , , /

T h e r e are of c o u r s e (~) (~') such pa ths . Hence ,

P o : = P r ( R a n d o m w a l k s t a r t s a t the o r ig in wi th a ver t ica l s tep and reaches

(m + 1, n + 1) in a h o r i z o n t a l s tep) = poPhl,,p"h'lhp~l,W~*'l(m)

if we de f ine /~ = PL, IhPhl,JPhlaP,'It,, a n d w~,"l(x) as in (11). Po can be used as a s t a r t i n g

p o i n t for fu r the r c a l cu l a t i ons . F o r e x a m p l e ,

Pk := P r ( R a n d o m w a l k s t a r t s a t the o r ig in wi th exac t ly k h o r i z o n t a l s teps before

the first ve r t i ca l s tep a n d reaches (m + 1,n + 1} in a h o r i z o n t a l

s tep) (1 Po)P<hPhl " J " ~"~ = -- ,~'Phhh P,,I,,W, ( m - - k).

Th is s impl i f ies if we let Po = P,qh, a n d hence Ph,h = 1 - - PO. T h e n

m n ,(#1 Pk p, . ihph!vphlhp, , iv~,~n ( m - - k )

for all k = 0 . . . . . m. S u m m i n g ove r k gives

P r ( R a n d o m w a l k s t a r t s a t the o r ig in a n d reaches (m + 1,n + 1) in a h o r i z o n t a l

s t e p ) = '~+ lp~l+,l l ' + 1~i{+ l Phlh ~./>~0 (t]) t / + l J

96 H. Niederhausen /Journal of Statistical Planning and Inference 54 (1996) 87 100

3.2. Linear boundaries

We saw that unrestricted paths with weight # for each left turn are counted by Du(m,n + 1) = w~.")(m) = ~ ' : o (~) ('~)#~. Now we keep the paths strictly underneath the line y = x + L + 1 for some integer L ~> 0. The weighted counts D,(m, n + 1) of these paths are represented by Sheffer polynomials {~.} that satisfy the same weighted path recurrence ~21")#.(m)=#._l(m) as before, but have the initial values Du(0, n + 1 ) = ~ . ( 0 ) = l for all n = 0, ... ,L - l, a n d D u ( n - L , n + l ) = ~ . ( n - L ) = 0 for all n/> L. In the same way as in the unweighted case (4), the symmetry of {wt.~l(x)} immediately leads to the answer # . ( x ) = w~.") (x) - w~.~'~L(x + L). So we find the probabili ty P~,. that a Markov chain with transition probabilities (18) starts at the origin with a vertical step, reaches (m + 1,n + 1) in a horizontal step, and does not

touch the line y = x + L + 1:

1 1 -m . p = ~ . (m) :- wt.")(m) -- w ~ L ( m + L) Po PhlvPhlhPvl~ m,n

for all n ~> 1, m ~> max{0, n - L} (# = PvlhPhl~/(PhlhP~l~))" A detailed discussion of this problem is given in Mohanty (1979, Ch. 5.2) for random walks reaching the boundary line for the first time after n steps (first passage probabilities). Formula (2) gives 'closed expressions' for boundary lines with positive integer slope larger than one (see also Mohanty, 1966). Use the basic sequence

in formula (2). A generating function approach to such problems can be found in Krattenthaler (1989). Piecewise linear boundaries can be treated by repeated applica- tion of formula (2), as described in Niederhausen (1980).

4. Intersecting weighted lattice paths

In this section we will work with bivariate Sheffer sequences {r~,.(x, y)} associated with partial delta operators (recurrences) B = fl I(D) and G = ),- I(D), say, on ~ Ix] such that

Bxrm,.(x, y) = r~- l , . (x , y) and Gyrm,.(x, y) = rm,.- l (x, y)

for all positive integers m and n. {rm,.(x, y)} has the generating function

Z rm,n(X, y)smt n = p(S, t)e ~"~+y~'"), (19) m,n>~O

H. Niederhausen /Journal c)f Statistical Planning and hl[erence 54 (1996) 87 100 9 .7

where p(s, t) is a bivariate power series of order 0, and [3 and ;, have order 1 as ill

Section 2. Bivariate Sheffer sequences are uniquely defined if they satisfy the abow:

partial opera tor equat ions and take on a given initial value r .... (x,,, y,,) for each

m, n c No. Of course, the explicit representat ion of such polynomials can be complic-

ated, even if the boundaries are piecewise linear. More about multivariate Umbral

Calculus can be found in R o m a n (1979).

We ment ioned nonintersecting pairs of lattice paths in the introduction, and

indicated how their enumerat ion can be seen as a bivariate boundary value problem.

This approach is no longer valid if we give weights tt to the left turns of the upper path,

and weights v, say, to the right turns of the lower path. To see thal let

N ( a , m + 1; n + 1,d) denote the number of such pairs of paths reaching (a,m + 1)

from (0, 1), and (n + 1, d) from (1,0). The weighted path recurrence relation (17t holds

for the upper and for the lower path if their end points are far enough aparl:

N ( a , m + 1;n + 1,d) = N(a - l , m + 1;n + 1,d)

+ (/i - 1)N(a - 1,m; n + 1,d) + N(a ,m: n + 1,d)

for all m > d ~> 0 on the interval 1 <~ a -'~ n, and

N ( a , m + 1;n + 1 , d ) = N ( a , m + 1;n + 1 , d - 1)

+ (v - 1 )N(a ,m + 1; n,d - 1) + N ( a , m + 1; n,d)

on the interval 1 ~ < d ~ m , n > a ~ > 0 .

But Fig. 6 shows that the obvious boundary value N (n + 1, m; n + 1, m) = 0 can not

validate the recurrence for the upper path, if it comes too close to the lower path:

N(n + 1,m + 1; n + 1,m) ~ N ( n , m + 1; n + 1,m)

+ (/i - 1)N(n,m; n + 1,m) + N(n + 1,m; n + 1,m)

because we cannot pass f rom (n,m) through the (black) end point (n + 1, m) of the lower path. We will obtain more useful initial values from the following lemma. Let

l(a, m + 1; n + 1, d) denote the number of weighted intersecting pairs of paths reach-

ing (a ,m + 1) from (0,1) with /l-weighted left turns, and (n + 1,d) from (1,0) with

m + l o -~ o

I d - m o ~ •

n a r~ F1

Fig. 6.

98 H. Niederhausen / Journal of Statistical Planning and Inference 54 (1996) 87-100

~s+l=3 o o ~-~ o --+ o

T T 2 0 0 ---~ 0 ---+ 0

T T T d=l • --+ o ~ o ~ o

0 • =)> o =~> o

0 1 2 3 ~2

o

4 n + l

Fig. 7. All intersecting paths reaching (3, 3) and (4, 1).

v-weighted right turns, where 0 ~< a ~< n and 0 ~< d ~< m. Of course, l(a, m + 1; n + 1, d) and N(a, m + 1; n + 1, d) sum up to the number of 'unrestricted' paths,

(see (16)).

L e m m a 2. Let {gt, u)} denote the Sheffer sequence for ~2 tu) with polynomials

gn(x):= /~ l - - 1 l + l / = 1

(they are obtained from (10) by setting k = - 1 and c~ = 1//~).

For all m >~ d >~ 0 and n >~ a >>- 0,

I (1 ,m+ 1 ; n + 1 , d ) = , '(~) ( d + 1) ~ n + l

and

l(a, m + 1; n + 1, 1) = vg,,+l(a~") + 1).

Proof. We show the second identity (the first follows in the same way).

I(O,m + 1; n + 1, 1) = 0 proves the case a = 0. Let a > 0. The lower path must turn

right at some point (k, 1) for k = 1, 2 . . . . , a ~< n (see Fig. 7). In order to intersect, the upper path must first go right, meeting the lower path, and then leave the lower path in a left turn at some point (j, 1) for j = k . . . . , a. Thereafter, no restrictions apply.

Hence,

I (a ,m+ 1 ; n + 1 , 1 ) = ~ v ~ pG(j, 2la, m + 1), (20) k = l j - k

where G(j, 2[a, m + 1) is the number of unrestricted paths with/~-weighted left turns from (j, 2) to (a,m + 1):

G(j, 2 ] a , m + l ) = ~ I f l ( m - - 1 ) ( a T J ) / = 0 l "

H. Niederhausen /Journal q['Stalistical Planning and ln[i)rence 54 (1996) X7 lot) 9 9

Substituting this into (20) gives

(,7 l ( a , m + 1 ; n + 1 , 1 ) = v ¢(t I = 1 - - - -}-

a f t e r two trivial summations. [ ]

We want to show the existence of a ShelTer sequence {s ..... ix, y)} for (f2!('), f21.')) such

that

l ( a , m + 1; n + 1 ,d) = Sm+l,n~ l(a,d) for all m > d > 0 a n d n >~ a > O.

Lemma 2 can be written in the very suggestive form

l ( l . m + 1; n + 1.d) = ,I.I (,') u . ,+l(2)g, ,+l(d + 1)

and

l (a ,m + 1; n + 1.1) = ~.,+"(") l(a + 1)g~"+)1(2)

for all m >~ d >~ 0 and n ~> a >~ 0. Obviously, s2.2(a,d):= g~z")(a + 1)g(z"l(d + 1}: a -F 1 d + 1 ( 2 )( 2 )/~2v: supports 1(a, 2; 2, dl, because a = I d is the only admissible argu-

ment. F rom this starting point, the existence of {s ..... (x,y)} follows by induction if we

apply the next theorem with Q.~ = f2!(l and R>, = o i,,) using the initial polynomials / . ~ - y

from Lemma 2.

Theorem 3. Let n and m be positive integers, u < u._ 1 <~ u. and b < b., 1 <~ b., be re,JI

numbers, and let Q and R be linear operators on ~ Ix] such that deg(Qx") = deg (Rx') :

n - l for all positive integers n, aml ker(Q) = ker(R) = {constant functions I. Suppo,~e.

p .... (x. y) is a polynomial in x o f degree m and in y o f degree n, and f ( x . y ) a piecewisc

polynomial function in both variables .such that

(a) Q ~ f (x, y) = Q~p.,, .(x, y) on the interval u ~ x <~ u., b ~ y <~ b.,_ l. and

(b) R, , f ( x , y) = R~,p.,,.(x, y) on the interval u <~ x <~ u. 1, b <~ y <~ b,..

I f there are numbers a and d, u -<. a <~; u., b <~ d <~ b.,, such that

(c) f ( a , y) is a polynomial o f degree n in y for all b ~ y <~ b,., and

(d) f ( x , d ) is a polynomial o f degree m in x fin" all u <~ x <~ u., then f (x , yl is

a polynomial in x o f degree m and in y o f degree n on u <~ x <~ u., b <~ y <~ bin.

The initial polynomial parts o f f in (c) and (d) uniquely determine the solution q ! t t w

partial operator equations (a) and (b).

Proof. Only constants are in the kernels of Q and R. It follows from (a) that

f '(x. y) - pm,n(X, y) is a constant in x that could be a piecewise polynomial function in y on u ~ x <~ u., b ~ y <~ b.,_ 1. The condi t ion (c) shows that this term is a (unique)

polynomial piece, even on the larger interval b ~< y-G< bin. Also, f ( x , y ) and p .... (x, y)

must have the same degree m in x on u ~< x ~< u.. In the same way, f ( x , y ) - p .... (x, y) is a (unique) polynomial piece in x by condit ions (b) and (d) on u ~< x ~< u., and , f(x. y)

a n d p ..... (x,y) m u s t h a v e t h e s a m e d e g r e e n i n y o n b ~ < y ~ < b . , . [ ]

100 I~ Niederhausen / Journal of Statistical Planning and Inference 54 (1996) 87 100

O n l y n o w after we have s h o w n that I(a, m + 1; n + 1, d) is s u p p o r t e d by a Sheffer

sequence, we can use tha t ~,~u) ,.) ) g , + l ( d + ~ , , + l ( a + 1 1) has the r ight in i t ia l va lues to

deduce f rom the u n i q u e n e s s pa r t of T h e o r e m 3:

l ( a , m + 1; n + 1,d) = ~,,+"~u) l (a + L~,+t~'~v) l (d + 1)

m - - 1 a + -- +

for all m > /d ~> 0 a n d n ~> a ~> 0. S u l a n k e a n d K r a t t e n t h a l e r (1996) o b t a i n e d this resul t

u s ing the ' r o t a t i o n me thod ' . If/~ = v = 1 then the f o r m u l a simplifies to

I(a'm÷l;n+l'd)=(m÷~)(:÷~)m+ +

(see the N a r a y a n a n u m b e r s (6)). This is the in t e r sec t ing c o n t r i b u t i o n to the 'Ref ine-

m e n t of N a r a y a n a n u m b e r s ' c o n s t r u c t e d in S u l a n k e (1993).

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