CHINl>SL JOURNAL 01 PHYSICS
VOL. 29. NO. 1
I~L’BR~J,4RY
1991
Perimeter and Area Generatine Function of Pyramid Polygons
K. Y. Lin (& $5 %)
Department
of Physics,
National Tsitg Hua Utliversity
Hsiwhu, Taiwan 30043, R. 0. C.
(Received Nov. 10. 1990; revised manuscript received Nov. 27, 1990)
An explicit expression is derived for the perimeter and area generating function for
pyramid polygons.
I. INTRODUCTION
Self-avoiding walk is a model for linear polymers.’ A self-avoiding loop is a selfavoiding walk returning to the starting position. One of the outstanding unsolved problems
in Statistical Physics is to derive the generating function for the number of self-avoiding
loops (polygons) on the square lattice with a fixed length of perimeter and a fixed size of
area.
The two-variable perimeter and area generating function of polygons on a lattice is
defined by
P(s,z)
=
5
C,f,,Tl
,,.,,r=l
w”ere c,r.,,r
x”
9”
(1)
is the number of polygons with perimeter n and area m. Exact solution for the
staircase polygons on the square lattice was first obtained by Polya.’ Solutions for the
staircase and row-convex polygons on the square lattice were given by Brak and Guttmann.3
Recently Lin and Tzeng4 generalized their results to the rectangular lattice.
A convex polygon on a square lattice is a special case of the self-avoiding loops such
that the number of steps equals the perimeter of the bounding rectangle. The generating
function for the number of convex polygons on a square lattice was first derived by Delest
and Viennot’ and then rederived by simpler methods. 6-8 The result was generalized to a
rectangular lattice by Lin and Chang.’ Enting and Guttmann’ proposed two formulae for
the generating functions for the area-weighted moments of convex polygons on a square
Lin”-” generalized the results of
lattice. Their conjecture were verified by Lin.“.”
Enting and Guttmann to a rectangular lattice.
A pyramid polygon on a square lattice is a special case of the convex polygons such
that the width at the bottom equals the width of the bounding rectangle. In this paper we
shall derive the three-variable generating function on the square lattice
is the number of pyramid polygons with 2n horizontal steps. 2m vertical steps
and area r. ‘In the special case of x = y. (‘3) reduces to (1 ).
Pyramid polygon was first considered by Temperley” as a model of crystal growth on
a plane substrate. He gave the explicit expression G( I .I .z) M,hich is a special case of (2 ).
The asymptotic form of the number of pyramid polyrgons with area n for large n was determined by Auluck. l6 Recently this model was considered again by Priiman and S\,rakic’7
as one of the lattice animal models.
mJhere c,,,,,* i
II. PYRAMID POLYGONS
Consider a pyramid polygon on the square lattice as shown in Fig. 1 where the width m
at the bottom equals the width of the bounding rectangle. The generating function (2) can
FIG. 1. A pyramid polygon on the square lattice with m=7 and k=2.
be written in the form
(3)
where s,,, generates polygons whose bottom width is m and F’ generates polygons with
exactly k squares in the left column. It was shown by Lin14 that
g1
= 9)s Z/(1 -y22)
g2
= .,4,.222 (1
(1 -
+yZ-‘),(l
(4)
-JvNl -J?)
pZ”+qg,,+* - 2.u2z g,l+l +
X4T2
g,l =
0
(5)
(6)
K. Y. LIN
9
The special case of z = I was solved by Lin and Chang7 and they found
g,,l (x2, 1) = (vZ /Mu:” + u: )
(7)
G(x,y,l) = x*y*(l -x*)/[(l -x2)* -y*l
(8)
where
11,
x2/(1
=
-fy)
It is convenient to define
go = y*
(9)
such that Eqs. (6) and (7) are valid for n = 0 also.
The special case of x = y = 1 was considered by Tcmperley.” He proved that
G( I, 1 ,z) = ,,fI z” (1 - 2” ,/q,: = C c,, z”
I,
= z/(1 -z)+z2/(1 -?)*(I -z’)++3/(1
(10)
-:)2(1 -z2)2(1 -z3)+. .
where
q,, = kGI(l -zk)
f o r n>O
and c,! is the number of pyramid polygons with area n. The function G( 1 ,l J) has an
essential singularity at z = I. It was shown by Auluck16 that for large n
= 8-l 3-3 ’ n-5.4 exp[27+2/3)“*]
Cn
(11)
Privman and Svrakic17 used the identity’*
[
mto (1 - tz”’ >I -I = ,Fo
t’ q,,+,(z)/q,,(:)qp
(12)
to prove that
(13)
where
tk.n
=
,=$ q,+i_~(Z)q,+k_,-~(r)iqi(;)qk-~‘~)
u-..
The generating function (3) can be derived bl. the method of ’ Trmpcrlcv as follows
We write
where Hli generates all pl’ramid polygons ~rhose height is n (see Fig. 2) and
FIG. 7. A pyramid polygon with three steps
H, = J.2
z (X2i)0 = y22x2/(l -x22)
a=1
H2 = ~3~ y (~~2)~ ;
h=O
a=0
(15)
(x2@ ? (x2z2 )c
1=1
= y4z2x2/(1 - .x22)2(1 - .x2z2)
When z = 1. we have
In this special cast the infinite series on the right hand side of eq. (14) can be summed up
exactly to give
G(x,y,l) = x2(1
--x2)i1
[_),/(I
-x2)]2’7
(17)
= x2J,2(l
-.x2)/1(1 -x2)2 -y2]
which agrees with (8). Eq. (14) reduces to (10) when x = y = 1. Similarly we write
(18)
where h,,!,] generates polygons with bottom width m and height n. We find
K. Y. LIN
h ),I 1
h ,,I 2
=
y2 (.Y2Z)”
=
y4_y2t~l
11
(19)
Za+h+2c
c
a+h+c=m
h rn,l =
y2r~X2m
c
ZaI+h,+..+(rl-l)~a,l_l+hn-l)+n~
cl, f.. +c=I?I
Y !?,,,n t”‘. ’
(y2z)“x2(1
=
,p,= 1
-x*tz’>/Q,:
where
Qt, =
$1 -AZk)
Q,’ = (1 - x2r)i;o (XW q,,+,(z)/4,,(z) 4,(z)
The generating function g,n can be derived also by different methods. Consider first
the method of Privman and Svrakic.” We define
gn = G,l
(x’z)‘))‘y*
(20)
and rewrite (6) in the form
G !I+7 - 3-G,,+, + G,, = G,,+* y2z”+ *
(21)
where
G, = 1
G, = (1 -y*zj-1
G,
=
(1 +y*z)/(l -y*z)(j -y2?)
Gb,r,z)
=
y2 ,F=,
G,,, (x*z)”
We define
P(t) = : G
k=l
k
tk-’
(22)
It follows from (21) that
P(r) = ,f, tk- 1 [~*z~+'G~+~ + 2Gk+l - Gk+2 I
= ziy/t)* [
- [P(r) -
2,
(zt)‘-’
Gr
-
G1 - tG, 1 t*
G,
-
ztG2]
+ 2[P(t) -
123)
G, l/t
PEKIMtTEK AND ARE.4 GENERATING F’UNCTlOh 01 PJ’KAMID POLYGONS
17
Eq. (23) can be rearranged in the form
P ( r ) = :~.?(l -~~P(f:)+Gr(l -2r-zj.‘)(l -T)-~ +G,t(l ->.*z2)(1 -T)-~ ( 2 4 )
= a(f) + b(riP(tz)
where
a(t) = (1 - t)-’
b ( t ) = ‘$/(l - r)2
The solution of (24) can be obtained by iteration and we get
=
F [(l’2z)n
(1 - rz”)iktO (1 - tzy 1
n=O
which agrees with (18) and (19).
The recursion relation (6) can also be solved by the method of Brak and Guttmann3 as
follows. We try
St, = Pn ,;=,
Y,,’
CT) z”‘”
(26)
where r. = 1. Substituting (26) into (6), we obtain
p2 - 2&p +x422
nEI 2”“’
+
[Y,,’
(27)
@2$J” - 2x2zpzT)I +x42*> - y2p2z2”’ Y,,,_* I = 0 .
We choose p such that
p2 - 2.&p +x422 = (p -x22)2 = 0
yn, (pQ2”’ - 2x2zpzm + x422) -y*p2z2n1Y”*_1
(28)
= 0
(29)
It follows from (28) and (29) that p = x*z and
Y,,’
Y
/r,,7_1 =
II
y2z2”’
(jjp)2”f
“7
= ))2rn
z
/( 1 - z”’ )*
‘n(‘n+l) /
z’?l(“‘+l)
k=l
/ii
Af7I (x2z - pz” )2
(l-z’;)2
(30)
-.
K. Y. LIN
13
Since eq. (28) has a double root, we return to (6) and try a solution of the form
$,I ) = n gJ1’ ) + tn
(3 1)
where
+ t,1+2) - 2x2,‘@;; + t,,+l > + x4z2tn = 0 .
( 1 - yZz”+2)(2g1;1+3
(32)
Substituting
t ,I
=
ST
(.u2 z)”
tfr=O
r
“’
2”‘”
1’,,1
(33)
with v0 = 1 into (32), we get
F z”“’ [r,,I vnl ( 1 - z”‘)2 -y2z2”’ rm_l unl_l + 2rm (z”’ - l)] = 0
,I* = 1
.
(34)
which implies
r,,! u,),
(1 - zm >2 - y2 z2?‘* r,rl_l vln = 2rm (1 - zm )
(35)
Substituting (30) into (35). we obtain (m > 0)
u ),I
-1’
,,I - 1
=
2/(1 - 2”‘)
.
(36)
-z”‘)-1
(37)
The solution of (36) is (m > 0)
1’
1
=
?,I
+2
171
2
(1
k=l
The general solution of (6) is
g 112
=
A, gi’ +A, g;;’
(38)
where the coefficients A 1 and A, are determined by the boundary conditions
A,
gi’)
A
r g(,‘)
+ A, gh2’ = g, = y2
+ A2 g:‘)
(39)
= g, = x2y2z/(l - y2z)
It is not clear in its present form that eqs. (26) (38), (39) and eqs. (18). (19) represent
the same result. This fact was noticed by Brak and Guttmann3 in their work on staircase
.
14
PEKIMETI;K AND AKk.A GLNERATING I,UNC’TION 01 PYRAMID POLYGONS
polygon. The perimeter and area generating function was first obtained by Polya” and then
rederived by Brak and Guttmann’ using a different method. Brak and Guttmann noticed
that the form of their result is very different to that given by Polya’s result and they were
unable to demonstate explicitly the equivalence of their expression to Polya’s result. They
also pointed out that the Polya’s result contains the more natural terms which generate areas
under zig-zag paths. We are facing the same situation here. The expression (18) is more
natural in the sense that each term generates polygons with given bottom width and height.
The expression (38) is very complicated and we are unable to prove explicitly the
equivalence of these two results.
III. SUMMARY
.
The problem of pyramid polygon was suggested by Temperley” as a two-dimensional
model of the growth of a crystal on a plane substrate. He derived the one-variable generating function for the number of polygons with fixed size. In this paper we generalize his
result to the three-variable generating function for the number of polygons with fixed size,
height, and width. We derive our result by three different methods.3*15t17 The graphical
method of Temperley is the simplest.
ACKNOWLEDGMENT
The research is supported by the National Science Council of ROC under grant
NSCBO-0208-M007-43.
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