LAGRANGE
COMBINATORICS
PROBLEM SET 3
INVERSION AND CATALAN NUMBERS
SE MIN A R P R oB LE MS
Problem 3.1. Solve the following ordinary differential equations using generating functions:
a) f'(r) - o
b) f'(") - f (r)
c) (r - r)f'(r) + /(r) - s
Problem 3.2. Compute the following sums using generatingfunctions:
O
(a)
\ - - /f
L
n:0
C
l
-
A, (b)
, - , \2r,
L
n:0
'
o
o
n
n' t'Ln
: (c)
\ - / \-'
L
2n,
nl
n:0
Problem 3.3. Apply the Lagrange inversion to compute the inverseof the seriesf (") - r.
Problem 3.4. Considerthe degreefiue equationar' _ r+I - 0. Using the Lagrangeinversion,
find the root r(a) of this equation for which r(0) - 1.
Problem 3.5. Let Cn, fr ) 0 (by agreement,Cs - 1) be the number of expressionscontaining
n pairs of parentheseswhich are correctly matched. (Here and below in similar problems we
illustrate the objects for n - 3.)
((0)) 000
(0)0 0(0) (00)
(a) Find the recurrencerelation f,or Cn;
(b) From it, find the equation for the generatingfunctron C(r) '- DLoCntr";
(c) Using the Lagrange inversion, find an explicit formula for Cn's.
The numbersC" (n:0, I,2,... ) are calledthe Catalannumbers.
Problem 3.6. In Problem 3.5, find the formula for Cn's by solving the equation for the generating function C (*) and expanding the solution as a power series.
Problem 3.7. A Dyck path of length 2n is a path in the (*,a) plane from (0,0) to (0,2n) with
steps (1,1) and (1, -1) which never passesbelow the r-axis. Show that the number of Dyck
paths of length 2n is Cn.
^A
Problem 3.8. Show that the number of binary trees with n verticesis C,.
/\
)
Problem 3.9. Showthat the nu mber of plane trees with n * I vertices is Cn.
1
2
COMBINATORICS. PROBLEM SET 3. LAGRANGE INVERSION AND CATALAN NUMBERS
I
l
d
I
I
,
'
l
/A\
\ /
{b \
\ { ry ?
t
l
/[\
Problem 3.10. Let an be the numberof all partitionsof the set {I,2,. . .,2n- I,2n} into pairs;
the order of pairs and the order of elementsin the pairs is inessential.E.g., f.orn - 3 we have
the following partitions:
{1,2}u {3,4}
{ 1 ,3 } u { 2 , 4 }
{ 1 , 4 } u { 2 ,3 } .
1, 2, . . . in a simple form (at least as a product of some numbers).
Find a, for alI n:
Problem 3.11. Aty partition into pairs in Problem 3.10 can be representedin a graphicalway
by putting the numbersI,2,...,2n - 1,2n rn a horizontalline and connectingthem abovethis
line. Show that the number of partitions as in Problem 3.10 in which there are no intersections
of the linesis c"
,,^,
/-.
a^]
(
no\
^
r
n
\
,
/
n
\
n
/rA\
n^o
Problem 3.L2. Show that the number of lattice paths from (0,0) to (n,n) with steps(0,1) or
(1,0) which do not go above the diagonal is Cn.
Problem 3.13. Show that the number of plane binary trees with 2n * 1 vertices (i.e.,, n I I
endpoints)is C".
dfi
HonrpwoRK
Problem 3.L4. Solve the following ordinary differential equations using generatingfunctions:
a) Q1 Trnl@)- 0 for somek - I,2,... (the kth derivative)
b) (1) f'(") - of (r) (* # 0 is a constant)
c ) ( / ) ( L- , ) f ' ( r ) + / ( r ) - o
d) (1) f'(") - 2rf (r) - g
e) (1) f'(r) - 4rtf(") - o
f ) ( 1 )f " ( r ) - f ( r ) - o
Problem 3.15. Compute the following sums using generatingfunctions:
( o () 1 ) i Y # 9 ;
n
:
o
n
( b )( 1 ) i t # ;
l
\
/
\
r
t
:
( c () 1 )i @ 3 + 4 n22")+ ( z n + ts) "
o
'
:
o
n
l
'
Problem 3.16. Find the radius of convergence
of the series:(u) ( t) f ("): DL, notrn (a e IR
is a givenconstant;the radiusmay dependon a); (b) (/) 1 * ra * 18 * rL2 * 176+...; (c) (t)
r + 5 r 2 * 2 5 1 4* 1 2 5 1 6+ . . . ; ( d ) ( / ) 1 * 2 b 2 * 4 b a * 6 t r 6 + . . . ; ( " ) ( / ) D n o r n !
COMBINATORICS. PROBLE1VISET 3. LAGRANGE INVERSION AND CATALAN NUMBERS
3
Problem 3.L7 (2). Let F(r) b" a unique power serieswith rational coefficientssuch that for
all n ) 0, the coefficientof rn in (F(r))'*t i. 1. Show that then F(r) - fi-".
Problem 3.18 (3). Using the Lagrangeinversion,find the inverseto the series/(r) : re-r t
that is, find a seriesg(r) such that g@)e-s(*): tr.
Problem 3.19 (3). Considerthe equationrs - r - a,- 0. Using the Lagrangeinversion,show
that its root r(a) for which r(0) : 1, has the form
,(a) if1-)gA
fu\k)4k+1'
Problem 3.2O (3). Show that the number of triangulations of (n + 2)-polygon with n - I
diagonalsis C,.
fr
Problem 3.2L (3). Show that the number of all collectionsof n nonintersectingchordsjoining
2n points on the circumferenceof a circle is Cn.
/ \ //z
d--1
F-l
\ /
qr/
*-4l
\\
((-!))
s suchthat
having n "11"s and n
Problem 3.22 (2). Show that the number of sequences
all their partial sums are nonnegative,is Cn.
1- 1- 11-11-1 1- - 1 1 1- 1 - 1 1 11 ( a1
Problem 3.23 (2).Show that the number of integersequences
for all i, is Cn.
111
rr2
113
122
r23
P r o b l e m 3 . 2 4 ( 3 ) . S h o w t h a t t h e n u m b e r o f i n t e g e r S e q u e n c e S 0. a1 n<-. r. w i t h L ( - a i < 2 i
for all n rs Cn.
23
24
14
13
12
o1)...)a,, suchthat &r:0
Problem 3.25 (l). Showthat the numberof integersequences
01a+r l at * 1 for all i is Cn.
000
001
010
011
and
0r2
o1i...i&n-r such that ai I I
Problem 3.26 (/r). Show that the number of integersequenc€s
and all their partial sums are nonnegative,is Cn.
1 ,1
1,0
1 ,- 1
0 ,1
0,0
Problem 3.27 (3). Showthat the numberof permutationsof {1 ,2,...,2n - 1,2n} suchthat
( 1 ) t h e e l e m e n t s1 , 3 , . . . , 2 n - 1 a r e i n c r e a s i n g ;
(2) the elements2,4,. . . ,2n are also increasing;
(3) eachelement2i - 1 is precedingthe element2z (i - L,. . . ,n),
is Cn.
123456
123546
132456
132564
735246
COMBINATORICS. PROBLEM SET 3. LAGRANGE INVERSION AND CATALAN NUMBERS
Problem 3.28 (2). Show that the number of all ways to stack coins in the plane such that the
bottom row has n consecutivecoins is Cn
wo
Problem 3.29 (D. Considerthe tableauxof the form 2 x n with entriesI,2,...,,2n suchthat
the entriesare increasingalong both rows and also along all the columns. Show that the number
of such tableaux rs Cn
' 1
L 2 3
2 4
I 2 5
1 3 4
1 3 5
4 5 6
3 5 6
3 4 6
2 5 6
2 4 6
SupplnMENTARYPRoBLEMS
Problem 3.30 (l). Use the Lagrangeinversionto show that the series
, \ - S / l rI _ + 1 \ 4 _ NI r n _ r
)_
n'L /N*+1'
-:o\
r\a )
where A > 1 and I is a positive integer, satisfiesthe equation trN - Ar * 1 : 0.
Problem3.31(3).Aparti,t,ion)isanintegerSequence.trr}\z>..
decreasingand has a finite number of elements.Partitions are representedby Young diagrams.
A Young dr,agramis a collection of boxes such that the the first row has )1 boxes, the second
row has )2 boxes,etc. For example,the Young diagram of the partition ) : (6,4, 1) is
Showthat the number of Young diagramswhich are inside the staircaseshapedYoung diagram
( n - I , n - 2 , . . . , 1 ) i s C n ( n o t et h a t w e c o u n tt h e e m p t y d i a g r a m ) .
l--l
g
[T_l
E E -
Problem 3.32 (/,'). Show that the number of Dyck paths from (0, 0) to (0, 4n) in which every
descenthas length 2, is C,.
-A
/ \ / \
P r o b l e m 3 . 3 3 ( l ) . S h o wt h a t t h e n u m b e ro f t i l i n g o f t h e s t a i r c a sseh a p e( r , r - T , n - z , .
with n rectanglesis C,.
..,2,L)