COMBINATORICS
PROBLEM SET 4
PATHS, NONINTERSECTING PATHS AND MORE
p'out.* 4.1.consider
thesquare
;"#jl";;ff:ffi'
onthislatticearepathswhichgo
o n l yu p o r r i g h t . D i . a g o n ai sl t h e l i n e t h a t g o e s t h r o u g h t h e p o i n t( s0 , 0 ) , ( 1 , 1 ) , ( 2 , 2 ) , . . . .
We know that the Catalan number C, is the number of up-right paths from (0,0) to (n, n)
which nevergo abovethe diagonal(seeProblem3.12)
1
I
v
-
a
r
a
r
i
l
e---a-+t
?!
'
r
I
-
t
t
r
\
f
t
t-----+
ry'lnelvn'
(a) Usethe paths reflecti,onpri,nci,pleto show that the number of up-right paths from (0, 0) to
(n, n) which cross (: go above) the diagonal is the same as the number of ang upright paths
on the lattice from (0,0) to (n - 1,n i 1). (This is a bijection argument.)
(b) Using (a) and the definition of the binomial coefficientsas numbers of certain up-right
paths, deduceC": H - (3r) Simplify this to get the known explicit expressionfor C".
Problem 4.2. Compute the number of up-right paths on the square lattice from (0,0) to
(n + l,n) which never go above the diagonal. Argue as follows:
(a) Use the paths reflection principle to show that the number of up-right paths from (0,0)
to (n * 1, n) which cross (: go above) the diagonal is the same as the number of all up-right
paths on the lattice from (0,0) to (n - I,n -t 2).
(b) The number of paths we want ir ('1*t) - (?]rt) Simplify this to get the answer.
Problem 4.3. Write KMGLGV matrix X;1 (X;i is the number of paths from ui to o3) for the
following graph:
(a)
(4/
o(r
vl
-_>_
I
ut
q
,.
-r
Y
v.
\
14,
I
4
?t\-/,
t
()
i
'
4y
Compute the determinant of the matrix. What doesit represent?
1
u.
7
COMBINATORICS. PROBLEIVI SET 4. PATHS, NONINTERSECTING
PATHS AND MORE
Problem 4.4. Write KMGLGV matrix X;i (X;3 is the sum of weights of all paths from u; to
T.,;)for the following graph tuzlhweights:
{.t1
-4
LreirL(-+-)
'
v4_
7t,
7/
^?
n ,
.
l ' , '
T a e t r / l L (t Y )
={2
v 2
Computethe deterninant of the matrix. What doesit represent?
v2
by hand:O"rl "": "i
Problem 4.5. Computethe deterrninant
Lc' c"
Cs
l
Problem 4.6. A matrix in which each ijth element dependsonly on i' -t j, is ca'll.eda' Han'
kel matrir. Use KN'IGLGV lemma (: nonintersectingpaths lemma) to compute the following
determinantsof Hankel matrices (C,, is the nth Catalan number):
: I
(b) det[Ci11tJir-, : r
(a) det[C,a1-z1T,i:t
(foralln:1,2,...).
Problem 4.7. Let Xii be a KMGLGV matrix of somegraph;take its squaresubmatrix(formed
by somerows and columns). What does the determinant of this submatrix (called a minor of
the initial matrix) represent?
HoinpwoRx
Problem 4.8 (2). Let /(r) : ao + arr 1 a2r2+. . . be the generatingfunction of a sequence
oi}:., i.e., the sequencesta,rtsas as,asI a1,a6t
{o" . . .}Lo. Considerthe sequence{ tt.
a1* a2,. . . . Showthat ihe generatingfunction of this new sequetce is S.
Problem 4.9. Let {F'"}Po : {0, 1, 1,2,3,5,B,13,. . . } be the Fibonaccinumbers,their generating function is
_
ln=0F ' " :
Show(usinggeneratingfunctions)that
(a) (2) Fs+ 4 +' " + Fn - Fny2- 1;
Fzn-*Fz-+t-l;
(b) (2) Fr-t Fzl. Fqt...l
(.) (2) F' + F3 + Fb + " + Fzn-': Fzn;
7-x-12
COMBINATORICS.PROBLEM SET 4. PATHS, NONINTERSECTING PATHS AND MORE
3
Problem 4.10 (Motzkin numbers(3)). Motzkin paths are the sameas Dyck paths (seeProblen3.7) but can containsteps (0, 1):
A
v, '.1, --- a
"-/\.*
J
n n
*-t/
.A-_-.=-
/ V \
b
n
#
\
b
c---ffi
- the number of Motzkin paths (by definition,paths
.Let Mn be the nth Motzkin number
-t)
and nevergoingbelowthe z-axis)from (0,0) to (n, n). Show
with steps(1,1), (1,0) and (t,
that the Motzkin numbers satisfv the recurrence
n-2
Mn: Mn-t+LMkM,-2-k,
n)
2
A:O
(the argument is very similar to the Catalan recurrence). Find the equation on the generating
function 1 + Dl, Mnr" for NIn, and then find the generatingfunction itself.
Problem 4.ll (2). Show that the nth Motzkin number is the number of differentways of
drawing non-intersectingchords on a circle betweenn points:
)L,=4
Problem 4.12. Write KMGLGV matrix Xu (Xri is the number of paths from z; to r.r,) for the
following graphs ((l ) for eachgraph and determinant):
("t),,
(ql ut
-->{ u fl, :
vl
'
l
,
Y
:
\
.
.'.
V1
t/z
-\
V3
\
llt
l
1
uq
v4
Comoute the determinants of these matrices. Whai do these determinants represent?
COMBINATORICS.PROBLEM SET 4, PATHS,NONINTERSECTINGPATHS AND MORE
4
Problem 4.13. Write KMGLGV matrix X1i (&i is the sum of weightsof all paths from u; to
ui) for the following graphs tuilb ueights ((1) for eachgraph and determinant):
ti u,
ll
,
h,t\+/=1
*f (+-)=l
t\
2l
v2
l,
llz
of(\)= a-r
v4
(f)=
v2
(>o
?/
ue
= B>g' Qq
I
rc)
ut
u.-
ut
'5
:
a?
4
(/,,
vt/
Compute the determinants of these matrices. What do these determinants represent?
Problem 4.14. Compute the following determinantsby hand (C" is the nth Catalan number,
C o : C , : L ,C z : 2 , . . . ) :
I c, c, ctl
(b)
(a) (,1)det I C2 CB C4 |
Lq
c" c" cnl
I
(1)det
I Cr C4 Cb I
lc4 c5 c6 )
C4Cs)
Problem 4.15, Compute the following determinantsby hand (M, is the nth Motzkin number,
s e eP r o b l e m4 . 1 0 ,M o - M t : L , M 2 : 2 ' . . . 1 ,
I Mo u, M, Mtf
(a)
(7)det
lY,:Y; Yr:Y,:l
lM,
M, M. 1
(b)(1)der.
I m) u' m^ |
LM3 M4 M5 l
LM, Mn Ms Ma )
Problem 4.f6 (4). UseKMGLGV lemmato showthat the Hankeldeterminantsdet[M;+i-dT,i:,
(M," is the nth Motzkin number)are equalto onefor all n: L,2,....
Problem 4.17 (3). Let ft > 0. Arguing as in Problems4.1 and 4.2, computethe number of
up-right paths on the lattice from (0,0) to (n -| k, n) which never go above the diagonal:
Problem 4.18 U). Considerpaths on the squarelattice with steps (0, 1) (up), (1,0) (right),
and (-1,0) (left). Let P' be the number of such paths of length n (n : 7,2,. . . ) with no
self-intersections(hint: this meansthat,the left and right steps cannot be neighbors):
Find a recurrencerelation for the P,''s and show that their generating function is
-t
+ \ - P -- ''--" :
^/---r
1+1f
1 2r - 12'
(Suchpaths serveas a simple model for polEmers.)
COMBINATORICS.PROBLEM SET 4. PATHS,NONINTERSECTINGPATHS AND MORE
Suppr-sN{pNraRYPRoBLEMS
Problem 4.lg (3). Showthat (in notation of Problem4.9) F& + Fr2+ "'+
F: -- FnFn+t.
Problem 4.2O (5). Let a", be the number of ways in which the recta,ngle3 x 2n can be tiled
bvlx2dominoes:
Write a recurrencefor a,", and then find the generatingfunction I +Dp.anr".
Problem 4,2L (3). Use KMGLGV Iemma (- nonintersectingpaths lemma) to computethe
followingdeterminant:deLlCilli,t:r: n + 7 (for ail n:1,2,...).
P r o b l e m 4 , 2 2 .L e t T : I , 2 , . . . , l e t 0 ( t l 1( " ' < o , "a n d0 ( b r < " ' l b n . S h o w t h a t t h e
determinant
*, [(' I,ut =,
",'')]",,,
is nonnegative.
Here (as usual) we use the agreement (f) : O for k < 0 or & > N. For example,the above
determinant of order 3 has the form
det
lT+b1-a!\
\ br-or ,l
lT+br-a2\
\ br-az )
-43\
/?+b L
\ br-os /
lT+b2-aL\
\ bz-or I
lT+bz-a2\
\ bz-az J
1Ttb2-a3\
\ bz-as /
(T+}3-q\
\ bg-ar
(T+bs-a2\
\ h,-az )
-43\
tr?+63
\ b3-43 /
/
Hint. Use the KMGLGV lemma for the squa,relattice:
e7
A.
ta)
Iz
L,