1. Find the number of paths from (0,0) to (2n, 0) with steps (1, 1), (1, – 1), and (2,0) that never pass below the x-axis. (You can find a sum in terms of Catalan numbers, a generating function, a recurrence, etc.)

1. Find the number of paths from (0,0) to (2n, 0) with steps (1, 1), (1, – 1), and (2,0) that never pass below the x-axis. (You can find a sum in terms of Catalan numbers, a generating function, a recurrence, etc.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Catalan numbers
= # of Dyckk paths
Def
A poth in the plene from Co,0)
to (2n,0) which wres stps
U= (!,1) (NorthEnst) or D=(i,-1) (Samth Earst)
and never goes
below X- axi5
Dyck peth.
Claim:
Cn x # of Dyck path of semilength n
then
Cn=
I,l,2,5, 14,42,132,
(2 proofs
a combinatorial
)
→ vecurrena & G.F.
n=1
X-axis
n=2
or
Note: # uyps
Jowns. =

1. Find the number of paths from (0,0) to (2n, 0) with steps (1, 1), (1, – 1), and (2,0) that
never pass below the x-axis. (You can find a sum in terms of Catalan numbers, a generating
function, a recurrence, etc.)
2. Consider the paths from Problem 1. Show that the number of such paths with no 'peaks',
i.e. no up-step followed immediately by a down-step, is given by the Catalan numbers. (Hint:
Try to construct a bijection with Dyck paths.)

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