#M1.txt: Maple Code for Lecture 1 of Math 454,Combinatorics, Rutgers University, taught by Dr. Z. (Doron Zeilberger)

Help1:=proc():print(`Walks(m,n), NuWalks(m,n) `): end:

#Walks(m,n): inputs two non-negative integers m and n
#and outputs the SET of all walks {E,N} from [0,0] to [m,n]
#all the rearramgemets of E^m N^n [ Added 9/10/20: Correcting a typo noticed by Jehan  Abdul-Jabbar]
Walks:=proc(m,n) local W1,W2,w1,w2:
option remember:

if m<0 or n<0 then
 RETURN({}):
fi:


if m=0 and n=0 then
 RETURN({[]}):
fi:

W1:=Walks(m-1,n):
W2:=Walks(m,n-1):

{seq([op(w1),E],w1 in W1), seq([op(w2),N],w2 in W2)  }:

end:


#NuWalks(m,n): inputs two non-negative integers m and n
#and outputs the SET of all walks {E,N} from [0,0] to [m,n]
#all the rearramgemets of E^nN^m
NuWalks:=proc(m,n) local W1,W2,w1,w2:
option remember:

if m<0 or n<0 then
 RETURN(0):
fi:


if m=0 and n=0 then
 RETURN(1):
fi:

W1:=NuWalks(m-1,n):
W2:=NuWalks(m,n-1):


W1+W2:
end: