#C15.txt, March 13, 2025
Help15:=proc(n): print(`A(n), IsID(pi),UD(n), UDset(n) `):end:

A:=proc(n)
1/(binomial(2*n,n)*sqrt(n)/4^n)^2;
end:

#IsUD(pi): Is the permutation pi up-down
IsUD:=proc(pi) local n,i:
n:=nops(pi):
for i from 1 to n-1 do

if i mod 2=1 then
if pi[i]>pi[i+1] then
  RETURN(false):
fi:
fi:

if i mod 2=0 then
if pi[i]<pi[i+1] then
  RETURN(false):
fi:
fi:

od:
true:
end:

with(combinat):
#UD(n): the number of up-down permutations of length n the stupid way
UD:=proc(n) local s,S:
S:=permute(n):
coeff(add(IsUD(s), s in S),true,1):
end:

#UDset(n): the set of up-down permutations of length n the stupid way
UDset:=proc(n) local s,S,G:
S:=permute(n):
G:={}:

for s in S do
 if IsUD(s) then
  G:=G union {s}:
 fi:
od:
G:
end: