Help:=proc(): print(`WhatType(pi), FindBlock(pi,a) `): end:

#WhatType(pi): inputs a permutation and outputs its
#type (a triple (or pair)) of pairs indicating
#whether the first block (block #0) is a singleton (0) increasing
#or decreasing, and similarly for block #1 (the block
#of integers immediately larger than the first block etc.
#for example, WhatType([2,3,5,4,1,6])  should be
#[ [0,1],[1,-1],[-1,0]], and retunrs the "parsing"
#[[2,3],[5,4],[1]]
WhatType:=proc(pi) local i,t,BZ,B1,Bm1,rebecca, wei,s,
s1,sm1:

rebecca:=[]:
wei:=[]:

if nops(pi)=1 then
 RETURN([[0,0]]):
fi:

t:=pi[1]:

if pi[2]=t+1 then
 
 for i from 2 to nops(pi) while pi[i]-pi[i-1]=1 do od:
  
 BZ:=[op(1..i-1,pi)]:
rebecca:=[op(rebecca),[0,1]]:
wei:=[op(wei), BZ]:

elif pi[2]=t-1 then

for i from 2 to nops(pi) while pi[i]-pi[i-1]=-1 do od:
BZ:=[op(1..i-1,pi)]:

rebecca:=[op(rebecca),[0,-1]]:
wei:=[op(wei), BZ]:

else

BZ:=[t] :
  
rebecca:=[op(rebecca),[0,0]]:
wei:=[op(wei), BZ]:
fi:

s:=BZ[nops(BZ)]:

if rebecca[1][2]=0 then
s1:=s+1:
sm1:=s-1:

elif rebecca[1][2]=1 then
s1:=s+1:
sm1:=t-1:

elif rebecca[1][2]=-1 then
s1:=t+1:
sm1:=s-1:
fi:







rebecca, wei,s1,sm1:


end:


#inputs a perm. pi and an integer a finds
# the block that it belogs to and whether 
# it is a sinleton (0), increasing (1) or decreasing (-1)
#For example,FindBlock([2,3,1,4,5,6,7,8],5); should return
#[4,5,6,7,8],1
FindBlock:=proc(pi,a) local i,ty,n,b,e,j:
n:=nops(pi):

for i from 1 to nops(pi) while pi[i]<>a do od:

i:

if (i>1 and abs(pi[i-1]-pi[i])<>1) and 
  (i<n and abs(pi[i+1]-pi[i])<>1) then
   ty:=0:
elif
  (i>1 and abs(pi[i]-pi[i-1])=1) and 
  
  (i<n and abs(pi[i]-pi[i+1])<>1) then
     e:=a:
   
 if pi[i-1]-pi[i]=1 then
   ty:=-1:
   
   for j from i to 2 by -1 while pi[j-1]-pi[j]=1 do od:
       RETURN([op(j..i,pi)],ty):

elif pi[i-1]-pi[i]=-1 then
   ty:=1:
   
   for j from i to 2 by -1 while pi[j-1]-pi[j]=-1 do od:
       RETURN([op(j..i,pi)],ty):


  fi:

fi:

    

end: