#C28.txt, Dec. 12, 2016, Planning the class paper
Help:=proc(): print(` An(n), PatA1(pi,k), PatA(S,k) `): end:
with(combinat):
#inputs a list of numbers, and outputs the reduction as a permutation of nops(L)
#redu([e,pi,gamma,phi, h])=[ 4, 5 , 2 , 3    ,1]
redu:=proc(L) local L1,T,k,i:
L1:=sort(sort(L,'output'='permutation'),'output'='permutation'):

end:


#The set of permutations of {1...n} defined by Big(i-1) n Small(n-i)
An:=proc(n) local S,i,S1,S2,pi,pi1,pi2:
option remember:
 if n=0 then
    RETURN({[]}):
 fi:

S:={}:

 for i from 1 to n do
   S1:=An(i-1):
   S1:={seq(pi+[(n-i)$(i-1)],pi in S1)}:
   S2:=An(n-i):

   S:=S union {seq(seq([op(pi1),n,op(pi2)],pi1 in S1), pi2 in S2)}:
  od:
   
S:
end:

  #PatA1(pi,k): all the patterns of size k avoided in the permutation pi
  PatA1:=proc(pi,k) local S,n:

   n:=nops(pi):
   S:=choose([seq(i,i=1..n)],k):

    convert(permute(k),set) minus {seq(redu(pi[s]),s in S)}:
  end:

  #PatA(S,k): the set of patterns of length k avoided by ALL the members of S
  PatA:=proc(S,k) local pi:
   `intersect`(seq(PatA1(pi,k),pi in S)):
 end: