#IsGood1(T): Given a potential trapezoid (in the right-angled format)
# that you know that its horiz. ,  vertical 
#and its NW->SE diagonals for the truncated version (where you chopped
#the last row), returns true iff the new one is also OK
#Try:
#IsGood1([[1,2,3],[2,1]]);
IsGood1:=proc(T) local n,k,i:
k:=nops(T):
n:=nops(T[1]):
for i from 1 to nops(T[k]) do
#checking that all NW->SW diagonals are OK
 
 if member(T[k][i],{seq(T[k-j][i+j],j=1..k-1)}) then
   RETURN(false):
 fi:

#Checking that all vertical lines are distinct
if member(T[k][i],{seq(T[j][i],j=1..k-1)}) then
   RETURN(false):
 fi:
od:

true:

end:


#LT(n,k): The SET OF REDUCED n by k LATIN TRAPEZOIDS
#The bottom line is 1,...,n, and each floor above it
#of length n-1,n-2, .., n-k+1 has all DIFFERENT entries
#also VERTICALLY all different also A[i][i],A[i+1][i+1], ... are
#all different
LT:=proc(n,k) local S, Sold,PNF,old1,f,new1:
option remember:

if k=1 then
  RETURN({[[seq(i,i=1..n)]]}):
fi:

Sold:=LT(n,k-1):


PNF:=permute(n,n-k+1):
S:={}:
for old1 in Sold do
  for f in PNF do
    new1:=[op(old1),f]:


   if IsGood1(new1) then
      S:=S union {new1}:
   fi:
  od:
od:
S:
end: