3320.sagews

#THIS is the code to create the list of hess permutations and to
#list the neighborhoods of the fixed points.

#Enter in the length of the permutations you want to consider
n=5;
#you can enter in any Hessenberg function below
h = [4,5,5,5,5];
print "h =" ,h

perm = permutations(n); #this just lists the permutations of length n

x1=var('x1');
x2=var('x2');
x3=var('x3');
x4=var('x4');
x5=var('x5');
t=var('t');
a=var('a');
b=var('b');
c=var('c');
d=var('d');
e=var('e');
f=var('f');
R=QQ[x1,x2,x3,x4,x5,t,a,b,c,d,e,f]; # this defines the ring 
    


       
######### this gives the permutation representation of the cell X^o_w 
Schubideallist=[];
###################


inversepermlist=[];
for pindex in range (factorial(n)):
    pin=[];
    p = perm[pindex];
    for index2 in range(n):
        for index1 in range(n):
            if p[index1]== index2+1:
                pin.append(index1+1)       
    inversepermlist.append(pin)
#print inversepermlist;


# this loop picks out which permutations are fixed points in the Hessenberg variety for 
# the given h function



#print "Fixed points in Hessenberg variety for h =", h;

# this loop will only work for n=5 you will have to add more if's for n>5



hessperm=[];

for index1 in range (factorial(n)):
    append=1;
    for index2 in range(n-1):
        if inversepermlist[index1][index2] > h[inversepermlist[index1][index2+1]-1]:
            append=0;
    if append==1:
        hessperm.append(perm[index1])

# number of permutations satisfying hessenberg condition
#print "Number of fixed points:",len(hessperm); 

#for index in range (len(hessperm)):
#    print hessperm[index];
    
#this loop generates the list of inverse permutations to the hessenberg allowable ones
inversehessperm=[];
for index in range (len(hessperm)):
    inversehessperm.append(hessperm[index].inverse())
 
# Now let's try and find the inversions

#print "Inversions:"
for index in range(len(hessperm)):
    #print hessperm[index]
    inversionlist=[];
    for index1 in range (n):
        for index2 in range (index1+1,n):
            if inversehessperm[index][index1] >= inversehessperm[index][index2]:
                inversionlist.append((inversehessperm[index][index2],inversehessperm[index][index1]));
    inversionlist;
    



print "Permutations | Permutation inverse | Allowable inversions";
for index in range(len(hessperm)):
    perminversions =[]
    perminv=[]
    perminverse=hessperm[index].inverse()
    for index1 in range(n):
        for index2 in range(index1+1,n):
            if perminverse[index1] > perminverse[index2]:
                #print perminverse[index1], perminverse[index2], h[perminverse[index2]-1], perminverse[index1]
                if h[perminverse[index2]-1]>=perminverse[index1]:
                    perminversions.append([perminverse[index1],perminverse[index2]])
    print hessperm[index], " | " , hessperm[index].inverse() , " | " , perminversions ;





    # We can list the dimension pairs and non-dimension pairs as well
#print "h =", h;

#print "Permutations| Permutation inverse";

#for index in range(len(hessperm)):
#    print hessperm[index], "|", hessperm[index].inverse() ;

h = [4, 5, 5, 5, 5] Permutations | Permutation inverse | Allowable inversions [1, 2, 3, 4, 5] | [1, 2, 3, 4, 5] | [] [1, 2, 3, 5, 4] | [1, 2, 3, 5, 4] | [[5, 4]] [1, 2, 4, 3, 5] | [1, 2, 4, 3, 5] | [[4, 3]] [1, 2, 4, 5, 3] | [1, 2, 5, 3, 4] | [[5, 3], [5, 4]] [1, 2, 5, 3, 4] | [1, 2, 4, 5, 3] | [[4, 3], [5, 3]] [1, 2, 5, 4, 3] | [1, 2, 5, 4, 3] | [[5, 4], [5, 3], [4, 3]] [1, 3, 2, 4, 5] | [1, 3, 2, 4, 5] | [[3, 2]] [1, 3, 2, 5, 4] | [1, 3, 2, 5, 4] | [[3, 2], [5, 4]] [1, 3, 4, 2, 5] | [1, 4, 2, 3, 5] | [[4, 2], [4, 3]] [1, 3, 4, 5, 2] | [1, 5, 2, 3, 4] | [[5, 2], [5, 3], [5, 4]] [1, 3, 5, 2, 4] | [1, 4, 2, 5, 3] | [[4, 2], [4, 3], [5, 3]] [1, 3, 5, 4, 2] | [1, 5, 2, 4, 3] | [[5, 2], [5, 4], [5, 3], [4, 3]] [1, 4, 2, 3, 5] | [1, 3, 4, 2, 5] | [[3, 2], [4, 2]] [1, 4, 2, 5, 3] | [1, 3, 5, 2, 4] | [[3, 2], [5, 2], [5, 4]] [1, 4, 3, 2, 5] | [1, 4, 3, 2, 5] | [[4, 3], [4, 2], [3, 2]] [1, 4, 3, 5, 2] | [1, 5, 3, 2, 4] | [[5, 3], [5, 2], [5, 4], [3, 2]] [1, 4, 5, 2, 3] | [1, 4, 5, 2, 3] | [[4, 2], [4, 3], [5, 2], [5, 3]] [1, 4, 5, 3, 2] | [1, 5, 4, 2, 3] | [[5, 4], [5, 2], [5, 3], [4, 2], [4, 3]] [1, 5, 2, 3, 4] | [1, 3, 4, 5, 2] | [[3, 2], [4, 2], [5, 2]] [1, 5, 2, 4, 3] | [1, 3, 5, 4, 2] | [[3, 2], [5, 4], [5, 2], [4, 2]] [1, 5, 3, 2, 4] | [1, 4, 3, 5, 2] | [[4, 3], [4, 2], [3, 2], [5, 2]] [1, 5, 3, 4, 2] | [1, 5, 3, 4, 2] | [[5, 3], [5, 4], [5, 2], [3, 2], [4, 2]] [1, 5, 4, 2, 3] | [1, 4, 5, 3, 2] | [[4, 3], [4, 2], [5, 3], [5, 2], [3, 2]] [1, 5, 4, 3, 2] | [1, 5, 4, 3, 2] | [[5, 4], [5, 3], [5, 2], [4, 3], [4, 2], [3, 2]] [2, 1, 3, 4, 5] | [2, 1, 3, 4, 5] | [[2, 1]] [2, 1, 3, 5, 4] | [2, 1, 3, 5, 4] | [[2, 1], [5, 4]] [2, 1, 4, 3, 5] | [2, 1, 4, 3, 5] | [[2, 1], [4, 3]] [2, 1, 4, 5, 3] | [2, 1, 5, 3, 4] | [[2, 1], [5, 3], [5, 4]] [2, 1, 5, 3, 4] | [2, 1, 4, 5, 3] | [[2, 1], [4, 3], [5, 3]] [2, 1, 5, 4, 3] | [2, 1, 5, 4, 3] | [[2, 1], [5, 4], [5, 3], [4, 3]] [2, 3, 1, 4, 5] | [3, 1, 2, 4, 5] | [[3, 1], [3, 2]] [2, 3, 1, 5, 4] | [3, 1, 2, 5, 4] | [[3, 1], [3, 2], [5, 4]] [2, 3, 4, 1, 5] | [4, 1, 2, 3, 5] | [[4, 1], [4, 2], [4, 3]] [2, 3, 5, 1, 4] | [4, 1, 2, 5, 3] | [[4, 1], [4, 2], [4, 3], [5, 3]] [2, 4, 1, 3, 5] | [3, 1, 4, 2, 5] | [[3, 1], [3, 2], [4, 2]] [2, 4, 1, 5, 3] | [3, 1, 5, 2, 4] | [[3, 1], [3, 2], [5, 2], [5, 4]] [2, 4, 3, 1, 5] | [4, 1, 3, 2, 5] | [[4, 1], [4, 3], [4, 2], [3, 2]] [2, 4, 5, 1, 3] | [4, 1, 5, 2, 3] | [[4, 1], [4, 2], [4, 3], [5, 2], [5, 3]] [2, 5, 1, 3, 4] | [3, 1, 4, 5, 2] | [[3, 1], [3, 2], [4, 2], [5, 2]] [2, 5, 1, 4, 3] | [3, 1, 5, 4, 2] | [[3, 1], [3, 2], [5, 4], [5, 2], [4, 2]] [2, 5, 3, 1, 4] | [4, 1, 3, 5, 2] | [[4, 1], [4, 3], [4, 2], [3, 2], [5, 2]] [2, 5, 4, 1, 3] | [4, 1, 5, 3, 2] | [[4, 1], [4, 3], [4, 2], [5, 3], [5, 2], [3, 2]] [3, 1, 2, 4, 5] | [2, 3, 1, 4, 5] | [[2, 1], [3, 1]] [3, 1, 2, 5, 4] | [2, 3, 1, 5, 4] | [[2, 1], [3, 1], [5, 4]] [3, 1, 4, 2, 5] | [2, 4, 1, 3, 5] | [[2, 1], [4, 1], [4, 3]] [3, 1, 5, 2, 4] | [2, 4, 1, 5, 3] | [[2, 1], [4, 1], [4, 3], [5, 3]] [3, 2, 1, 4, 5] | [3, 2, 1, 4, 5] | [[3, 2], [3, 1], [2, 1]] [3, 2, 1, 5, 4] | [3, 2, 1, 5, 4] | [[3, 2], [3, 1], [2, 1], [5, 4]] [3, 2, 4, 1, 5] | [4, 2, 1, 3, 5] | [[4, 2], [4, 1], [4, 3], [2, 1]] [3, 2, 4, 5, 1] | [5, 2, 1, 3, 4] | [[5, 2], [5, 3], [5, 4], [2, 1]] [3, 2, 5, 1, 4] | [4, 2, 1, 5, 3] | [[4, 2], [4, 1], [4, 3], [2, 1], [5, 3]] [3, 2, 5, 4, 1] | [5, 2, 1, 4, 3] | [[5, 2], [5, 4], [5, 3], [2, 1], [4, 3]] [3, 4, 1, 2, 5] | [3, 4, 1, 2, 5] | [[3, 1], [3, 2], [4, 1], [4, 2]] [3, 4, 2, 1, 5] | [4, 3, 1, 2, 5] | [[4, 3], [4, 1], [4, 2], [3, 1], [3, 2]] [3, 4, 2, 5, 1] | [5, 3, 1, 2, 4] | [[5, 3], [5, 2], [5, 4], [3, 1], [3, 2]] [3, 4, 5, 2, 1] | [5, 4, 1, 2, 3] | [[5, 4], [5, 2], [5, 3], [4, 1], [4, 2], [4, 3]] [3, 5, 1, 2, 4] | [3, 4, 1, 5, 2] | [[3, 1], [3, 2], [4, 1], [4, 2], [5, 2]] [3, 5, 2, 1, 4] | [4, 3, 1, 5, 2] | [[4, 3], [4, 1], [4, 2], [3, 1], [3, 2], [5, 2]] [3, 5, 2, 4, 1] | [5, 3, 1, 4, 2] | [[5, 3], [5, 4], [5, 2], [3, 1], [3, 2], [4, 2]] [3, 5, 4, 2, 1] | [5, 4, 1, 3, 2] | [[5, 4], [5, 3], [5, 2], [4, 1], [4, 3], [4, 2], [3, 2]] [4, 1, 2, 3, 5] | [2, 3, 4, 1, 5] | [[2, 1], [3, 1], [4, 1]] [4, 1, 3, 2, 5] | [2, 4, 3, 1, 5] | [[2, 1], [4, 3], [4, 1], [3, 1]] [4, 1, 3, 5, 2] | [2, 5, 3, 1, 4] | [[2, 1], [5, 3], [5, 4], [3, 1]] [4, 1, 5, 3, 2] | [2, 5, 4, 1, 3] | [[2, 1], [5, 4], [5, 3], [4, 1], [4, 3]] [4, 2, 1, 3, 5] | [3, 2, 4, 1, 5] | [[3, 2], [3, 1], [2, 1], [4, 1]] [4, 2, 3, 1, 5] | [4, 2, 3, 1, 5] | [[4, 2], [4, 3], [4, 1], [2, 1], [3, 1]] [4, 2, 3, 5, 1] | [5, 2, 3, 1, 4] | [[5, 2], [5, 3], [5, 4], [2, 1], [3, 1]] [4, 2, 5, 3, 1] | [5, 2, 4, 1, 3] | [[5, 2], [5, 4], [5, 3], [2, 1], [4, 1], [4, 3]] [4, 3, 1, 2, 5] | [3, 4, 2, 1, 5] | [[3, 2], [3, 1], [4, 2], [4, 1], [2, 1]] [4, 3, 1, 5, 2] | [3, 5, 2, 1, 4] | [[3, 2], [3, 1], [5, 2], [5, 4], [2, 1]] [4, 3, 2, 1, 5] | [4, 3, 2, 1, 5] | [[4, 3], [4, 2], [4, 1], [3, 2], [3, 1], [2, 1]] [4, 3, 2, 5, 1] | [5, 3, 2, 1, 4] | [[5, 3], [5, 2], [5, 4], [3, 2], [3, 1], [2, 1]] [4, 3, 5, 1, 2] | [4, 5, 2, 1, 3] | [[4, 2], [4, 1], [4, 3], [5, 2], [5, 3], [2, 1]] [4, 3, 5, 2, 1] | [5, 4, 2, 1, 3] | [[5, 4], [5, 2], [5, 3], [4, 2], [4, 1], [4, 3], [2, 1]] [4, 5, 1, 3, 2] | [3, 5, 4, 1, 2] | [[3, 1], [3, 2], [5, 4], [5, 2], [4, 1], [4, 2]] [4, 5, 2, 3, 1] | [5, 3, 4, 1, 2] | [[5, 3], [5, 4], [5, 2], [3, 1], [3, 2], [4, 1], [4, 2]] [4, 5, 3, 1, 2] | [4, 5, 3, 1, 2] | [[4, 3], [4, 1], [4, 2], [5, 3], [5, 2], [3, 1], [3, 2]] [4, 5, 3, 2, 1] | [5, 4, 3, 1, 2] | [[5, 4], [5, 3], [5, 2], [4, 3], [4, 1], [4, 2], [3, 1], [3, 2]] [5, 1, 2, 4, 3] | [2, 3, 5, 4, 1] | [[2, 1], [3, 1], [5, 4], [4, 1]] [5, 1, 3, 4, 2] | [2, 5, 3, 4, 1] | [[2, 1], [5, 3], [5, 4], [3, 1], [4, 1]] [5, 1, 4, 2, 3] | [2, 4, 5, 3, 1] | [[2, 1], [4, 3], [4, 1], [5, 3], [3, 1]] [5, 1, 4, 3, 2] | [2, 5, 4, 3, 1] | [[2, 1], [5, 4], [5, 3], [4, 3], [4, 1], [3, 1]] [5, 2, 1, 4, 3] | [3, 2, 5, 4, 1] | [[3, 2], [3, 1], [2, 1], [5, 4], [4, 1]] [5, 2, 3, 4, 1] | [5, 2, 3, 4, 1] | [[5, 2], [5, 3], [5, 4], [2, 1], [3, 1], [4, 1]] [5, 2, 4, 1, 3] | [4, 2, 5, 3, 1] | [[4, 2], [4, 3], [4, 1], [2, 1], [5, 3], [3, 1]] [5, 2, 4, 3, 1] | [5, 2, 4, 3, 1] | [[5, 2], [5, 4], [5, 3], [2, 1], [4, 3], [4, 1], [3, 1]] [5, 3, 1, 4, 2] | [3, 5, 2, 4, 1] | [[3, 2], [3, 1], [5, 2], [5, 4], [2, 1], [4, 1]] [5, 3, 2, 4, 1] | [5, 3, 2, 4, 1] | [[5, 3], [5, 2], [5, 4], [3, 2], [3, 1], [2, 1], [4, 1]] [5, 3, 4, 1, 2] | [4, 5, 2, 3, 1] | [[4, 2], [4, 3], [4, 1], [5, 2], [5, 3], [2, 1], [3, 1]] [5, 3, 4, 2, 1] | [5, 4, 2, 3, 1] | [[5, 4], [5, 2], [5, 3], [4, 2], [4, 3], [4, 1], [2, 1], [3, 1]] [5, 4, 1, 2, 3] | [3, 4, 5, 2, 1] | [[3, 2], [3, 1], [4, 2], [4, 1], [5, 2], [2, 1]] [5, 4, 1, 3, 2] | [3, 5, 4, 2, 1] | [[3, 2], [3, 1], [5, 4], [5, 2], [4, 2], [4, 1], [2, 1]] [5, 4, 2, 1, 3] | [4, 3, 5, 2, 1] | [[4, 3], [4, 2], [4, 1], [3, 2], [3, 1], [5, 2], [2, 1]] [5, 4, 2, 3, 1] | [5, 3, 4, 2, 1] | [[5, 3], [5, 4], [5, 2], [3, 2], [3, 1], [4, 2], [4, 1], [2, 1]] [5, 4, 3, 1, 2] | [4, 5, 3, 2, 1] | [[4, 3], [4, 2], [4, 1], [5, 3], [5, 2], [3, 2], [3, 1], [2, 1]] [5, 4, 3, 2, 1] | [5, 4, 3, 2, 1] | [[5, 4], [5, 3], [5, 2], [4, 3], [4, 2], [4, 1], [3, 2], [3, 1], [2, 1]]