Intro to research based coding in Sage.sagews

Intro to research based coding in Sage

Create the set (list?) of 3x3 alternating sign matrices

A3 = AlternatingSignMatrices(3)

A3

Use a for loop to display all the 3x3 alternating sign matrices

for a in A3:
    print a

#tab complete and define 4x4 alternating sign matrices
A4 = Altern

unrank(28) gives us the 28th (starting at 0) alternating sign matrix in A4

asm = A4.unrank(28); asm

asm.height_function()

In the below block, put your cursor at the end and hit tab to see what alternating sign matrix methods start with 'to'

asm.to

asm.to_fully_packed_loop()

asm

asm.gyration()

asm.gyration??

animate(a2.to_fully_packed_loop() for a2 in asm.gyration_orbit())

Construct the lattice of 4x4 alternating sign matrices

l4 = A4.lattice()

l4.cardinality()

l4.plot(label_elements = False)

j4 = l4.join_irreducibles_poset()

plot(j4)

Construct the same poset using the tetrahedral poset code, then test it is isomorphic to our previous construction

a4 = posets.TetrahedralPoset(4, 'green','yellow','blue','orange')
a4.is_isomorphic(j4)

Construct the TSSCPP tetrahedral poset

t4 = posets.TetrahedralPoset(4, 'green','yellow','red','orange')
t4.plot()

oit4 = t4.order_ideals_lattice()
oit4.cardinality()

ll = [len(orb) for orb in j4.rowmotion_orbits()]

sorted(ll)

A4.gyration_orbit_sizes()

def row_gyr_orb_size(n):
    A = AlternatingSignMatrices(n)
    lat = A.lattice()
    j = lat.join_irreducibles_poset()
    rowlist = [len(orb) for orb in j.rowmotion_orbits()]
    rowlist = sorted(rowlist)
    gyrlist = A.gyration_orbit_sizes()
    gyrlist.sort()
    print gyrlist
    return rowlist==gyrlist

for i in [1..5]:
    row_gyr_orb_size(i)

myasm = AlternatingSignMatrix([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 1, -1, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0],
[0, 0, 1, -1, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 1],
[1, 0, -1, 0, 0, 0, 1, 0, 0, -1, 1, 0, 0, 0, 0, -1, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 1, 0, -1, 0, 1, -1, 1, -1, 1, -1, 1, 0, 0, 0, 0],
[0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, -1, 1, -1, 1, 0, 0, -1, 1, 0],
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 1, -1, 0, 0, 0, 1, 0, 0, 0],
[0, 1, 0, 0, -1, 0, 0, 1, 0, -1, 1, -1, 1, 0, 0, 0, -1, 1, 0, 0],
[0, 0, 1, -1, 1, 0, -1, 0, 0, 1, -1, 1, -1, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]])

ld = myasm.to_dyck_word(algorithm = 'last_diagonal')

print(ld)

lp = myasm.to_dyck_word??

type(lp)

myasm.to_monotone_triangle().pp()

myasm.to_fully_packed_loop()

myasm.

print(fpl)

def random_asm(n):
    toprow = [n-i for i in range(n)]
    gt = GelfandTsetlinPatterns(top_row = toprow, strict = True)
    randomgt = gt.random_element()
    A = AlternatingSignMatrices(n)
    return A.from_monotone_triangle(randomgt)
    

random_asm(14)