def getMat(k):
    vs = [(m, M) for m in range(k) for M in range(m,k)]
    vToInd = {v:i for i,v in enumerate(vs)}
    mat = matrix(QQ, len(vs))
    for i1, (m,M) in enumerate(vs):
        for x1 in range(m, k):
            for x2 in range(m, k):
                if max(x1,x2)<M: continue
                v2 = (min(x1,x2), max(x1,x2))
                mat.add_to_entry(i1, vToInd[v2], 1)
    return mat

def getGraph(k):
    g = DiGraph()
    g.allow_loops(True)
    mat = getMat(k)
    vs = [(m, M) for m in range(k) for M in range(m,k)]
    for i1, v1 in enumerate(vs):
        for i2, v2 in enumerate(vs):
            if mat[i1,i2]>0:
                g.add_edge(v1,v2,mat[i1,i2])
    return g


def f(n, k):
    mat = getMat(k)
    return sum((mat^n)[0])


g = getGraph(3)
show(g.plot(figsize=6))
show(getMat(3))

#a = matrix([[f(n,k) for n in range(1, 10)] for k in range(1, 10)])
#show(a)