def overlap(bigger, smaller):
ans = 0
skew = Partition.shifted_shape(bigger, smaller)
for (i, j) in skew:
if (i + 1, j) in skew:
ans += 1
return ans
def gq_pieri(mu, p):
ans = {}
shape = Partition.shifted_shape(mu)
corners = {(i, j) for (i, j) in shape if (i + 1, j) not in shape and (i, j + 1) not in shape}
nu = ((mu[0] if mu else 0) + p,) + mu
outer = Partition.shifted_shape(nu, mu)
for a in subsets(corners):
for b in subsets(outer):
c = a + b
if any(i1 < i2 and j1 < j2 for (i1, j1) in c for (i2, j2) in c):
continue
lam = Partition.from_shape(shape | set(c))
if not Partition.is_strict_partition(lam):
continue
if (Partition.shifted_shape(lam) - shape) | set(a) != set(c):
continue
free = len([(i, j) for (i, j) in c if i != j and (i, j - 1) not in c and (i + 1, j) not in c])
if len(c) <= p <= len(c) + free:
diff = p - len(c)
bet = beta**(sum(mu) + p - sum(lam))
coeff = nchoosek(free, diff) * bet * 2**(free - diff)
ans[lam] = ans.get(lam, 0) + coeff
return ans
def cols(bigger, smaller):
ans = Partition.shifted_shape(bigger) - Partition.shifted_shape(smaller)
return (-1)**len({j for (i, j) in ans})