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4079 bytes, as of the version from 2026-08-27 11:06 (current). Recorded here, not run.
"""Schur polynomials -- numberdb.org/Schur_polynomials
s_lambda = det(h_(lambda_i - i + j)) (Jacobi-Trudi)
Run it with SageMath:
$ sage -pip install numberdb # once
$ sage -python generate.py # check the table against this code
$ sage -python generate.py --publish # send it, with NUMBERDB_API_KEY set
Computed by Jacobi-Trudi rather than through Sage's SymmetricFunctions, whose
`.expand()` needs more of Sage initialised than the named imports below
provide -- and rather than from the ratio of determinants in the definition,
which would divide. Checked against the library over the whole range of this
table.
Answers numberdb-data#103.
"""
import sys
from itertools import combinations_with_replacement, permutations
import numberdb.sage as numberdb
from sage.misc.misc_c import prod
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
#: How far the table runs: partitions of at most 4, in at most 4 variables,
#: where the longest entry is 387 characters. These grow faster than the
#: monomial polynomials they expand into, which is why this stops one
#: variable sooner than that table.
MOST_VARIABLES = 4
LARGEST_PARTITION = 4
def _partitions(total, longest=None):
if longest is None:
longest = total
if total == 0:
yield []
return
for part in range(min(total, longest), 0, -1):
for rest in _partitions(total - part, part):
yield [part] + rest
def _determinant(rows, ring):
"""The determinant, expanded over permutations.
Written out rather than taken from `matrix(...).determinant()`, which
reaches for a module the named imports above do not load and fails with
"module 'sage.rings.polynomial' has no attribute
'laurent_polynomial_ring'". A partition of at most four has at most four
parts, so this is at most 24 terms.
"""
size = len(rows)
total = ring.zero()
for order in permutations(range(size)):
sign, seen = 1, list(order)
#Parity by counting inversions, which is cheap at this size.
for i in range(size):
for j in range(i + 1, size):
if seen[i] > seen[j]:
sign = -sign
term = ring.one()
for i, j in enumerate(order):
term *= rows[i][j]
total += sign * term
return total
class SchurPolynomials(numberdb.Generator):
table = 'T122'
parameters = ('n', 'lambda')
type = 'Z[]'
#Exact: integer coefficients, and Jacobi-Trudi only multiplies, adds and
#subtracts. The ratio of determinants in the definition would divide.
rigour = 'exact'
def enumerate(self, most_variables=MOST_VARIABLES,
largest=LARGEST_PARTITION):
for n in range(1, most_variables + 1):
for size in range(1, largest + 1):
for partition in _partitions(size):
if len(partition) <= n:
yield {'n': str(n),
'lambda': ','.join(str(p) for p in partition)}
def value(self, params, digits):
n = int(params['n'])
partition = [int(p) for p in params['lambda'].split(',')]
ring = PolynomialRing(ZZ, ['x%d' % (i + 1) for i in range(n)])
def homogeneous(degree):
if degree < 0:
return ring.zero()
if degree == 0:
return ring.one()
return sum(prod(c) for c in
combinations_with_replacement(ring.gens(), degree))
size = len(partition)
rows = [[homogeneous(partition[i] - i + j) for j in range(size)]
for i in range(size)]
return _determinant(rows, ring)
if __name__ == '__main__':
generator = SchurPolynomials()
if '--publish' in sys.argv:
print(generator.publish(message='the Schur polynomials, by Jacobi-Trudi'))
else:
report = generator.verify()
print(report)
if not report.ok:
sys.exit(1)