public void StirlingNumber2SpecialCases()
{
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber2(0, 0) == 1);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber2(1, 0) == 0);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber2(1, 1) == 1);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber2(2, 0) == 0);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber2(2, 1) == 1);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber2(2, 2) == 1);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber2(3, 0) == 0);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber2(3, 1) == 1);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber2(3, 2) == 3);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber2(3, 3) == 1);
foreach (int m in TestUtilities.GenerateIntegerValues(2, 100, 4))
{
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber2(m, 0) == 0);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber2(m, 1) == 1);
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedIntegerMath.StirlingNumber2(m, 2),
MoreMath.Pow(2, m - 1) - 1
));
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber2(m, m) == 1);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber2(m, m - 1) == AdvancedIntegerMath.BinomialCoefficient(m, 2));
}
}
//[TestMethod]
// Cancellation too bad to be useful
public void BernoulliStirlingRelationship()
{
// This involves significant cancellation, so don't pick n too high
foreach (int n in TestUtilities.GenerateIntegerValues(2, 16, 4))
{
double S = 0.0;
for (int k = 0; k <= n; k++)
{
double dS = AdvancedIntegerMath.Factorial(k) / (k + 1) * AdvancedIntegerMath.StirlingNumber2(n, k);
if (k % 2 != 0)
{
dS = -dS;
}
S += dS;
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(S, AdvancedIntegerMath.BernoulliNumber(n)));
}
}
public void StirlingNumbers2ColumnSum()
{
foreach (int n in TestUtilities.GenerateIntegerValues(1, 100, 4))
{
foreach (int k in TestUtilities.GenerateIntegerValues(1, n, 4))
{
double sum = 0.0;
for (int m = k; m <= n; m++)
{
sum +=
AdvancedIntegerMath.BinomialCoefficient(n, m) *
AdvancedIntegerMath.StirlingNumber2(m, k);
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(sum, AdvancedIntegerMath.StirlingNumber2(n + 1, k + 1)));
}
}
}
public void StirlingNumber2Parity()
{
// Sterling numbers of the 2nd kind provably have the same parity
// as a particular related binomial coefficient.
// This is actually quite a difficult test because it tests the least
// significant bits of the numbers. We can't operate on values
// with more than 52 bits, since for those numbers we will loose
// the least significant bits.
foreach (int n in TestUtilities.GenerateIntegerValues(1, 50, 4))
{
foreach (int k in TestUtilities.GenerateIntegerValues(1, n, 4))
{
long B = (long)AdvancedIntegerMath.BinomialCoefficient(n - k / 2 - 1, n - k);
long S = (long)AdvancedIntegerMath.StirlingNumber2(n, k);
Assert.IsTrue(B % 2 == S % 2);
}
}
}
public void StirlingNumberInequality()
{
// Rows are log-concave
foreach (int n in TestUtilities.GenerateIntegerValues(10, 100, 4))
{
foreach (int k in TestUtilities.GenerateRealValues(2, n - 1, 4))
{
Assert.IsTrue(
MoreMath.Sqr(AdvancedIntegerMath.StirlingNumber1(n, k)) >=
AdvancedIntegerMath.StirlingNumber1(n, k - 1) *
AdvancedIntegerMath.StirlingNumber1(n, k + 1)
);
Assert.IsTrue(
MoreMath.Sqr(AdvancedIntegerMath.StirlingNumber2(n, k)) >=
AdvancedIntegerMath.StirlingNumber2(n, k - 1) *
AdvancedIntegerMath.StirlingNumber2(n, k + 1)
);
}
}
}