public void StirlingNumber1SpecialCases()
{
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber1(0, 0) == 1);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber1(1, 0) == 0);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber1(1, 1) == 1);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber1(2, 0) == 0);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber1(2, 1) == 1);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber1(2, 2) == 1);
foreach (int m in TestUtilities.GenerateIntegerValues(2, 100, 4))
{
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber1(m, 0) == 0);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber1(m, 1) == AdvancedIntegerMath.Factorial(m - 1));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedIntegerMath.StirlingNumber1(m, 2),
AdvancedIntegerMath.Factorial(m - 1) * AdvancedIntegerMath.HarmonicNumber(m - 1)
));
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber1(m, m) == 1);
Assert.IsTrue(AdvancedIntegerMath.StirlingNumber1(m, m - 1) == AdvancedIntegerMath.BinomialCoefficient(m, 2));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedIntegerMath.StirlingNumber1(m, m - 2),
(3 * m - 1) * AdvancedIntegerMath.BinomialCoefficient(m, 3) / 4
));
}
}
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)
);
}
}
}