The Stirling numbers of the first kind are the coefficients in the expansion
where (a Pochhammer symbol) denotes the falling factorial,
Note that (x)0 = 1 because it is an empty product. Combinatorialists also sometimes use the notation for the falling factorial, and for the rising factorial.
(Confusingly, the Pochhammer symbol that many use for falling factorials is used in special functions for rising factorials.)
The unsigned Stirling numbers of the first kind,
(with a lower-case "s"), count the number of permutations of n elements with k disjoint cycles.
Read more about this topic: Stirling Number
Famous quotes containing the words stirling, numbers and/or kind:
“Oh, if thy pride did not our joys control,
What world of loving wonders shouldst thou see!
For if I saw thee once transformed in me,
Then in thy bosom I would pour my soul;”
—William Alexander, Earl O Stirling (1580?–1640)
“Individually, museums are fine institutions, dedicated to the high values of preservation, education and truth; collectively, their growth in numbers points to the imaginative death of this country.”
—Robert Hewison (b. 1943)
“They that are discontented under monarchy, call it tyranny; and they that are displeased with aristocracy, call it oligarchy: so also, they which find themselves grieved under a democracy, call it anarchy, which signifies the want of government; and yet I think no man believes, that want of government, is any new kind of government.”
—Thomas Hobbes (1588–1679)