The signed Stirling numbers of the first kind are variously denoted (Riordan 1980, Roman 1984), (Fort 1948, Abramowitz and Stegun 1972), (Jordan 1950). Abramowitz and Stegun (1972, p. 822)
summarize the various notational conventions, which can be a bit confusing (especially
since an unsigned version is also in common use). The signed Stirling
number of the first kind is are returned by StirlingS1[n,
m] in the Wolfram Language,
where they are denoted .
The signed Stirling numbers of the first kind are defined such that the number of permutations
of
elements which contain exactly permutation cycles is
the nonnegative number
(1)
This means that for and . A related set of numbers is known as the associated
Stirling numbers of the first kind. Both these and the usual Stirling numbers of
the first kind are special cases of a general function which is related to the number of cycles in a permutation.
The triangle of signed Stirling numbers of the first kind is
The Stirling numbers of the first kind are connected with the Stirling
numbers of the second kind . For example, the matrices and are inverses of
each other, where denotes the matrix with th entry for , ..., (G. Helms, pers. comm., Apr. 28, 2006).
Other formulas include
(18)
(19)
(Roman 1984, p. 67), as well as
(20)
(21)
(22)
(23)
A nonnegative (unsigned) version of the Stirling numbers gives the number of permutations of
objects having permutation cycles (with
cycles in opposite directions counted as distinct) and is obtained by taking the
absolute value of the signed version. The nonnegative
Stirling numbers of the first kind are variously denoted
(24)
(Graham et al. 1994). Diagrams illustrating , , , and (Dickau) are shown above.
The unsigned Stirling numbers of the first kind satisfy
(25)
and can be generalized to noninteger arguments (a sort of "Stirling polynomial") using the identity