Maxima Function
stirling1 (n, m)
Represents the Stirling number of the first kind.
When n and m are nonnegative
integers, the magnitude of stirling1 (n, m)
is the number of
permutations of a set with n members that have m cycles.
For details, see Graham, Knuth and Patashnik Concrete Mathematics.
Maxima uses a recursion relation to define stirling1 (n, m)
for
m less than 0; it is undefined for n less than 0 and for non-integer
arguments.
stirling1
is a simplifying function.
Maxima knows the following identities.
stirling1(0, n) = kron_delta(0, n) (Ref. [1])
stirling1(n, n) = 1 (Ref. [1])
stirling1(n, n - 1) = binomial(n, 2) (Ref. [1])
stirling1(n + 1, 0) = 0 (Ref. [1])
stirling1(n + 1, 1) = n! (Ref. [1])
stirling1(n + 1, 2) = 2^n - 1 (Ref. [1])
These identities are applied when the arguments are literal integers
or symbols declared as integers, and the first argument is nonnegative.
stirling1
does not simplify for non-integer arguments.
References:
[1] Donald Knuth, The Art of Computer Programming, third edition, Volume 1, Section 1.2.6, Equations 48, 49, and 50.
Examples:
(%i1) declare (n, integer)$ (%i2) assume (n >= 0)$ (%i3) stirling1 (n, n); (%o3) 1
stirling1
does not simplify for non-integer arguments.
Maxima applies identities to stirling1
.
(%i1) declare (n, integer)$ (%i2) assume (n >= 0)$ (%i3) stirling1 (n + 1, n); n (n + 1) (%o3) --------- 2 (%i4) stirling1 (n + 1, 1); (%o4) n!