combinat::stirling1
--
Stirling numbers of the first kind
combinat::stirling1
(n,k) computes the Stirling numbers of the first kind.
combinat::stirling1(n,k)
n,k | - | nonnegative integers |
an integer.
n
symbols that have exactly k
cycles. Then combinat::stirling1
(n,k)
computes -1^(n+k)*S(n,k).sum(S1(n,k)*x^k, k=0...n) = x*(x-1) ... (x-n+1)
Let us have a look what's the result of x*(x-1)*(x-2)*(x-3)*(x-4)*(x-5) written as a sum.
>> expand(x*(x-1)*(x-2)*(x-3)*(x-4)*(x-5))
2 3 4 5 6 274 x - 120 x - 225 x + 85 x - 15 x + x
Now let us ``prove'' the formula mentioned in the ``Details'' section by calculating the proper Stirling numbers:
>> combinat::stirling1(6,1);
combinat::stirling1(6,2);
combinat::stirling1(6,3);
combinat::stirling1(6,4);
combinat::stirling1(6,5);
combinat::stirling1(6,6)
-120 274 -225 85 -15 1
>> combinat::stirling1(3,-1)
Error: Arguments must be nonnegative integers. [combinat::sti\ rling1]
J.J. Rotman, An Introduction to the Theory of Groups, 3rd Edition, Wm. C. Brown Publishers, Dubuque, 1988
MuPAD Combinat, an open source algebraic combinatorics package