stirling SciMax Toolbox stirling2

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stirling1

Maxima Function

Calling Sequence

stirling1 (n, m)

Description

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.

  1. stirling1(0, n) = kron_delta(0, n) (Ref. [1])

  2. stirling1(n, n) = 1 (Ref. [1])

  3. stirling1(n, n - 1) = binomial(n, 2) (Ref. [1])

  4. stirling1(n + 1, 0) = 0 (Ref. [1])

  5. stirling1(n + 1, 1) = n! (Ref. [1])

  6. 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.

(%i1) stirling1 (sqrt(2), sqrt(2));
(%o1)              stirling1(sqrt(2), sqrt(2))

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!
stirling SciMax Toolbox stirling2