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stirling2

Maxima Function

Calling Sequence

stirling2 (n, m)

Description

Represents the Stirling number of the second kind.

When n and m are nonnegative integers, stirling2 (n, m) is the number of ways a set with cardinality n can be partitioned into m disjoint subsets. Maxima uses a recursion relation to define stirling2 (n, m) for m less than 0; it is undefined for n less than 0 and for non-integer arguments.

stirling2 is a simplifying function. Maxima knows the following identities.

  1. stirling2(0, n) = kron_delta(0, n) (Ref. [1])

  2. stirling2(n, n) = 1 (Ref. [1])

  3. stirling2(n, n - 1) = binomial(n, 2) (Ref. [1])

  4. stirling2(n + 1, 1) = 1 (Ref. [1])

  5. stirling2(n + 1, 2) = 2^n - 1 (Ref. [1])

  6. stirling2(n, 0) = kron_delta(n, 0) (Ref. [2])

  7. stirling2(n, m) = 0 when m > n (Ref. [2])

  8. stirling2(n, m) = sum((-1)^(m - k) binomial(m k) k^n,i,1,m) / m! when m and n are integers, and n is nonnegative. (Ref. [3])

These identities are applied when the arguments are literal integers or symbols declared as integers, and the first argument is nonnegative. stirling2 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.

[2] Graham, Knuth, and Patashnik. Concrete Mathematics, Table 264.

[3] Abramowitz and Stegun. Handbook of Mathematical Functions, Section 24.1.4.

Examples:

(%i1) declare (n, integer)$
(%i2) assume (n >= 0)$
(%i3) stirling2 (n, n);
(%o3)                           1

stirling2 does not simplify for non-integer arguments.

(%i1) stirling2 (%pi, %pi);
(%o1)                  stirling2(%pi, %pi)

Maxima applies identities to stirling2.

(%i1) declare (n, integer)$
(%i2) assume (n >= 0)$
(%i3) stirling2 (n + 9, n + 8);
                         (n + 8) (n + 9)
(%o3)                    ---------------
                                2
(%i4) stirling2 (n + 1, 2);
                              n
(%o4)                        2  - 1
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