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The Maxima on-line user's manual

Algebra Calculator

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Stirling2 Calculator

Stirling2

Function: stirling2 (<n>, <m>) Represents the Stirling number of the second kind.

stirling2(1001,5);

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

(%o1)                                true
(%i2) 

Stirling2 Example

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