Maxima – 199 – Insiemi – 8

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Continuo da qui, copio dal Reference Manual, PDF scaricabile da qui, sono a p.601.

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.

stirling1 is a simplifying function. Maxima knows the following identities:

  • stirling1(1, k) = kron_delta(1, k), k >= 0,(see here)
  • stirling1(n, n) = 1, n >= 0 (see here)
  • stirling1(n, n − 1) = −binomial(n, 2), n >= 1, (see here)
  • stirling1(n, 0) = kron d elta(n, 0), n >= 0 (see here and here)
  • stirling1(n, 1) = (−1) ( n − 1)(n − 1)!, n >= 1 (see here)
  • stirling1(n, k) = 0, n >= 0 and k > n.

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.

(%i1) declare (n, integer)$

(%i2) assume (n >= 0)$

(%i3) stirling1 (n, n);
(%o3)                                  1

stirling2 (n, m)
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.

  • stirling2 is a simplifying function. Maxima knows the following identities:
  • stirling2(n, 0) = 1, n >= 1 (see here) and stirling2(0,0) = 1)
  • stirling2(n, n) = 1, n >= 0, (see here)
  • stirling2(n, 1) = 1, n >= 1, (see here) and stirling2(0,1) = 0)
  • stirling2(n, 2) = 2 ( n − 1) − 1, n >= 1, (see here)
  • stirling2(n, n − 1) = binomial(n, 2), n >= 1 (see here)
  • stirling2(n, k) = 0, n >= 0 and k > n.

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.

(%i1) declare (n, integer)$

(%i2) assume (n >= 0)$

(%i3) stirling2 (n, n);
(%o3)                                  1

stirling2 does not simplify for non-integer arguments.

(%i4) stirling2 (%pi, %pi);
(%o4)                         stirling2(%pi, %pi)

subset (a, f)
Returns the subset of the set a that satisfies the predicate f.

subset returns a set which comprises the elements of a for which f returns anything other than false. subset does not apply is to the return value of f.

subset complains if a is not a literal set.

See also partition_set.

(%i5) subset ({1, 2, x, x + y, z, x + y + z}, atom);
(%o5)                            {1, 2, x, z}
(%i6) subset ({1, 2, 7, 8, 9, 14}, evenp);
(%o6)                             {2, 8, 14}

subsetp (a, b)
Returns true if and only if the set a is a subset of b.

subsetp complains if either a or b is not a literal set.

(%i7) subsetp ({1, 2, 3}, {a, 1, b, 2, c, 3});
(%o7)                                true
(%i8) subsetp ({a, 1, b, 2, c, 3}, {1, 2, 3});
(%o8)                                false

symmdifference (a_1, ..., a_n)
Returns the symmetric difference of sets a_1, ..., a_n.

Given two arguments, symmdifference (a, b) is the same as union (setdifference (a, b), setdifference (b, a)).

symmdifference complains if any argument is not a literal set.

(%i9) S_1 : {a, b, c};
(%o9)                              {a, b, c}
(%i10) S_2 : {1, b, c};
(%o10)                             {1, b, c}
(%i11) S_3 : {a, b, z};
(%o11)                             {a, b, z}
(%i12) symmdifference ();
(%o12)                                {}
(%i13) symmdifference (S_1);
(%o13)                             {a, b, c}
(%i14) symmdifference (S_1, S_2);
(%o14)                              {1, a}
(%i15) symmdifference (S_1, S_2, S_3);
(%o15)                             {1, b, z}
(%i16) symmdifference ({}, S_1, S_2, S_3);
(%o16)                             {1, b, z}

union (a_1, ..., a_n)
Returns the union of the sets a_1 through a_n.

union() (with no arguments) returns the empty set.

union complains if any argument is not a literal set.

(%i17) S_1 : {a, b, c + d, %e};
(%o17)                         {%e, a, b, d + c}
(%i18) S_2 : {%pi, %i, %e, c + d};
(%o18)                       {%e, %i, %pi, d + c}
(%i19) S_3 : {17, 29, 1729, %pi, %i};
(%o19)                      {17, 29, 1729, %i, %pi}
(%i20) union ();
(%o20)                                {}
(%i21) union (S_1);
(%o21)                         {%e, a, b, d + c}
(%i22) union (S_1, S_2);
(%o22)                    {%e, %i, %pi, a, b, d + c}
(%i23) union (S_1, S_2, S_3);
(%o23)             {17, 29, 1729, %e, %i, %pi, a, b, d + c}
(%i24) union ({}, S_1, S_2, S_3);
(%o24)             {17, 29, 1729, %e, %i, %pi, a, b, d + c}

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