### The Maxima on-line user's manual

Algebra Calculator

#### Integer_partitions

Function: integer_partitions (<n>)

Function: integer_partitions (<n>, <len>) Returns integer partitions of <n>, that is, lists of integers which sum to <n>.

`integer_partitions(<n>)` returns the set of all partitions of the integer <n>. Each partition is a list sorted from greatest to least.

`integer_partitions(<n>, <len>)` returns all partitions that have length <len> or less; in this case, zeros are appended to each partition with fewer than <len> terms to make each partition have exactly <len> terms. Each partition is a list sorted from greatest to least.

A list [a_1, ..., a_m] is a partition of a nonnegative integer n when (1) each a_i is a nonzero integer, and (2) a_1 + ... + a_m = n. Thus 0 has no partitions.

Examples:

```          (%i1) integer_partitions (3);
(%o1)               {[1, 1, 1], [2, 1], [3]}
(%i2) s: integer_partitions (25)\$
(%i3) cardinality (s);
(%o3)                         1958
(%i4) map (lambda ([x], apply ("+", x)), s);
(%o4)                         {25}
(%i5) integer_partitions (5, 3);
(%o5) {[2, 2, 1], [3, 1, 1], [3, 2, 0], [4, 1, 0], [5, 0, 0]}
(%i6) integer_partitions (5, 2);
(%o6)               {[3, 2], [4, 1], [5, 0]}```

To find all partitions that satisfy a condition, use the function `subset`; here is an example that finds all partitions of 10 that consist of prime numbers.

```          (%i1) s: integer_partitions (10)\$
(%i2) cardinality (s);
(%o2)                          42
(%i3) xprimep(x) := integerp(x) and (x > 1) and primep(x)\$
(%i4) subset (s, lambda ([x], every (xprimep, x)));
(%o4) {[2, 2, 2, 2, 2], [3, 3, 2, 2], [5, 3, 2], [5, 5], [7, 3]}```

```(%o1)                                true
(%i2) ```

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