i1 : compositions(4, 2)
o1 = {{2, 0, 0, 0}, {1, 1, 0, 0}, {0, 2, 0, 0}, {1, 0, 1, 0}, {0, 1, 1, 0},
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{0, 0, 2, 0}, {1, 0, 0, 1}, {0, 1, 0, 1}, {0, 0, 1, 1}, {0, 0, 0, 2}}
o1 : List
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i2 : compositions(2, 4)
o2 = {{4, 0}, {3, 1}, {2, 2}, {1, 3}, {0, 4}}
o2 : List
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To find all unordered compositions, we can use sort or rsort to put each composition into a standard form, then use the function unique to remove duplicates.
i3 : unique apply(compositions(4, 10), comp -> rsort comp)
o3 = {{10, 0, 0, 0}, {9, 1, 0, 0}, {8, 2, 0, 0}, {7, 3, 0, 0}, {6, 4, 0, 0},
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{5, 5, 0, 0}, {8, 1, 1, 0}, {7, 2, 1, 0}, {6, 3, 1, 0}, {5, 4, 1, 0},
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{6, 2, 2, 0}, {5, 3, 2, 0}, {4, 4, 2, 0}, {4, 3, 3, 0}, {7, 1, 1, 1},
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{6, 2, 1, 1}, {5, 3, 1, 1}, {4, 4, 1, 1}, {5, 2, 2, 1}, {4, 3, 2, 1},
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{3, 3, 3, 1}, {4, 2, 2, 2}, {3, 3, 2, 2}}
o3 : List
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In the next example, we use select to find all the compositions of 18 into 5 parts, with each part greater than or equal to 3.
i4 : select(compositions(5, 18), comp -> all(comp, c -> c>=3))
o4 = {{6, 3, 3, 3, 3}, {5, 4, 3, 3, 3}, {4, 5, 3, 3, 3}, {3, 6, 3, 3, 3}, {5,
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3, 4, 3, 3}, {4, 4, 4, 3, 3}, {3, 5, 4, 3, 3}, {4, 3, 5, 3, 3}, {3, 4,
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5, 3, 3}, {3, 3, 6, 3, 3}, {5, 3, 3, 4, 3}, {4, 4, 3, 4, 3}, {3, 5, 3,
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4, 3}, {4, 3, 4, 4, 3}, {3, 4, 4, 4, 3}, {3, 3, 5, 4, 3}, {4, 3, 3, 5,
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3}, {3, 4, 3, 5, 3}, {3, 3, 4, 5, 3}, {3, 3, 3, 6, 3}, {5, 3, 3, 3, 4},
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{4, 4, 3, 3, 4}, {3, 5, 3, 3, 4}, {4, 3, 4, 3, 4}, {3, 4, 4, 3, 4}, {3,
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3, 5, 3, 4}, {4, 3, 3, 4, 4}, {3, 4, 3, 4, 4}, {3, 3, 4, 4, 4}, {3, 3,
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3, 5, 4}, {4, 3, 3, 3, 5}, {3, 4, 3, 3, 5}, {3, 3, 4, 3, 5}, {3, 3, 3,
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4, 5}, {3, 3, 3, 3, 6}}
o4 : List
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For partitions of n into unordered, strictly positive parts, use partitions instead.
If a negative integer is given for n, an empty list is returned. If a negative integer is given for k, it will cause an error.
The object compositions is a method function.