Example (3) Verify p(D,n)=p(O,n) for n = 9
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> with(combinat):
> read "FUNCS.txt":
> ptns:=partition(9); 
ptns := [[1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 2],

    [1, 1, 1, 1, 1, 2, 2], [1, 1, 1, 2, 2, 2], [1, 2, 2, 2, 2],

    [1, 1, 1, 1, 1, 1, 3], [1, 1, 1, 1, 2, 3], [1, 1, 2, 2, 3], [2, 2, 2, 3],

    [1, 1, 1, 3, 3], [1, 2, 3, 3], [3, 3, 3], [1, 1, 1, 1, 1, 4],

    [1, 1, 1, 2, 4], [1, 2, 2, 4], [1, 1, 3, 4], [2, 3, 4], [1, 4, 4],

    [1, 1, 1, 1, 5], [1, 1, 2, 5], [2, 2, 5], [1, 3, 5], [4, 5], [1, 1, 1, 6],

    [1, 2, 6], [3, 6], [1, 1, 7], [2, 7], [1, 8], [9]]

> ptns1:=select(ptnDP,ptns);
 ptns1 := [[2, 3, 4], [1, 3, 5], [4, 5], [1, 2, 6], [3, 6], [2, 7], [1, 8], [9]]

> ptns2:=select(ptnOP,ptns);
ptns2 := [[1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 3], [1, 1, 1, 3, 3],

    [3, 3, 3], [1, 1, 1, 1, 5], [1, 3, 5], [1, 1, 7], [9]]

> printptns(ptns1);
9
`8 + 1`
`7 + 2`
`6 + 3`
`6 + 2 + 1`
`5 + 4`
`5 + 3 + 1`
`4 + 3 + 2`
> printptns(ptns2);
9
`7 + 1 + 1`
`5 + 3 + 1`
`5 + 1 + 1 + 1 + 1`
`3 + 3 + 3`
`3 + 3 + 1 + 1 + 1`
`3 + 1 + 1 + 1 + 1 + 1 + 1`
`1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1`
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The partitions of 9 into distinct parts are
9
8 + 1
7 + 2
6 + 3
6 + 2 + 1
5 + 4
5 + 3 + 1
4 + 3 + 2

Hence p(D,9)=8

The partitions of 9 into odd parts are
9
7 + 1 + 1
5 + 3 + 1
5 + 1 + 1 + 1 + 1
3 + 3 + 3
3 + 3 + 1 + 1 + 1
3 + 1 + 1 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

Hence p(O,9)=8
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