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I do not want to use Select to find the solution. I do not need to use Select. I can generate all partitions of an integer into odd parts, then keep merging odd parts until there are no duplicate odd parts left like {3,3,3,1,1,1,1} to {6,3,2,2,} to {6,3,4} to {6,4,3}. Select generates partitions I do not need. That is why I do not want to use it. The partition of an integer into odd parts can be generated with IntegerPartitions[n, Infinity,Range[1, n, 2]] for an integer n. For example,

IntegerPartitions[10, Infinity, Range[1, 10, 2]] 

returns

{{9,1},{7,3},{7,1,1,1},{5,5},{5,3,1,1},{5,1,1,1,1,1},{3,3,3,1},{3,3,1,1,1,1},{3,1,1,1,1,1,1,1},{1,1,1,1,1,1,1,1,1,1}} 

StrictIntegerPartitions[n_Integer?IntegerQ/;Not[n<=0]]:=OddPartitionsToStrictPartitions[IntegerPartitions[n,Infinity,{1,Infinity,2}] 

StrictIntegerPartitions[n_Integer?IntegerQ/;Not[n<=0]]:=OddPartitionsToStrictPartitions[IntegerPartitions[n,Infinity,{1,n,2}] 

I would like to design a function StrictIntegerPartitions that lists the strict integer partitions of a strictly positive integer 1 by 1 by one.

The function would be defined as something like this.

StrictIntegerPartitions[n_Integer?IntegerQ/;Not[n<=0]]:=OddPartitionsToStrictPartitions[IntegerPartitions[n,Infinity,{1,Infinity,2}]