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I am not a mathematician but am wanting to informally research some math topics related to order and combinatorics. Up to now unable to discover the right terms to use in a math context, so that I can research more. Attempts at terminology are bold italicized. Please correct my terminology errors.

Members/Elements are represented below with numbers. Sets and sequences are represented by enclosure in brackets {}. Partitions/groups are represented below by enclosure in parentheses ().

  • A sequence ...

    sequenceA = {1,2,3}

  • An inclusion order ...

    power set of sequenceA =

{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}

  • And then including the idea of partitioning the power set ...

    partitioned power set of sequenceA =

{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}, {(1,2,)3}, {1,(2,3)}

  • possibilities for sequence order ...

    set of permutations of sequenceA =

{1, 2, 3}, {1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}, {3,2,1}

  • Then I am wanting to bring in an operation of an inclusion order on the set of permutations of sequenceA, so the idea of including increasing quantity of elements, like a power set is built up... and yields possibilities in a case of having both sequential order and inclusion order ... seems like a union of power sets of permutations of sequenceA ...

    power set of permutations of sequenceA =

{},{1}, {2}, {3}, {1,2}, {2,1}, {1,3}, {3,1}, {2,3}, {3,2}, {1,2,3}, {1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}, {3,2,1}

  • finally I am trying to include partitioning as well as inclusion order and sequential order ...

    partitioned power set of permutations of sequenceA =

{},{1}, {2}, {3}, {1,2}, {2,1}, {1,3}, {3,1}, {2,3}, {3,2}, {1,2,3}, {1,3,2}, {2,1,3}, {2,3,1}, {3,1,2}, {3,2,1}, {(1,2),3}, {(2,1),3}, {3,(1,2)}, {3,(2,1)}, {1,(2,3)}, {1,(3,2)},{(2,3),1}, {(3,2),1}, {2,(1,3)}, {2,(3,1)}, {(1,3),2}, {(3,1),2}

Thank you very much for your help.

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{a,b,c} is a set. It is never a sequence.
Sequences are ordered n-tuples.
(a,b,c) is a sequence.
(b,c,a) is a different sequence.
{b,c,a} is not a different set.
It is the same as {a,b,c},

There is no such thing as the power set of a sequence.
The power set of the set A = {1,2,3} is
{ {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }.
{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
is not the powerset of A.
It is a list of the elements of the power set of A.

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