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Cell[TextData[StyleBox["Number of chains in the sorting permutation of the \
(a,b) 2D finite sequence \
{(1,1),(1,2),...,(1,n),...,(k,k),(k,k+1),...,(k,n),...,(n-1,n-1),(n-1,n),(n,n)\
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Cell["\<\
For a specific positive integer n, we define two 2D finite sequences A and B \
as follows:

A={(a,b) where a,b \[Element]{1,2,...,n} , a\[LessEqual]b and it is sorted \
a-ascending and then b-ascending}
B=(A sorted b-ascending and then a-ascending)

One can easily deduce that:
B={(a,b) where a,b \[Element]{1,2,...,n} , a\[LessEqual]b and it is sorted \
b-ascending and then a-ascending}

Since A and B sequences contain exactly the same members there is a 1-1 \
correspondence between them that describes the effect of the sorting A to get \
B.
This correspondence is in fact a permutation of their positions in A and B.

A235647(n) equals the number of chains (including 1-cycles) in the \
permutation that corresponds A elements to B elements\
\>", "Text"],

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Example: A and B sequences for n=6\
\>", "Text",
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Cell["\<\
Lemma: (a,b) element in B sequence is at position (b-1)b/2+a.
Proof
B is sorted b-ascending and then a-ascending, thus (a,b) is a places after \
(b-1,b-1)\[CloseCurlyQuote]s position.
But (b-1,b-1) is at position 1+2+...+(b-1)= (b-1)b/2
QED \
\>", "Text"],

Cell["\<\
Concequently the permutation for n is
P={(b-1)b/2+a where a,b \[Element]{1,2,...,n} , a\[LessEqual]b and it is \
sorted a-ascending and then b-ascending (as in A)} \
\>", "Text"],

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Example: Permutation calculation for n=6\
\>", "Text",
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We then calculate the number of cycles in that permutation including 1-cycles\
\>", "Text"],

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The number of chains (the length of the above list) equals A235647(6)\
\>", "Text"],

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